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Theorem hoidmv1lelem1 41311
Description: The supremum of 𝑈 belongs to 𝑈. This is the last part of step (a) and the whole step (b) in the proof of Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Hypotheses
Ref Expression
hoidmv1lelem1.a (𝜑𝐴 ∈ ℝ)
hoidmv1lelem1.b (𝜑𝐵 ∈ ℝ)
hoidmv1lelem1.l (𝜑𝐴 < 𝐵)
hoidmv1lelem1.c (𝜑𝐶:ℕ⟶ℝ)
hoidmv1lelem1.d (𝜑𝐷:ℕ⟶ℝ)
hoidmv1lelem1.r (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ)
hoidmv1lelem1.u 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))}
hoidmv1lelem1.s 𝑆 = sup(𝑈, ℝ, < )
Assertion
Ref Expression
hoidmv1lelem1 (𝜑 → (𝑆𝑈𝐴𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
Distinct variable groups:   𝐴,𝑗,𝑧   𝑦,𝐴   𝑥,𝐵,𝑦   𝑧,𝐵   𝑧,𝐶   𝑧,𝐷   𝑆,𝑗,𝑧   𝑈,𝑗,𝑧   𝑥,𝑈,𝑦   𝜑,𝑗,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑗)   𝐶(𝑥,𝑦,𝑗)   𝐷(𝑥,𝑦,𝑗)   𝑆(𝑥,𝑦)

Proof of Theorem hoidmv1lelem1
StepHypRef Expression
1 hoidmv1lelem1.s . . . . . 6 𝑆 = sup(𝑈, ℝ, < )
2 hoidmv1lelem1.a . . . . . . 7 (𝜑𝐴 ∈ ℝ)
3 hoidmv1lelem1.b . . . . . . 7 (𝜑𝐵 ∈ ℝ)
4 hoidmv1lelem1.u . . . . . . . . 9 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))}
5 ssrab2 3828 . . . . . . . . 9 {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ⊆ (𝐴[,]𝐵)
64, 5eqsstri 3776 . . . . . . . 8 𝑈 ⊆ (𝐴[,]𝐵)
76a1i 11 . . . . . . 7 (𝜑𝑈 ⊆ (𝐴[,]𝐵))
82rexrd 10281 . . . . . . . . . . . 12 (𝜑𝐴 ∈ ℝ*)
93rexrd 10281 . . . . . . . . . . . 12 (𝜑𝐵 ∈ ℝ*)
10 hoidmv1lelem1.l . . . . . . . . . . . . 13 (𝜑𝐴 < 𝐵)
112, 3, 10ltled 10377 . . . . . . . . . . . 12 (𝜑𝐴𝐵)
12 lbicc2 12481 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
138, 9, 11, 12syl3anc 1477 . . . . . . . . . . 11 (𝜑𝐴 ∈ (𝐴[,]𝐵))
142recnd 10260 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ ℂ)
1514subidd 10572 . . . . . . . . . . . 12 (𝜑 → (𝐴𝐴) = 0)
16 nfv 1992 . . . . . . . . . . . . 13 𝑗𝜑
17 nnex 11218 . . . . . . . . . . . . . 14 ℕ ∈ V
1817a1i 11 . . . . . . . . . . . . 13 (𝜑 → ℕ ∈ V)
19 volf 23497 . . . . . . . . . . . . . . 15 vol:dom vol⟶(0[,]+∞)
2019a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
21 hoidmv1lelem1.c . . . . . . . . . . . . . . . 16 (𝜑𝐶:ℕ⟶ℝ)
2221ffvelrnda 6522 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ)
23 hoidmv1lelem1.d . . . . . . . . . . . . . . . . . 18 (𝜑𝐷:ℕ⟶ℝ)
2423ffvelrnda 6522 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ)
252adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ ℝ)
2624, 25ifcld 4275 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴) ∈ ℝ)
2726rexrd 10281 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴) ∈ ℝ*)
28 icombl 23532 . . . . . . . . . . . . . . 15 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴) ∈ ℝ*) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)) ∈ dom vol)
2922, 27, 28syl2anc 696 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)) ∈ dom vol)
3020, 29ffvelrnd 6523 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))) ∈ (0[,]+∞))
3116, 18, 30sge0ge0mpt 41158 . . . . . . . . . . . 12 (𝜑 → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))))
3215, 31eqbrtrd 4826 . . . . . . . . . . 11 (𝜑 → (𝐴𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))))
3313, 32jca 555 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ (𝐴[,]𝐵) ∧ (𝐴𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))))))
34 oveq1 6820 . . . . . . . . . . . 12 (𝑧 = 𝐴 → (𝑧𝐴) = (𝐴𝐴))
35 breq2 4808 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝐴 → ((𝐷𝑗) ≤ 𝑧 ↔ (𝐷𝑗) ≤ 𝐴))
36 id 22 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝐴𝑧 = 𝐴)
3735, 36ifbieq2d 4255 . . . . . . . . . . . . . . . 16 (𝑧 = 𝐴 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))
3837oveq2d 6829 . . . . . . . . . . . . . . 15 (𝑧 = 𝐴 → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) = ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))
3938fveq2d 6356 . . . . . . . . . . . . . 14 (𝑧 = 𝐴 → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) = (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))
4039mpteq2dv 4897 . . . . . . . . . . . . 13 (𝑧 = 𝐴 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))))
4140fveq2d 6356 . . . . . . . . . . . 12 (𝑧 = 𝐴 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))))
4234, 41breq12d 4817 . . . . . . . . . . 11 (𝑧 = 𝐴 → ((𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ↔ (𝐴𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))))))
4342elrab 3504 . . . . . . . . . 10 (𝐴 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ↔ (𝐴 ∈ (𝐴[,]𝐵) ∧ (𝐴𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))))))
4433, 43sylibr 224 . . . . . . . . 9 (𝜑𝐴 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
4544, 4syl6eleqr 2850 . . . . . . . 8 (𝜑𝐴𝑈)
46 ne0i 4064 . . . . . . . 8 (𝐴𝑈𝑈 ≠ ∅)
4745, 46syl 17 . . . . . . 7 (𝜑𝑈 ≠ ∅)
482, 3, 7, 47supicc 12513 . . . . . 6 (𝜑 → sup(𝑈, ℝ, < ) ∈ (𝐴[,]𝐵))
491, 48syl5eqel 2843 . . . . 5 (𝜑𝑆 ∈ (𝐴[,]𝐵))
501a1i 11 . . . . . . 7 (𝜑𝑆 = sup(𝑈, ℝ, < ))
51 nfv 1992 . . . . . . . . 9 𝑧𝜑
522, 3iccssred 40230 . . . . . . . . . . . . 13 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
537, 52sstrd 3754 . . . . . . . . . . . 12 (𝜑𝑈 ⊆ ℝ)
5453sselda 3744 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → 𝑧 ∈ ℝ)
55 nfv 1992 . . . . . . . . . . . . . . . 16 𝑗(𝜑𝑧𝑈)
5617a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → ℕ ∈ V)
5719a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
5822adantlr 753 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ)
5924adantlr 753 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ)
6054adantr 472 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → 𝑧 ∈ ℝ)
6159, 60ifcld 4275 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ)
6261rexrd 10281 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ*)
63 icombl 23532 . . . . . . . . . . . . . . . . . 18 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ*) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ∈ dom vol)
6458, 62, 63syl2anc 696 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ∈ dom vol)
6557, 64ffvelrnd 6523 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ∈ (0[,]+∞))
6655, 56, 65sge0xrclmpt 41148 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ∈ ℝ*)
67 pnfxr 10284 . . . . . . . . . . . . . . . 16 +∞ ∈ ℝ*
6867a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑈) → +∞ ∈ ℝ*)
69 hoidmv1lelem1.r . . . . . . . . . . . . . . . . . 18 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ)
7069rexrd 10281 . . . . . . . . . . . . . . . . 17 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ*)
7170adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ*)
7224rexrd 10281 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ*)
73 icombl 23532 . . . . . . . . . . . . . . . . . . . 20 (((𝐶𝑗) ∈ ℝ ∧ (𝐷𝑗) ∈ ℝ*) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
7422, 72, 73syl2anc 696 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
7520, 74ffvelrnd 6523 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)(𝐷𝑗))) ∈ (0[,]+∞))
7675adantlr 753 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)(𝐷𝑗))) ∈ (0[,]+∞))
7774adantlr 753 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
7822rexrd 10281 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ*)
7978adantlr 753 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ*)
8072adantlr 753 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ*)
8122leidd 10786 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ≤ (𝐶𝑗))
8281adantlr 753 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗) ≤ (𝐶𝑗))
83 min1 12213 . . . . . . . . . . . . . . . . . . . 20 (((𝐷𝑗) ∈ ℝ ∧ 𝑧 ∈ ℝ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ (𝐷𝑗))
8459, 60, 83syl2anc 696 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ (𝐷𝑗))
85 icossico 12436 . . . . . . . . . . . . . . . . . . 19 ((((𝐶𝑗) ∈ ℝ* ∧ (𝐷𝑗) ∈ ℝ*) ∧ ((𝐶𝑗) ≤ (𝐶𝑗) ∧ if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ (𝐷𝑗))) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
8679, 80, 82, 84, 85syl22anc 1478 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
87 volss 23501 . . . . . . . . . . . . . . . . . 18 ((((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ∈ dom vol ∧ ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol ∧ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗))) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
8864, 77, 86, 87syl3anc 1477 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
8955, 56, 65, 76, 88sge0lempt 41130 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
9069ltpnfd 12148 . . . . . . . . . . . . . . . . 17 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) < +∞)
9190adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) < +∞)
9266, 71, 68, 89, 91xrlelttrd 12184 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) < +∞)
9366, 68, 92xrltned 40071 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ≠ +∞)
9493neneqd 2937 . . . . . . . . . . . . 13 ((𝜑𝑧𝑈) → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = +∞)
95 eqid 2760 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))
9665, 95fmptd 6548 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑈) → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))):ℕ⟶(0[,]+∞))
9756, 96sge0repnf 41106 . . . . . . . . . . . . 13 ((𝜑𝑧𝑈) → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ∈ ℝ ↔ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = +∞))
9894, 97mpbird 247 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ∈ ℝ)
992adantr 472 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → 𝐴 ∈ ℝ)
10098, 99readdcld 10261 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) + 𝐴) ∈ ℝ)
10152, 49sseldd 3745 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑆 ∈ ℝ)
102101adantr 472 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
10324, 102ifcld 4275 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ)
104103rexrd 10281 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*)
105 icombl 23532 . . . . . . . . . . . . . . . . . . 19 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol)
10622, 104, 105syl2anc 696 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol)
10720, 106ffvelrnd 6523 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ∈ (0[,]+∞))
10816, 18, 107sge0xrclmpt 41148 . . . . . . . . . . . . . . . 16 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ*)
10967a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → +∞ ∈ ℝ*)
110 min1 12213 . . . . . . . . . . . . . . . . . . . . 21 (((𝐷𝑗) ∈ ℝ ∧ 𝑆 ∈ ℝ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ≤ (𝐷𝑗))
11124, 102, 110syl2anc 696 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ≤ (𝐷𝑗))
112 icossico 12436 . . . . . . . . . . . . . . . . . . . 20 ((((𝐶𝑗) ∈ ℝ* ∧ (𝐷𝑗) ∈ ℝ*) ∧ ((𝐶𝑗) ≤ (𝐶𝑗) ∧ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ≤ (𝐷𝑗))) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
11378, 72, 81, 111, 112syl22anc 1478 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
114 volss 23501 . . . . . . . . . . . . . . . . . . 19 ((((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol ∧ ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol ∧ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗))) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
115106, 74, 113, 114syl3anc 1477 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
11616, 18, 107, 75, 115sge0lempt 41130 . . . . . . . . . . . . . . . . 17 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
117108, 70, 109, 116, 90xrlelttrd 12184 . . . . . . . . . . . . . . . 16 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) < +∞)
118108, 109, 117xrltned 40071 . . . . . . . . . . . . . . 15 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ≠ +∞)
119118neneqd 2937 . . . . . . . . . . . . . 14 (𝜑 → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) = +∞)
120 eqid 2760 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
121107, 120fmptd 6548 . . . . . . . . . . . . . . 15 (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))):ℕ⟶(0[,]+∞))
12218, 121sge0repnf 41106 . . . . . . . . . . . . . 14 (𝜑 → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ ↔ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) = +∞))
123119, 122mpbird 247 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ)
124123, 2readdcld 10261 . . . . . . . . . . . 12 (𝜑 → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ∈ ℝ)
125124adantr 472 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ∈ ℝ)
1264eleq2i 2831 . . . . . . . . . . . . . . . 16 (𝑧𝑈𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
127126biimpi 206 . . . . . . . . . . . . . . 15 (𝑧𝑈𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
128127adantl 473 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑈) → 𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
129 rabid 3254 . . . . . . . . . . . . . 14 (𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ↔ (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))))
130128, 129sylib 208 . . . . . . . . . . . . 13 ((𝜑𝑧𝑈) → (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))))
131130simprd 482 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))))
13254, 99, 98lesubaddd 10816 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → ((𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ↔ 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) + 𝐴)))
133131, 132mpbid 222 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) + 𝐴))
134123adantr 472 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ)
135107adantlr 753 . . . . . . . . . . . . 13 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ∈ (0[,]+∞))
136106adantlr 753 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol)
137104adantlr 753 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*)
13861adantr 472 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ)
139 eqidd 2761 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) = (𝐷𝑗))
140 iftrue 4236 . . . . . . . . . . . . . . . . . . 19 ((𝐷𝑗) ≤ 𝑧 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = (𝐷𝑗))
141140adantl 473 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = (𝐷𝑗))
14259adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ∈ ℝ)
14360adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → 𝑧 ∈ ℝ)
144101ad3antrrr 768 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → 𝑆 ∈ ℝ)
145 simpr 479 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ≤ 𝑧)
14653adantr 472 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → 𝑈 ⊆ ℝ)
14747adantr 472 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → 𝑈 ≠ ∅)
1482, 3jca 555 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
149 iccsupr 12459 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑈 ⊆ (𝐴[,]𝐵) ∧ 𝐴𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
150148, 7, 45, 149syl3anc 1477 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
151150simp3d 1139 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥)
152151adantr 472 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥)
153128, 126sylibr 224 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → 𝑧𝑈)
154 suprub 11176 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥) ∧ 𝑧𝑈) → 𝑧 ≤ sup(𝑈, ℝ, < ))
155146, 147, 152, 153, 154syl31anc 1480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑧𝑈) → 𝑧 ≤ sup(𝑈, ℝ, < ))
156155, 1syl6breqr 4846 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑧𝑈) → 𝑧𝑆)
157156ad2antrr 764 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → 𝑧𝑆)
158142, 143, 144, 145, 157letrd 10386 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ≤ 𝑆)
159158iftrued 4238 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = (𝐷𝑗))
160139, 141, 1593eqtr4d 2804 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
161138, 160eqled 10332 . . . . . . . . . . . . . . . 16 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
16260adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → 𝑧 ∈ ℝ)
16359adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ∈ ℝ)
164 simpr 479 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → ¬ (𝐷𝑗) ≤ 𝑧)
165162, 163ltnled 10376 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → (𝑧 < (𝐷𝑗) ↔ ¬ (𝐷𝑗) ≤ 𝑧))
166164, 165mpbird 247 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → 𝑧 < (𝐷𝑗))
167162, 163, 166ltled 10377 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → 𝑧 ≤ (𝐷𝑗))
168167adantr 472 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → 𝑧 ≤ (𝐷𝑗))
169 iffalse 4239 . . . . . . . . . . . . . . . . . . . 20 (¬ (𝐷𝑗) ≤ 𝑧 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = 𝑧)
170169ad2antlr 765 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = 𝑧)
171 iftrue 4236 . . . . . . . . . . . . . . . . . . . 20 ((𝐷𝑗) ≤ 𝑆 → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = (𝐷𝑗))
172171adantl 473 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = (𝐷𝑗))
173170, 172breq12d 4817 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → (if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ↔ 𝑧 ≤ (𝐷𝑗)))
174168, 173mpbird 247 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
175156ad3antrrr 768 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → 𝑧𝑆)
176169ad2antlr 765 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = 𝑧)
177 iffalse 4239 . . . . . . . . . . . . . . . . . . . 20 (¬ (𝐷𝑗) ≤ 𝑆 → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = 𝑆)
178177adantl 473 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = 𝑆)
179176, 178breq12d 4817 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → (if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ↔ 𝑧𝑆))
180175, 179mpbird 247 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
181174, 180pm2.61dan 867 . . . . . . . . . . . . . . . 16 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
182161, 181pm2.61dan 867 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
183 icossico 12436 . . . . . . . . . . . . . . 15 ((((𝐶𝑗) ∈ ℝ* ∧ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*) ∧ ((𝐶𝑗) ≤ (𝐶𝑗) ∧ if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))
18479, 137, 82, 182, 183syl22anc 1478 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))
185 volss 23501 . . . . . . . . . . . . . 14 ((((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ∈ dom vol ∧ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol ∧ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ≤ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
18664, 136, 184, 185syl3anc 1477 . . . . . . . . . . . . 13 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ≤ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
18755, 56, 65, 135, 186sge0lempt 41130 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))))
18898, 134, 99, 187leadd1dd 10833 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) + 𝐴) ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
18954, 100, 125, 133, 188letrd 10386 . . . . . . . . . 10 ((𝜑𝑧𝑈) → 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
190189ex 449 . . . . . . . . 9 (𝜑 → (𝑧𝑈𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴)))
19151, 190ralrimi 3095 . . . . . . . 8 (𝜑 → ∀𝑧𝑈 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
192 suprleub 11181 . . . . . . . . 9 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥) ∧ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ∈ ℝ) → (sup(𝑈, ℝ, < ) ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ↔ ∀𝑧𝑈 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴)))
19353, 47, 151, 124, 192syl31anc 1480 . . . . . . . 8 (𝜑 → (sup(𝑈, ℝ, < ) ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ↔ ∀𝑧𝑈 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴)))
194191, 193mpbird 247 . . . . . . 7 (𝜑 → sup(𝑈, ℝ, < ) ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
19550, 194eqbrtrd 4826 . . . . . 6 (𝜑𝑆 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
196101, 2, 123lesubaddd 10816 . . . . . 6 (𝜑 → ((𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ↔ 𝑆 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴)))
197195, 196mpbird 247 . . . . 5 (𝜑 → (𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))))
19849, 197jca 555 . . . 4 (𝜑 → (𝑆 ∈ (𝐴[,]𝐵) ∧ (𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))))))
199 oveq1 6820 . . . . . 6 (𝑧 = 𝑆 → (𝑧𝐴) = (𝑆𝐴))
200 breq2 4808 . . . . . . . . . . 11 (𝑧 = 𝑆 → ((𝐷𝑗) ≤ 𝑧 ↔ (𝐷𝑗) ≤ 𝑆))
201 id 22 . . . . . . . . . . 11 (𝑧 = 𝑆𝑧 = 𝑆)
202200, 201ifbieq2d 4255 . . . . . . . . . 10 (𝑧 = 𝑆 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
203202oveq2d 6829 . . . . . . . . 9 (𝑧 = 𝑆 → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) = ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))
204203fveq2d 6356 . . . . . . . 8 (𝑧 = 𝑆 → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) = (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
205204mpteq2dv 4897 . . . . . . 7 (𝑧 = 𝑆 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))))
206205fveq2d 6356 . . . . . 6 (𝑧 = 𝑆 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))))
207199, 206breq12d 4817 . . . . 5 (𝑧 = 𝑆 → ((𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ↔ (𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))))))
208207elrab 3504 . . . 4 (𝑆 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ↔ (𝑆 ∈ (𝐴[,]𝐵) ∧ (𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))))))
209198, 208sylibr 224 . . 3 (𝜑𝑆 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
210209, 4syl6eleqr 2850 . 2 (𝜑𝑆𝑈)
211210, 45, 1513jca 1123 1 (𝜑 → (𝑆𝑈𝐴𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  wss 3715  c0 4058  ifcif 4230   class class class wbr 4804  cmpt 4881  dom cdm 5266  wf 6045  cfv 6049  (class class class)co 6813  supcsup 8511  cr 10127  0cc0 10128   + caddc 10131  +∞cpnf 10263  *cxr 10265   < clt 10266  cle 10267  cmin 10458  cn 11212  [,)cico 12370  [,]cicc 12371  volcvol 23432  Σ^csumge0 41082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-q 11982  df-rp 12026  df-xadd 12140  df-ioo 12372  df-ico 12374  df-icc 12375  df-fz 12520  df-fzo 12660  df-fl 12787  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-clim 14418  df-rlim 14419  df-sum 14616  df-xmet 19941  df-met 19942  df-ovol 23433  df-vol 23434  df-sumge0 41083
This theorem is referenced by:  hoidmv1lelem3  41313
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