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Theorem itg2addnc 33594
Description: Alternate proof of itg2add 23571 using the "buffer zone" definition from the first lemma, in which every simple function in the set is divided into to by dividing its buffer by a third and finding the largest allowable function locked to a grid laid out in increments of the new, smaller buffer up to the original simple function. The measurability of this function follows from that of the augend, and subtracting it from the original simple function yields another simple function by i1fsub 23520, which is allowable by the fact that the grid must have a mark between one third and two thirds the original buffer. This has two advantages over the current approach: first, eliminating ax-cc 9295, and second, weakening the measurability hypothesis to only the augend. (Contributed by Brendan Leahy, 31-Oct-2017.) (Revised by Brendan Leahy, 13-Mar-2018.)
Hypotheses
Ref Expression
itg2addnc.f1 (𝜑𝐹 ∈ MblFn)
itg2addnc.f2 (𝜑𝐹:ℝ⟶(0[,)+∞))
itg2addnc.f3 (𝜑 → (∫2𝐹) ∈ ℝ)
itg2addnc.g2 (𝜑𝐺:ℝ⟶(0[,)+∞))
itg2addnc.g3 (𝜑 → (∫2𝐺) ∈ ℝ)
Assertion
Ref Expression
itg2addnc (𝜑 → (∫2‘(𝐹𝑓 + 𝐺)) = ((∫2𝐹) + (∫2𝐺)))

Proof of Theorem itg2addnc
Dummy variables 𝑡 𝑠 𝑢 𝑥 𝑦 𝑧 𝑓 𝑔 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprr 811 . . . . . . 7 ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))) → 𝑥 = (∫1𝑓))
2 itg1cl 23497 . . . . . . . 8 (𝑓 ∈ dom ∫1 → (∫1𝑓) ∈ ℝ)
32adantr 480 . . . . . . 7 ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))) → (∫1𝑓) ∈ ℝ)
41, 3eqeltrd 2730 . . . . . 6 ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))) → 𝑥 ∈ ℝ)
54rexlimiva 3057 . . . . 5 (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓)) → 𝑥 ∈ ℝ)
65abssi 3710 . . . 4 {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ⊆ ℝ
76a1i 11 . . 3 (𝜑 → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ⊆ ℝ)
8 i1f0 23499 . . . . . 6 (ℝ × {0}) ∈ dom ∫1
9 3nn 11224 . . . . . . . 8 3 ∈ ℕ
10 nnrp 11880 . . . . . . . 8 (3 ∈ ℕ → 3 ∈ ℝ+)
11 ne0i 3954 . . . . . . . 8 (3 ∈ ℝ+ → ℝ+ ≠ ∅)
129, 10, 11mp2b 10 . . . . . . 7 + ≠ ∅
13 itg2addnc.f2 . . . . . . . . . . . . 13 (𝜑𝐹:ℝ⟶(0[,)+∞))
1413ffvelrnda 6399 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) ∈ (0[,)+∞))
15 elrege0 12316 . . . . . . . . . . . 12 ((𝐹𝑧) ∈ (0[,)+∞) ↔ ((𝐹𝑧) ∈ ℝ ∧ 0 ≤ (𝐹𝑧)))
1614, 15sylib 208 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℝ) → ((𝐹𝑧) ∈ ℝ ∧ 0 ≤ (𝐹𝑧)))
1716simprd 478 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℝ) → 0 ≤ (𝐹𝑧))
1817ralrimiva 2995 . . . . . . . . 9 (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐹𝑧))
19 reex 10065 . . . . . . . . . . 11 ℝ ∈ V
2019a1i 11 . . . . . . . . . 10 (𝜑 → ℝ ∈ V)
21 c0ex 10072 . . . . . . . . . . 11 0 ∈ V
2221a1i 11 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℝ) → 0 ∈ V)
23 eqidd 2652 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ ℝ ↦ 0) = (𝑧 ∈ ℝ ↦ 0))
2413feqmptd 6288 . . . . . . . . . 10 (𝜑𝐹 = (𝑧 ∈ ℝ ↦ (𝐹𝑧)))
2520, 22, 14, 23, 24ofrfval2 6957 . . . . . . . . 9 (𝜑 → ((𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐹𝑧)))
2618, 25mpbird 247 . . . . . . . 8 (𝜑 → (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹)
2726ralrimivw 2996 . . . . . . 7 (𝜑 → ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹)
28 r19.2z 4093 . . . . . . 7 ((ℝ+ ≠ ∅ ∧ ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹) → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹)
2912, 27, 28sylancr 696 . . . . . 6 (𝜑 → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹)
30 fveq2 6229 . . . . . . . . . 10 (𝑓 = (ℝ × {0}) → (∫1𝑓) = (∫1‘(ℝ × {0})))
31 itg10 23500 . . . . . . . . . 10 (∫1‘(ℝ × {0})) = 0
3230, 31syl6req 2702 . . . . . . . . 9 (𝑓 = (ℝ × {0}) → 0 = (∫1𝑓))
3332biantrud 527 . . . . . . . 8 (𝑓 = (ℝ × {0}) → (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ↔ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ 0 = (∫1𝑓))))
34 fveq1 6228 . . . . . . . . . . . . 13 (𝑓 = (ℝ × {0}) → (𝑓𝑧) = ((ℝ × {0})‘𝑧))
3521fvconst2 6510 . . . . . . . . . . . . 13 (𝑧 ∈ ℝ → ((ℝ × {0})‘𝑧) = 0)
3634, 35sylan9eq 2705 . . . . . . . . . . . 12 ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) = 0)
3736iftrued 4127 . . . . . . . . . . 11 ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) = 0)
3837mpteq2dva 4777 . . . . . . . . . 10 (𝑓 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ 0))
3938breq1d 4695 . . . . . . . . 9 (𝑓 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ↔ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹))
4039rexbidv 3081 . . . . . . . 8 (𝑓 = (ℝ × {0}) → (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ↔ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹))
4133, 40bitr3d 270 . . . . . . 7 (𝑓 = (ℝ × {0}) → ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ 0 = (∫1𝑓)) ↔ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹))
4241rspcev 3340 . . . . . 6 (((ℝ × {0}) ∈ dom ∫1 ∧ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹) → ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ 0 = (∫1𝑓)))
438, 29, 42sylancr 696 . . . . 5 (𝜑 → ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ 0 = (∫1𝑓)))
44 eqeq1 2655 . . . . . . . 8 (𝑥 = 0 → (𝑥 = (∫1𝑓) ↔ 0 = (∫1𝑓)))
4544anbi2d 740 . . . . . . 7 (𝑥 = 0 → ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓)) ↔ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ 0 = (∫1𝑓))))
4645rexbidv 3081 . . . . . 6 (𝑥 = 0 → (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓)) ↔ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ 0 = (∫1𝑓))))
4721, 46elab 3382 . . . . 5 (0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ↔ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ 0 = (∫1𝑓)))
4843, 47sylibr 224 . . . 4 (𝜑 → 0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))})
49 ne0i 3954 . . . 4 (0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ≠ ∅)
5048, 49syl 17 . . 3 (𝜑 → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ≠ ∅)
51 icossicc 12298 . . . . . . 7 (0[,)+∞) ⊆ (0[,]+∞)
52 fss 6094 . . . . . . 7 ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℝ⟶(0[,]+∞))
5351, 52mpan2 707 . . . . . 6 (𝐹:ℝ⟶(0[,)+∞) → 𝐹:ℝ⟶(0[,]+∞))
54 eqid 2651 . . . . . . 7 {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} = {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}
5554itg2addnclem 33591 . . . . . 6 (𝐹:ℝ⟶(0[,]+∞) → (∫2𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
5613, 53, 553syl 18 . . . . 5 (𝜑 → (∫2𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
57 itg2addnc.f3 . . . . 5 (𝜑 → (∫2𝐹) ∈ ℝ)
5856, 57eqeltrrd 2731 . . . 4 (𝜑 → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ∈ ℝ)
59 ressxr 10121 . . . . . . 7 ℝ ⊆ ℝ*
606, 59sstri 3645 . . . . . 6 {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ⊆ ℝ*
61 supxrub 12192 . . . . . 6 (({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ⊆ ℝ*𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}) → 𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
6260, 61mpan 706 . . . . 5 (𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} → 𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
6362rgen 2951 . . . 4 𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < )
64 breq2 4689 . . . . . 6 (𝑎 = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) → (𝑏𝑎𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < )))
6564ralbidv 3015 . . . . 5 (𝑎 = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) → (∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}𝑏𝑎 ↔ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < )))
6665rspcev 3340 . . . 4 ((sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ∈ ℝ ∧ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < )) → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}𝑏𝑎)
6758, 63, 66sylancl 695 . . 3 (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}𝑏𝑎)
68 simprr 811 . . . . . . 7 ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))) → 𝑥 = (∫1𝑔))
69 itg1cl 23497 . . . . . . . 8 (𝑔 ∈ dom ∫1 → (∫1𝑔) ∈ ℝ)
7069adantr 480 . . . . . . 7 ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))) → (∫1𝑔) ∈ ℝ)
7168, 70eqeltrd 2730 . . . . . 6 ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))) → 𝑥 ∈ ℝ)
7271rexlimiva 3057 . . . . 5 (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔)) → 𝑥 ∈ ℝ)
7372abssi 3710 . . . 4 {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ⊆ ℝ
7473a1i 11 . . 3 (𝜑 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ⊆ ℝ)
75 itg2addnc.g2 . . . . . . . . . . . . 13 (𝜑𝐺:ℝ⟶(0[,)+∞))
7675ffvelrnda 6399 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) ∈ (0[,)+∞))
77 elrege0 12316 . . . . . . . . . . . 12 ((𝐺𝑧) ∈ (0[,)+∞) ↔ ((𝐺𝑧) ∈ ℝ ∧ 0 ≤ (𝐺𝑧)))
7876, 77sylib 208 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℝ) → ((𝐺𝑧) ∈ ℝ ∧ 0 ≤ (𝐺𝑧)))
7978simprd 478 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℝ) → 0 ≤ (𝐺𝑧))
8079ralrimiva 2995 . . . . . . . . 9 (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐺𝑧))
8175feqmptd 6288 . . . . . . . . . 10 (𝜑𝐺 = (𝑧 ∈ ℝ ↦ (𝐺𝑧)))
8220, 22, 76, 23, 81ofrfval2 6957 . . . . . . . . 9 (𝜑 → ((𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐺𝑧)))
8380, 82mpbird 247 . . . . . . . 8 (𝜑 → (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺)
8483ralrimivw 2996 . . . . . . 7 (𝜑 → ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺)
85 r19.2z 4093 . . . . . . 7 ((ℝ+ ≠ ∅ ∧ ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺) → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺)
8612, 84, 85sylancr 696 . . . . . 6 (𝜑 → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺)
87 fveq2 6229 . . . . . . . . . 10 (𝑔 = (ℝ × {0}) → (∫1𝑔) = (∫1‘(ℝ × {0})))
8887, 31syl6req 2702 . . . . . . . . 9 (𝑔 = (ℝ × {0}) → 0 = (∫1𝑔))
8988biantrud 527 . . . . . . . 8 (𝑔 = (ℝ × {0}) → (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ↔ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ∧ 0 = (∫1𝑔))))
90 fveq1 6228 . . . . . . . . . . . . 13 (𝑔 = (ℝ × {0}) → (𝑔𝑧) = ((ℝ × {0})‘𝑧))
9190, 35sylan9eq 2705 . . . . . . . . . . . 12 ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) = 0)
9291iftrued 4127 . . . . . . . . . . 11 ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) = 0)
9392mpteq2dva 4777 . . . . . . . . . 10 (𝑔 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ 0))
9493breq1d 4695 . . . . . . . . 9 (𝑔 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ↔ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺))
9594rexbidv 3081 . . . . . . . 8 (𝑔 = (ℝ × {0}) → (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ↔ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺))
9689, 95bitr3d 270 . . . . . . 7 (𝑔 = (ℝ × {0}) → ((∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ∧ 0 = (∫1𝑔)) ↔ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺))
9796rspcev 3340 . . . . . 6 (((ℝ × {0}) ∈ dom ∫1 ∧ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺) → ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ∧ 0 = (∫1𝑔)))
988, 86, 97sylancr 696 . . . . 5 (𝜑 → ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ∧ 0 = (∫1𝑔)))
99 eqeq1 2655 . . . . . . . 8 (𝑥 = 0 → (𝑥 = (∫1𝑔) ↔ 0 = (∫1𝑔)))
10099anbi2d 740 . . . . . . 7 (𝑥 = 0 → ((∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔)) ↔ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ∧ 0 = (∫1𝑔))))
101100rexbidv 3081 . . . . . 6 (𝑥 = 0 → (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ∧ 0 = (∫1𝑔))))
10221, 101elab 3382 . . . . 5 (0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ↔ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ∧ 0 = (∫1𝑔)))
10398, 102sylibr 224 . . . 4 (𝜑 → 0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))})
104 ne0i 3954 . . . 4 (0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ≠ ∅)
105103, 104syl 17 . . 3 (𝜑 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ≠ ∅)
106 fss 6094 . . . . . . 7 ((𝐺:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐺:ℝ⟶(0[,]+∞))
10751, 106mpan2 707 . . . . . 6 (𝐺:ℝ⟶(0[,)+∞) → 𝐺:ℝ⟶(0[,]+∞))
108 eqid 2651 . . . . . . 7 {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}
109108itg2addnclem 33591 . . . . . 6 (𝐺:ℝ⟶(0[,]+∞) → (∫2𝐺) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))
11075, 107, 1093syl 18 . . . . 5 (𝜑 → (∫2𝐺) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))
111 itg2addnc.g3 . . . . 5 (𝜑 → (∫2𝐺) ∈ ℝ)
112110, 111eqeltrrd 2731 . . . 4 (𝜑 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) ∈ ℝ)
11373, 59sstri 3645 . . . . . 6 {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ⊆ ℝ*
114 supxrub 12192 . . . . . 6 (({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ⊆ ℝ*𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}) → 𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))
115113, 114mpan 706 . . . . 5 (𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} → 𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))
116115rgen 2951 . . . 4 𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < )
117 breq2 4689 . . . . . 6 (𝑎 = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) → (𝑏𝑎𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))
118117ralbidv 3015 . . . . 5 (𝑎 = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) → (∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑏𝑎 ↔ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))
119118rspcev 3340 . . . 4 ((sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) ∈ ℝ ∧ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑏𝑎)
120112, 116, 119sylancl 695 . . 3 (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑏𝑎)
121 eqid 2651 . . 3 {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)} = {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}
1227, 50, 67, 74, 105, 120, 121supadd 11029 . 2 (𝜑 → (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ, < )) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ, < ))
123 supxrre 12195 . . . . 5 (({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ⊆ ℝ ∧ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ≠ ∅ ∧ ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}𝑏𝑎) → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ, < ))
1247, 50, 67, 123syl3anc 1366 . . . 4 (𝜑 → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ, < ))
12556, 124eqtrd 2685 . . 3 (𝜑 → (∫2𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ, < ))
126 supxrre 12195 . . . . 5 (({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ⊆ ℝ ∧ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ≠ ∅ ∧ ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑏𝑎) → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ, < ))
12774, 105, 120, 126syl3anc 1366 . . . 4 (𝜑 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ, < ))
128110, 127eqtrd 2685 . . 3 (𝜑 → (∫2𝐺) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ, < ))
129125, 128oveq12d 6708 . 2 (𝜑 → ((∫2𝐹) + (∫2𝐺)) = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ, < )))
130 ge0addcl 12322 . . . . . . 7 ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,)+∞))
13151, 130sseldi 3634 . . . . . 6 ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,]+∞))
132131adantl 481 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ (0[,]+∞))
133 inidm 3855 . . . . 5 (ℝ ∩ ℝ) = ℝ
134132, 13, 75, 20, 20, 133off 6954 . . . 4 (𝜑 → (𝐹𝑓 + 𝐺):ℝ⟶(0[,]+∞))
135 eqid 2651 . . . . 5 {𝑠 ∣ ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1))} = {𝑠 ∣ ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1))}
136135itg2addnclem 33591 . . . 4 ((𝐹𝑓 + 𝐺):ℝ⟶(0[,]+∞) → (∫2‘(𝐹𝑓 + 𝐺)) = sup({𝑠 ∣ ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1))}, ℝ*, < ))
137134, 136syl 17 . . 3 (𝜑 → (∫2‘(𝐹𝑓 + 𝐺)) = sup({𝑠 ∣ ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1))}, ℝ*, < ))
138 itg2addnc.f1 . . . . . . . 8 (𝜑𝐹 ∈ MblFn)
139138, 13, 57, 75, 111itg2addnclem3 33593 . . . . . . 7 (𝜑 → (∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1)) → ∃𝑡𝑢(∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢))))
140 simpl 472 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑓 ∈ dom ∫1)
141 simpr 476 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑔 ∈ dom ∫1)
142140, 141i1fadd 23507 . . . . . . . . . . . . 13 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑓𝑓 + 𝑔) ∈ dom ∫1)
143142ad3antlr 767 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → (𝑓𝑓 + 𝑔) ∈ dom ∫1)
144 reeanv 3136 . . . . . . . . . . . . . . . . 17 (∃𝑐 ∈ ℝ+𝑑 ∈ ℝ+ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺) ↔ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺))
145144biimpri 218 . . . . . . . . . . . . . . . 16 ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺) → ∃𝑐 ∈ ℝ+𝑑 ∈ ℝ+ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺))
146145ad2ant2r 798 . . . . . . . . . . . . . . 15 (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) → ∃𝑐 ∈ ℝ+𝑑 ∈ ℝ+ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺))
147 ifcl 4163 . . . . . . . . . . . . . . . . . . 19 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ+)
148147ad2antlr 763 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺)) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ+)
149 breq1 4688 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → (0 ≤ (𝐹𝑧) ↔ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧)))
150149anbi1d 741 . . . . . . . . . . . . . . . . . . . . . . 23 (0 = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → ((0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) ↔ (if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
151150imbi1d 330 . . . . . . . . . . . . . . . . . . . . . 22 (0 = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → (((0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))) ↔ ((if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))))
152 breq1 4688 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓𝑧) + 𝑐) = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ↔ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧)))
153152anbi1d 741 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓𝑧) + 𝑐) = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) ↔ (if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
154153imbi1d 330 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓𝑧) + 𝑐) = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → (((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))) ↔ ((if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))))
155 breq1 4688 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (0 ≤ (𝐺𝑧) ↔ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
156155anbi2d 740 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((0 ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) ↔ (0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
157156imbi1d 330 . . . . . . . . . . . . . . . . . . . . . . 23 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (((0 ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))) ↔ ((0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))))
158 breq1 4688 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑔𝑧) + 𝑑) = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧) ↔ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
159158anbi2d 740 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑔𝑧) + 𝑑) = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((0 ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) ↔ (0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
160159imbi1d 330 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑔𝑧) + 𝑑) = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (((0 ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))) ↔ ((0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))))
161 oveq12 6699 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓𝑧) = 0 ∧ (𝑔𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = (0 + 0))
162 00id 10249 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0 + 0) = 0
163161, 162syl6eq 2701 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓𝑧) = 0 ∧ (𝑔𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = 0)
164163iftrued 4127 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓𝑧) = 0 ∧ (𝑔𝑧) = 0) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) = 0)
165164adantll 750 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) = 0)
166 simpll 805 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝜑)
16715simplbi 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹𝑧) ∈ (0[,)+∞) → (𝐹𝑧) ∈ ℝ)
16814, 167syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) ∈ ℝ)
16977simplbi 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐺𝑧) ∈ (0[,)+∞) → (𝐺𝑧) ∈ ℝ)
17076, 169syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) ∈ ℝ)
171168, 170, 17, 79addge0d 10641 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑧 ∈ ℝ) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
172166, 171sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
173172ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
174165, 173eqbrtrd 4707 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
175174a1d 25 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → ((0 ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
176172ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
177 oveq1 6697 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓𝑧) = 0 → ((𝑓𝑧) + (𝑔𝑧)) = (0 + (𝑔𝑧)))
178 simplrr 818 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑔 ∈ dom ∫1)
179 i1ff 23488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑔 ∈ dom ∫1𝑔:ℝ⟶ℝ)
180179ffvelrnda 6399 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑔 ∈ dom ∫1𝑧 ∈ ℝ) → (𝑔𝑧) ∈ ℝ)
181178, 180sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) ∈ ℝ)
182181recnd 10106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) ∈ ℂ)
183182addid2d 10275 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (0 + (𝑔𝑧)) = (𝑔𝑧))
184177, 183sylan9eqr 2707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = (𝑔𝑧))
185184oveq1d 6705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
186185adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
187147rpred 11910 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ)
188187ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ)
189181, 188readdcld 10107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
190189adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
191166, 170sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) ∈ ℝ)
192191adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → (𝐺𝑧) ∈ ℝ)
193166, 168sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) ∈ ℝ)
194193, 191readdcld 10107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑧) + (𝐺𝑧)) ∈ ℝ)
195194adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝐹𝑧) + (𝐺𝑧)) ∈ ℝ)
196 simplrr 818 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ+)
197196rpred 11910 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ)
198 rpre 11877 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 ∈ ℝ+𝑐 ∈ ℝ)
199 rpre 11877 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑑 ∈ ℝ+𝑑 ∈ ℝ)
200 min2 12059 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑑)
201198, 199, 200syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑑)
202201ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑑)
203188, 197, 181, 202leadd2dd 10680 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑔𝑧) + 𝑑))
204181, 197readdcld 10107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) + 𝑑) ∈ ℝ)
205 letr 10169 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ ∧ ((𝑔𝑧) + 𝑑) ∈ ℝ ∧ (𝐺𝑧) ∈ ℝ) → ((((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑔𝑧) + 𝑑) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧)))
206189, 204, 191, 205syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑔𝑧) + 𝑑) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧)))
207203, 206mpand 711 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧)))
208207imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧))
209170, 168addge02d 10654 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ ℝ) → (0 ≤ (𝐹𝑧) ↔ (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧))))
21017, 209mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
211166, 210sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
212211adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
213190, 192, 195, 208, 212letrd 10232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
214213adantlr 751 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
215186, 214eqbrtrd 4707 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
216 breq1 4688 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0 = if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) → (0 ≤ ((𝐹𝑧) + (𝐺𝑧)) ↔ if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
217 breq1 4688 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) → ((((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)) ↔ if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
218216, 217ifboth 4157 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ≤ ((𝐹𝑧) + (𝐺𝑧)) ∧ (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧))) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
219176, 215, 218syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
220219ex 449 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → (((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
221220adantld 482 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → ((0 ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
222221adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ¬ (𝑔𝑧) = 0) → ((0 ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
223157, 160, 175, 222ifbothda 4156 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → ((0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
224155anbi2d 740 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) ↔ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
225224imbi1d 330 . . . . . . . . . . . . . . . . . . . . . . 23 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))) ↔ ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))))
226158anbi2d 740 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑔𝑧) + 𝑑) = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) ↔ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
227226imbi1d 330 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑔𝑧) + 𝑑) = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))) ↔ ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))))
228172ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
229 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑔𝑧) = 0 → ((𝑓𝑧) + (𝑔𝑧)) = ((𝑓𝑧) + 0))
230 simplrl 817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑓 ∈ dom ∫1)
231 i1ff 23488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑓 ∈ dom ∫1𝑓:ℝ⟶ℝ)
232231ffvelrnda 6399 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑓 ∈ dom ∫1𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
233230, 232sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
234233recnd 10106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℂ)
235234addid1d 10274 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + 0) = (𝑓𝑧))
236229, 235sylan9eqr 2707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = (𝑓𝑧))
237236oveq1d 6705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
238237adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
239233, 188readdcld 10107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
240239adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
241193adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝐹𝑧) ∈ ℝ)
242194adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝐹𝑧) + (𝐺𝑧)) ∈ ℝ)
243 simplrl 817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ+)
244243rpred 11910 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ)
245 min1 12058 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑐)
246198, 199, 245syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑐)
247246ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑐)
248188, 244, 233, 247leadd2dd 10680 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑓𝑧) + 𝑐))
249233, 244readdcld 10107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + 𝑐) ∈ ℝ)
250 letr 10169 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ ∧ ((𝑓𝑧) + 𝑐) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → ((((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑓𝑧) + 𝑐) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐹𝑧)))
251239, 249, 193, 250syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑓𝑧) + 𝑐) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐹𝑧)))
252248, 251mpand 711 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐹𝑧)))
253252imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐹𝑧))
254168, 170addge01d 10653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ ℝ) → (0 ≤ (𝐺𝑧) ↔ (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧))))
25579, 254mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
256166, 255sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
257256adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
258240, 241, 242, 253, 257letrd 10232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
259258adantlr 751 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
260238, 259eqbrtrd 4707 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
261228, 260, 218syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
262261ex 449 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
263262adantlr 751 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
264263adantrd 483 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
265172adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
266188recnd 10106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℂ)
267234, 182, 266addassd 10100 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑓𝑧) + ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑))))
268267adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑓𝑧) + ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑))))
269233, 243ltaddrpd 11943 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) < ((𝑓𝑧) + 𝑐))
270233, 249, 269ltled 10223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ≤ ((𝑓𝑧) + 𝑐))
271 letr 10169 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓𝑧) ∈ ℝ ∧ ((𝑓𝑧) + 𝑐) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → (((𝑓𝑧) ≤ ((𝑓𝑧) + 𝑐) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝑓𝑧) ≤ (𝐹𝑧)))
272233, 249, 193, 271syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) ≤ ((𝑓𝑧) + 𝑐) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝑓𝑧) ≤ (𝐹𝑧)))
273270, 272mpand 711 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) → (𝑓𝑧) ≤ (𝐹𝑧)))
274 le2add 10548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑓𝑧) ∈ ℝ ∧ ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ) ∧ ((𝐹𝑧) ∈ ℝ ∧ (𝐺𝑧) ∈ ℝ)) → (((𝑓𝑧) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧)) → ((𝑓𝑧) + ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
275233, 189, 193, 191, 274syl22anc 1367 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧)) → ((𝑓𝑧) + ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
276273, 207, 275syl2and 499 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑓𝑧) + ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
277276imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → ((𝑓𝑧) + ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
278268, 277eqbrtrd 4707 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
279265, 278, 218syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
280279ex 449 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
281280ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓𝑧) = 0) ∧ ¬ (𝑔𝑧) = 0) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
282225, 227, 264, 281ifbothda 4156 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓𝑧) = 0) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
283151, 154, 223, 282ifbothda 4156 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
284283ralimdva 2991 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (∀𝑧 ∈ ℝ (if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → ∀𝑧 ∈ ℝ if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
285 ovex 6718 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓𝑧) + 𝑐) ∈ V
28621, 285ifex 4189 . . . . . . . . . . . . . . . . . . . . . . . . 25 if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ∈ V
287286a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑧 ∈ ℝ) → if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ∈ V)
288 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))))
28920, 287, 14, 288, 24ofrfval2 6957 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ↔ ∀𝑧 ∈ ℝ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧)))
290 ovex 6718 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔𝑧) + 𝑑) ∈ V
29121, 290ifex 4189 . . . . . . . . . . . . . . . . . . . . . . . . 25 if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ∈ V
292291a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑧 ∈ ℝ) → if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ∈ V)
293 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))))
29420, 292, 76, 293, 81ofrfval2 6957 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ↔ ∀𝑧 ∈ ℝ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
295289, 294anbi12d 747 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺) ↔ (∀𝑧 ∈ ℝ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ ∀𝑧 ∈ ℝ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
296 r19.26 3093 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑧 ∈ ℝ (if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) ↔ (∀𝑧 ∈ ℝ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ ∀𝑧 ∈ ℝ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
297295, 296syl6bbr 278 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺) ↔ ∀𝑧 ∈ ℝ (if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
298297ad2antrr 762 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺) ↔ ∀𝑧 ∈ ℝ (if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
29919a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → ℝ ∈ V)
300 ovex 6718 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ V
30121, 300ifex 4189 . . . . . . . . . . . . . . . . . . . . . 22 if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ∈ V
302301a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ∈ V)
303 ovexd 6720 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑧) + (𝐺𝑧)) ∈ V)
304 ffn 6083 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓:ℝ⟶ℝ → 𝑓 Fn ℝ)
305231, 304syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 ∈ dom ∫1𝑓 Fn ℝ)
306305adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑓 Fn ℝ)
307306ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑓 Fn ℝ)
308 ffn 6083 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑔:ℝ⟶ℝ → 𝑔 Fn ℝ)
309179, 308syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔 ∈ dom ∫1𝑔 Fn ℝ)
310309adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑔 Fn ℝ)
311310ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑔 Fn ℝ)
312 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) = (𝑓𝑧))
313 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) = (𝑔𝑧))
314307, 311, 299, 299, 133, 312, 313ofval 6948 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑓 + 𝑔)‘𝑧) = ((𝑓𝑧) + (𝑔𝑧)))
315314eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑓 + 𝑔)‘𝑧) = 0 ↔ ((𝑓𝑧) + (𝑔𝑧)) = 0))
316314oveq1d 6705 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑓 + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) = (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)))
317315, 316ifbieq2d 4144 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑))) = if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))))
318317mpteq2dva 4777 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))) = (𝑧 ∈ ℝ ↦ if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)))))
319 ffn 6083 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:ℝ⟶(0[,)+∞) → 𝐹 Fn ℝ)
32013, 319syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐹 Fn ℝ)
321 ffn 6083 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐺:ℝ⟶(0[,)+∞) → 𝐺 Fn ℝ)
32275, 321syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐺 Fn ℝ)
323 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) = (𝐹𝑧))
324 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
325320, 322, 20, 20, 133, 323, 324offval 6946 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐹𝑓 + 𝐺) = (𝑧 ∈ ℝ ↦ ((𝐹𝑧) + (𝐺𝑧))))
326325ad2antrr 762 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (𝐹𝑓 + 𝐺) = (𝑧 ∈ ℝ ↦ ((𝐹𝑧) + (𝐺𝑧))))
327299, 302, 303, 318, 326ofrfval2 6957 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → ((𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ↔ ∀𝑧 ∈ ℝ if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
328284, 298, 3273imtr4d 283 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺) → (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺)))
329328imp 444 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺)) → (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺))
330 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = if(𝑐𝑑, 𝑐, 𝑑) → (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦) = (((𝑓𝑓 + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
331330ifeq2d 4138 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = if(𝑐𝑑, 𝑐, 𝑑) → if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦)) = if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑))))
332331mpteq2dv 4778 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = if(𝑐𝑑, 𝑐, 𝑑) → (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))))
333332breq1d 4695 . . . . . . . . . . . . . . . . . . 19 (𝑦 = if(𝑐𝑑, 𝑐, 𝑑) → ((𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ↔ (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺)))
334333rspcev 3340 . . . . . . . . . . . . . . . . . 18 ((if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ+ ∧ (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺)) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺))
335148, 329, 334syl2anc 694 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺)) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺))
336335ex 449 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺)))
337336rexlimdvva 3067 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (∃𝑐 ∈ ℝ+𝑑 ∈ ℝ+ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺)))
338146, 337syl5 34 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺)))
339338a1dd 50 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) → (𝑠 = (𝑡 + 𝑢) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺))))
340339imp31 447 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺))
341 oveq12 6699 . . . . . . . . . . . . . . 15 ((𝑡 = (∫1𝑓) ∧ 𝑢 = (∫1𝑔)) → (𝑡 + 𝑢) = ((∫1𝑓) + (∫1𝑔)))
342341ad2ant2l 797 . . . . . . . . . . . . . 14 (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) → (𝑡 + 𝑢) = ((∫1𝑓) + (∫1𝑔)))
343140, 141itg1add 23513 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (∫1‘(𝑓𝑓 + 𝑔)) = ((∫1𝑓) + (∫1𝑔)))
344343eqcomd 2657 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((∫1𝑓) + (∫1𝑔)) = (∫1‘(𝑓𝑓 + 𝑔)))
345344adantl 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → ((∫1𝑓) + (∫1𝑔)) = (∫1‘(𝑓𝑓 + 𝑔)))
346342, 345sylan9eqr 2707 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔)))) → (𝑡 + 𝑢) = (∫1‘(𝑓𝑓 + 𝑔)))
347 eqtr 2670 . . . . . . . . . . . . . 14 ((𝑠 = (𝑡 + 𝑢) ∧ (𝑡 + 𝑢) = (∫1‘(𝑓𝑓 + 𝑔))) → 𝑠 = (∫1‘(𝑓𝑓 + 𝑔)))
348347ancoms 468 . . . . . . . . . . . . 13 (((𝑡 + 𝑢) = (∫1‘(𝑓𝑓 + 𝑔)) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (∫1‘(𝑓𝑓 + 𝑔)))
349346, 348sylan 487 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (∫1‘(𝑓𝑓 + 𝑔)))
350 fveq1 6228 . . . . . . . . . . . . . . . . . . 19 ( = (𝑓𝑓 + 𝑔) → (𝑧) = ((𝑓𝑓 + 𝑔)‘𝑧))
351350eqeq1d 2653 . . . . . . . . . . . . . . . . . 18 ( = (𝑓𝑓 + 𝑔) → ((𝑧) = 0 ↔ ((𝑓𝑓 + 𝑔)‘𝑧) = 0))
352350oveq1d 6705 . . . . . . . . . . . . . . . . . 18 ( = (𝑓𝑓 + 𝑔) → ((𝑧) + 𝑦) = (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))
353351, 352ifbieq2d 4144 . . . . . . . . . . . . . . . . 17 ( = (𝑓𝑓 + 𝑔) → if((𝑧) = 0, 0, ((𝑧) + 𝑦)) = if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦)))
354353mpteq2dv 4778 . . . . . . . . . . . . . . . 16 ( = (𝑓𝑓 + 𝑔) → (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))))
355354breq1d 4695 . . . . . . . . . . . . . . 15 ( = (𝑓𝑓 + 𝑔) → ((𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ↔ (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺)))
356355rexbidv 3081 . . . . . . . . . . . . . 14 ( = (𝑓𝑓 + 𝑔) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺)))
357 fveq2 6229 . . . . . . . . . . . . . . 15 ( = (𝑓𝑓 + 𝑔) → (∫1) = (∫1‘(𝑓𝑓 + 𝑔)))
358357eqeq2d 2661 . . . . . . . . . . . . . 14 ( = (𝑓𝑓 + 𝑔) → (𝑠 = (∫1) ↔ 𝑠 = (∫1‘(𝑓𝑓 + 𝑔))))
359356, 358anbi12d 747 . . . . . . . . . . . . 13 ( = (𝑓𝑓 + 𝑔) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1‘(𝑓𝑓 + 𝑔)))))
360359rspcev 3340 . . . . . . . . . . . 12 (((𝑓𝑓 + 𝑔) ∈ dom ∫1 ∧ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1‘(𝑓𝑓 + 𝑔)))) → ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1)))
361143, 340, 349, 360syl12anc 1364 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1)))
362361exp31 629 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) → (𝑠 = (𝑡 + 𝑢) → ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1)))))
363362rexlimdvva 3067 . . . . . . . . 9 (𝜑 → (∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) → (𝑠 = (𝑡 + 𝑢) → ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1)))))
364363impd 446 . . . . . . . 8 (𝜑 → ((∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1))))
365364exlimdvv 1902 . . . . . . 7 (𝜑 → (∃𝑡𝑢(∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1))))
366139, 365impbid 202 . . . . . 6 (𝜑 → (∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1)) ↔ ∃𝑡𝑢(∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢))))
367 eqeq1 2655 . . . . . . . . . 10 (𝑥 = 𝑡 → (𝑥 = (∫1𝑓) ↔ 𝑡 = (∫1𝑓)))
368367anbi2d 740 . . . . . . . . 9 (𝑥 = 𝑡 → ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓)) ↔ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓))))
369368rexbidv 3081 . . . . . . . 8 (𝑥 = 𝑡 → (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓)) ↔ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓))))
370369rexab 3402 . . . . . . 7 (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)))
371 eqeq1 2655 . . . . . . . . . . . . 13 (𝑥 = 𝑢 → (𝑥 = (∫1𝑔) ↔ 𝑢 = (∫1𝑔)))
372371anbi2d 740 . . . . . . . . . . . 12 (𝑥 = 𝑢 → ((∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔)) ↔ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))))
373372rexbidv 3081 . . . . . . . . . . 11 (𝑥 = 𝑢 → (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))))
374373rexab 3402 . . . . . . . . . 10 (∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑢(∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔)) ∧ 𝑠 = (𝑡 + 𝑢)))
375374anbi2i 730 . . . . . . . . 9 ((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑢(∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))))
376 19.42v 1921 . . . . . . . . 9 (∃𝑢(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑢(∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))))
377 reeanv 3136 . . . . . . . . . . . 12 (∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))))
378377anbi1i 731 . . . . . . . . . . 11 ((∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) ↔ ((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
379 anass 682 . . . . . . . . . . 11 (((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))))
380378, 379bitr2i 265 . . . . . . . . . 10 ((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))) ↔ (∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
381380exbii 1814 . . . . . . . . 9 (∃𝑢(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))) ↔ ∃𝑢(∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
382375, 376, 3813bitr2i 288 . . . . . . . 8 ((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)) ↔ ∃𝑢(∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
383382exbii 1814 . . . . . . 7 (∃𝑡(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)) ↔ ∃𝑡𝑢(∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
384370, 383bitri 264 . . . . . 6 (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡𝑢(∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
385366, 384syl6bbr 278 . . . . 5 (𝜑 → (∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1)) ↔ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)))
386385abbidv 2770 . . . 4 (𝜑 → {𝑠 ∣ ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1))} = {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)})
387386supeq1d 8393 . . 3 (𝜑 → sup({𝑠 ∣ ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘𝑟 ≤ (𝐹𝑓 + 𝐺) ∧ 𝑠 = (∫1))}, ℝ*, < ) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ*, < ))
388 simpr 476 . . . . . . . . 9 (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (𝑡 + 𝑢))
3896sseli 3632 . . . . . . . . . . 11 (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} → 𝑡 ∈ ℝ)
390389ad2antrr 762 . . . . . . . . . 10 (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑡 ∈ ℝ)
39173sseli 3632 . . . . . . . . . . 11 (𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} → 𝑢 ∈ ℝ)
392391ad2antlr 763 . . . . . . . . . 10 (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑢 ∈ ℝ)
393390, 392readdcld 10107 . . . . . . . . 9 (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → (𝑡 + 𝑢) ∈ ℝ)
394388, 393eqeltrd 2730 . . . . . . . 8 (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 ∈ ℝ)
395394ex 449 . . . . . . 7 ((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}) → (𝑠 = (𝑡 + 𝑢) → 𝑠 ∈ ℝ))
396395rexlimivv 3065 . . . . . 6 (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢) → 𝑠 ∈ ℝ)
397396abssi 3710 . . . . 5 {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)} ⊆ ℝ
398397a1i 11 . . . 4 (𝜑 → {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)} ⊆ ℝ)
399162eqcomi 2660 . . . . . . . 8 0 = (0 + 0)
400 rspceov 6732 . . . . . . . 8 ((0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ∧ 0 = (0 + 0)) → ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}0 = (𝑡 + 𝑢))
401399, 400mp3an3 1453 . . . . . . 7 ((0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}) → ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}0 = (𝑡 + 𝑢))
40248, 103, 401syl2anc 694 . . . . . 6 (𝜑 → ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}0 = (𝑡 + 𝑢))
403 eqeq1 2655 . . . . . . . 8 (𝑠 = 0 → (𝑠 = (𝑡 + 𝑢) ↔ 0 = (𝑡 + 𝑢)))
4044032rexbidv 3086 . . . . . . 7 (𝑠 = 0 → (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}0 = (𝑡 + 𝑢)))
40521, 404spcev 3331 . . . . . 6 (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}0 = (𝑡 + 𝑢) → ∃𝑠𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢))
406402, 405syl 17 . . . . 5 (𝜑 → ∃𝑠𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢))
407 abn0 3987 . . . . 5 ({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)} ≠ ∅ ↔ ∃𝑠𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢))
408406, 407sylibr 224 . . . 4 (𝜑 → {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)} ≠ ∅)
40958, 112readdcld 10107 . . . . 5 (𝜑 → (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) ∈ ℝ)
410 simpr 476 . . . . . . . . 9 (((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))})) ∧ 𝑏 = (𝑡 + 𝑢)) → 𝑏 = (𝑡 + 𝑢))
411389ad2antrl 764 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))})) → 𝑡 ∈ ℝ)
412391ad2antll 765 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))})) → 𝑢 ∈ ℝ)
41358adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))})) → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ∈ ℝ)
414112adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))})) → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) ∈ ℝ)
415 supxrub 12192 . . . . . . . . . . . . 13 (({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ⊆ ℝ*𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}) → 𝑡 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
41660, 415mpan 706 . . . . . . . . . . . 12 (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} → 𝑡 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
417416ad2antrl 764 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))})) → 𝑡 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
418 supxrub 12192 . . . . . . . . . . . . 13 (({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ⊆ ℝ*𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}) → 𝑢 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))
419113, 418mpan 706 . . . . . . . . . . . 12 (𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} → 𝑢 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))
420419ad2antll 765 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))})) → 𝑢 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))
421411, 412, 413, 414, 417, 420le2addd 10684 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))})) → (𝑡 + 𝑢) ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))
422421adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))})) ∧ 𝑏 = (𝑡 + 𝑢)) → (𝑡 + 𝑢) ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))
423410, 422eqbrtrd 4707 . . . . . . . 8 (((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))})) ∧ 𝑏 = (𝑡 + 𝑢)) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))
424423ex 449 . . . . . . 7 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))})) → (𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))))
425424rexlimdvva 3067 . . . . . 6 (𝜑 → (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))))
426425alrimiv 1895 . . . . 5 (𝜑 → ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))))
427 breq2 4689 . . . . . . . 8 (𝑎 = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) → (𝑏𝑎𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))))
428427ralbidv 3015 . . . . . . 7 (𝑎 = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) → (∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏𝑎 ↔ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))))
429 eqeq1 2655 . . . . . . . . 9 (𝑠 = 𝑏 → (𝑠 = (𝑡 + 𝑢) ↔ 𝑏 = (𝑡 + 𝑢)))
4304292rexbidv 3086 . . . . . . . 8 (𝑠 = 𝑏 → (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑏 = (𝑡 + 𝑢)))
431430ralab 3400 . . . . . . 7 (∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) ↔ ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))))
432428, 431syl6bb 276 . . . . . 6 (𝑎 = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) → (∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏𝑎 ↔ ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))))
433432rspcev 3340 . . . . 5 (((sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) ∈ ℝ ∧ ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))) → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏𝑎)
434409, 426, 433syl2anc 694 . . . 4 (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏𝑎)
435 supxrre 12195 . . . 4 (({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)} ⊆ ℝ ∧ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)} ≠ ∅ ∧ ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏𝑎) → sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ*, < ) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ, < ))
436398, 408, 434, 435syl3anc 1366 . . 3 (𝜑 → sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ*, < ) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ, < ))
437137, 387, 4363eqtrd 2689 . 2 (𝜑 → (∫2‘(𝐹𝑓 + 𝐺)) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ, < ))
438122, 129, 4373eqtr4rd 2696 1 (𝜑 → (∫2‘(𝐹𝑓 + 𝐺)) = ((∫2𝐹) + (∫2𝐺)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  Vcvv 3231  wss 3607  c0 3948  ifcif 4119  {csn 4210   class class class wbr 4685  cmpt 4762   × cxp 5141  dom cdm 5143   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑓 cof 6937  𝑟 cofr 6938  supcsup 8387  cr 9973  0cc0 9974   + caddc 9977  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cn 11058  3c3 11109  +crp 11870  [,)cico 12215  [,]cicc 12216  MblFncmbf 23428  1citg1 23429  2citg2 23430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280  df-mbf 23433  df-itg1 23434  df-itg2 23435
This theorem is referenced by:  ibladdnclem  33596  itgaddnclem1  33598  iblabsnclem  33603  iblabsnc  33604  iblmulc2nc  33605  ftc1anclem4  33618  ftc1anclem5  33619  ftc1anclem6  33620  ftc1anclem8  33622
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