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Theorem ovolicc2lem5 23508
 Description: Lemma for ovolicc2 23509. (Contributed by Mario Carneiro, 14-Jun-2014.)
Hypotheses
Ref Expression
ovolicc.1 (𝜑𝐴 ∈ ℝ)
ovolicc.2 (𝜑𝐵 ∈ ℝ)
ovolicc.3 (𝜑𝐴𝐵)
ovolicc2.4 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ovolicc2.5 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ovolicc2.6 (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
ovolicc2.7 (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)
ovolicc2.8 (𝜑𝐺:𝑈⟶ℕ)
ovolicc2.9 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
ovolicc2.10 𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
Assertion
Ref Expression
ovolicc2lem5 (𝜑 → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
Distinct variable groups:   𝑢,𝑡,𝐴   𝑡,𝐵,𝑢   𝑡,𝐹   𝑡,𝐺   𝜑,𝑡   𝑡,𝑇   𝑡,𝑈,𝑢
Allowed substitution hints:   𝜑(𝑢)   𝑆(𝑢,𝑡)   𝑇(𝑢)   𝐹(𝑢)   𝐺(𝑢)

Proof of Theorem ovolicc2lem5
Dummy variables 𝑚 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovolicc2.7 . . . 4 (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)
2 ovolicc.1 . . . . . 6 (𝜑𝐴 ∈ ℝ)
32rexrd 10290 . . . . 5 (𝜑𝐴 ∈ ℝ*)
4 ovolicc.2 . . . . . 6 (𝜑𝐵 ∈ ℝ)
54rexrd 10290 . . . . 5 (𝜑𝐵 ∈ ℝ*)
6 ovolicc.3 . . . . 5 (𝜑𝐴𝐵)
7 lbicc2 12494 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
83, 5, 6, 7syl3anc 1475 . . . 4 (𝜑𝐴 ∈ (𝐴[,]𝐵))
91, 8sseldd 3751 . . 3 (𝜑𝐴 𝑈)
10 eluni2 4576 . . 3 (𝐴 𝑈 ↔ ∃𝑧𝑈 𝐴𝑧)
119, 10sylib 208 . 2 (𝜑 → ∃𝑧𝑈 𝐴𝑧)
12 ovolicc2.6 . . . . . . . 8 (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
13 elin 3945 . . . . . . . 8 (𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin) ↔ (𝑈 ∈ 𝒫 ran ((,) ∘ 𝐹) ∧ 𝑈 ∈ Fin))
1412, 13sylib 208 . . . . . . 7 (𝜑 → (𝑈 ∈ 𝒫 ran ((,) ∘ 𝐹) ∧ 𝑈 ∈ Fin))
1514simprd 477 . . . . . 6 (𝜑𝑈 ∈ Fin)
16 ovolicc2.10 . . . . . . 7 𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
17 ssrab2 3834 . . . . . . 7 {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} ⊆ 𝑈
1816, 17eqsstri 3782 . . . . . 6 𝑇𝑈
19 ssfi 8335 . . . . . 6 ((𝑈 ∈ Fin ∧ 𝑇𝑈) → 𝑇 ∈ Fin)
2015, 18, 19sylancl 566 . . . . 5 (𝜑𝑇 ∈ Fin)
211adantr 466 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝐴[,]𝐵) ⊆ 𝑈)
22 inss2 3980 . . . . . . . . . . . . 13 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
23 ovolicc2.8 . . . . . . . . . . . . . . 15 (𝜑𝐺:𝑈⟶ℕ)
24 ineq1 3956 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑡 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑡 ∩ (𝐴[,]𝐵)))
2524neeq1d 3001 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑡 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅))
2625, 16elrab2 3516 . . . . . . . . . . . . . . . 16 (𝑡𝑇 ↔ (𝑡𝑈 ∧ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅))
2726simplbi 479 . . . . . . . . . . . . . . 15 (𝑡𝑇𝑡𝑈)
28 ffvelrn 6500 . . . . . . . . . . . . . . 15 ((𝐺:𝑈⟶ℕ ∧ 𝑡𝑈) → (𝐺𝑡) ∈ ℕ)
2923, 27, 28syl2an 575 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑇) → (𝐺𝑡) ∈ ℕ)
30 ovolicc2.5 . . . . . . . . . . . . . . 15 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3130ffvelrnda 6502 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐺𝑡) ∈ ℕ) → (𝐹‘(𝐺𝑡)) ∈ ( ≤ ∩ (ℝ × ℝ)))
3229, 31syldan 571 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → (𝐹‘(𝐺𝑡)) ∈ ( ≤ ∩ (ℝ × ℝ)))
3322, 32sseldi 3748 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → (𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ))
34 xp2nd 7347 . . . . . . . . . . . 12 ((𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
3533, 34syl 17 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
364adantr 466 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → 𝐵 ∈ ℝ)
3735, 36ifcld 4268 . . . . . . . . . 10 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ ℝ)
3826simprbi 478 . . . . . . . . . . . . . 14 (𝑡𝑇 → (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅)
3938adantl 467 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅)
40 n0 4076 . . . . . . . . . . . . 13 ((𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
4139, 40sylib 208 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
422adantr 466 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ∈ ℝ)
43 simprr 748 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
44 elin 3945 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) ↔ (𝑦𝑡𝑦 ∈ (𝐴[,]𝐵)))
4543, 44sylib 208 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦𝑡𝑦 ∈ (𝐴[,]𝐵)))
4645simprd 477 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝐴[,]𝐵))
474adantr 466 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐵 ∈ ℝ)
48 elicc2 12442 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵)))
4942, 47, 48syl2anc 565 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵)))
5046, 49mpbid 222 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵))
5150simp1d 1135 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ℝ)
5233adantrr 688 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ))
5352, 34syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
5450simp2d 1136 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴𝑦)
5545simpld 476 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦𝑡)
5629adantrr 688 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐺𝑡) ∈ ℕ)
57 fvco3 6417 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐺𝑡) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = ((,)‘(𝐹‘(𝐺𝑡))))
5830, 57sylan 561 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝐺𝑡) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = ((,)‘(𝐹‘(𝐺𝑡))))
5956, 58syldan 571 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = ((,)‘(𝐹‘(𝐺𝑡))))
60 ovolicc2.9 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
6127, 60sylan2 572 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑡𝑇) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
6261adantrr 688 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
63 1st2nd2 7353 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ) → (𝐹‘(𝐺𝑡)) = ⟨(1st ‘(𝐹‘(𝐺𝑡))), (2nd ‘(𝐹‘(𝐺𝑡)))⟩)
6452, 63syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐹‘(𝐺𝑡)) = ⟨(1st ‘(𝐹‘(𝐺𝑡))), (2nd ‘(𝐹‘(𝐺𝑡)))⟩)
6564fveq2d 6336 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺𝑡))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑡))), (2nd ‘(𝐹‘(𝐺𝑡)))⟩))
66 df-ov 6795 . . . . . . . . . . . . . . . . . . . . 21 ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑡))), (2nd ‘(𝐹‘(𝐺𝑡)))⟩)
6765, 66syl6eqr 2822 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺𝑡))) = ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))))
6859, 62, 673eqtr3d 2812 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑡 = ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))))
6955, 68eleqtrd 2851 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))))
70 xp1st 7346 . . . . . . . . . . . . . . . . . . . 20 ((𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
7152, 70syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
72 rexr 10286 . . . . . . . . . . . . . . . . . . . 20 ((1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ → (1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ*)
73 rexr 10286 . . . . . . . . . . . . . . . . . . . 20 ((2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ → (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ*)
74 elioo2 12420 . . . . . . . . . . . . . . . . . . . 20 (((1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ* ∧ (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ*) → (𝑦 ∈ ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑡))) < 𝑦𝑦 < (2nd ‘(𝐹‘(𝐺𝑡))))))
7572, 73, 74syl2an 575 . . . . . . . . . . . . . . . . . . 19 (((1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ ∧ (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ) → (𝑦 ∈ ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑡))) < 𝑦𝑦 < (2nd ‘(𝐹‘(𝐺𝑡))))))
7671, 53, 75syl2anc 565 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑡))) < 𝑦𝑦 < (2nd ‘(𝐹‘(𝐺𝑡))))))
7769, 76mpbid 222 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑡))) < 𝑦𝑦 < (2nd ‘(𝐹‘(𝐺𝑡)))))
7877simp3d 1137 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 < (2nd ‘(𝐹‘(𝐺𝑡))))
7951, 53, 78ltled 10386 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ≤ (2nd ‘(𝐹‘(𝐺𝑡))))
8042, 51, 53, 54, 79letrd 10395 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡))))
8180expr 444 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → (𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡)))))
8281exlimdv 2012 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → (∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡)))))
8341, 82mpd 15 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡))))
846adantr 466 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → 𝐴𝐵)
85 breq2 4788 . . . . . . . . . . . 12 ((2nd ‘(𝐹‘(𝐺𝑡))) = if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) → (𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡))) ↔ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵)))
86 breq2 4788 . . . . . . . . . . . 12 (𝐵 = if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) → (𝐴𝐵𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵)))
8785, 86ifboth 4261 . . . . . . . . . . 11 ((𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡))) ∧ 𝐴𝐵) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵))
8883, 84, 87syl2anc 565 . . . . . . . . . 10 ((𝜑𝑡𝑇) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵))
89 min2 12225 . . . . . . . . . . 11 (((2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ≤ 𝐵)
9035, 36, 89syl2anc 565 . . . . . . . . . 10 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ≤ 𝐵)
91 elicc2 12442 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐴[,]𝐵) ↔ (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ ℝ ∧ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ≤ 𝐵)))
922, 4, 91syl2anc 565 . . . . . . . . . . 11 (𝜑 → (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐴[,]𝐵) ↔ (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ ℝ ∧ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ≤ 𝐵)))
9392adantr 466 . . . . . . . . . 10 ((𝜑𝑡𝑇) → (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐴[,]𝐵) ↔ (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ ℝ ∧ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ≤ 𝐵)))
9437, 88, 90, 93mpbir3and 1426 . . . . . . . . 9 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐴[,]𝐵))
9521, 94sseldd 3751 . . . . . . . 8 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑈)
96 eluni2 4576 . . . . . . . 8 (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑈 ↔ ∃𝑥𝑈 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
9795, 96sylib 208 . . . . . . 7 ((𝜑𝑡𝑇) → ∃𝑥𝑈 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
98 simprl 746 . . . . . . . . . . 11 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → 𝑥𝑈)
99 simprr 748 . . . . . . . . . . . 12 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
10094adantr 466 . . . . . . . . . . . 12 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐴[,]𝐵))
101 inelcm 4173 . . . . . . . . . . . 12 ((if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐴[,]𝐵)) → (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅)
10299, 100, 101syl2anc 565 . . . . . . . . . . 11 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅)
103 ineq1 3956 . . . . . . . . . . . . 13 (𝑢 = 𝑥 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑥 ∩ (𝐴[,]𝐵)))
104103neeq1d 3001 . . . . . . . . . . . 12 (𝑢 = 𝑥 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅))
105104, 16elrab2 3516 . . . . . . . . . . 11 (𝑥𝑇 ↔ (𝑥𝑈 ∧ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅))
10698, 102, 105sylanbrc 564 . . . . . . . . . 10 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → 𝑥𝑇)
107106, 99jca 495 . . . . . . . . 9 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → (𝑥𝑇 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥))
108107ex 397 . . . . . . . 8 ((𝜑𝑡𝑇) → ((𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥) → (𝑥𝑇 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)))
109108reximdv2 3161 . . . . . . 7 ((𝜑𝑡𝑇) → (∃𝑥𝑈 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥 → ∃𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥))
11097, 109mpd 15 . . . . . 6 ((𝜑𝑡𝑇) → ∃𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
111110ralrimiva 3114 . . . . 5 (𝜑 → ∀𝑡𝑇𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
112 eleq2 2838 . . . . . 6 (𝑥 = (𝑡) → (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥 ↔ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)))
113112ac6sfi 8359 . . . . 5 ((𝑇 ∈ Fin ∧ ∀𝑡𝑇𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥) → ∃(:𝑇𝑇 ∧ ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)))
11420, 111, 113syl2anc 565 . . . 4 (𝜑 → ∃(:𝑇𝑇 ∧ ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)))
115114adantr 466 . . 3 ((𝜑 ∧ (𝑧𝑈𝐴𝑧)) → ∃(:𝑇𝑇 ∧ ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)))
116 fveq2 6332 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (𝐺𝑥) = (𝐺𝑡))
117116fveq2d 6336 . . . . . . . . . . 11 (𝑥 = 𝑡 → (𝐹‘(𝐺𝑥)) = (𝐹‘(𝐺𝑡)))
118117fveq2d 6336 . . . . . . . . . 10 (𝑥 = 𝑡 → (2nd ‘(𝐹‘(𝐺𝑥))) = (2nd ‘(𝐹‘(𝐺𝑡))))
119118breq1d 4794 . . . . . . . . 9 (𝑥 = 𝑡 → ((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵 ↔ (2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵))
120119, 118ifbieq1d 4246 . . . . . . . 8 (𝑥 = 𝑡 → if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) = if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵))
121 fveq2 6332 . . . . . . . 8 (𝑥 = 𝑡 → (𝑥) = (𝑡))
122120, 121eleq12d 2843 . . . . . . 7 (𝑥 = 𝑡 → (if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥) ↔ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)))
123122cbvralv 3319 . . . . . 6 (∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥) ↔ ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡))
1242adantr 466 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐴 ∈ ℝ)
1254adantr 466 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐵 ∈ ℝ)
1266adantr 466 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐴𝐵)
127 ovolicc2.4 . . . . . . . . 9 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
12830adantr 466 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
12912adantr 466 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
1301adantr 466 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → (𝐴[,]𝐵) ⊆ 𝑈)
13123adantr 466 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐺:𝑈⟶ℕ)
13260adantlr 686 . . . . . . . . 9 (((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) ∧ 𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
133 simprrl 758 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → :𝑇𝑇)
134 simprrr 759 . . . . . . . . . 10 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥))
135122rspccva 3457 . . . . . . . . . 10 ((∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥) ∧ 𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡))
136134, 135sylan 561 . . . . . . . . 9 (((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) ∧ 𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡))
137 simprlr 757 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐴𝑧)
138 simprll 756 . . . . . . . . . 10 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝑧𝑈)
1398adantr 466 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐴 ∈ (𝐴[,]𝐵))
140 inelcm 4173 . . . . . . . . . . 11 ((𝐴𝑧𝐴 ∈ (𝐴[,]𝐵)) → (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)
141137, 139, 140syl2anc 565 . . . . . . . . . 10 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)
142 ineq1 3956 . . . . . . . . . . . 12 (𝑢 = 𝑧 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑧 ∩ (𝐴[,]𝐵)))
143142neeq1d 3001 . . . . . . . . . . 11 (𝑢 = 𝑧 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅))
144143, 16elrab2 3516 . . . . . . . . . 10 (𝑧𝑇 ↔ (𝑧𝑈 ∧ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅))
145138, 141, 144sylanbrc 564 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝑧𝑇)
146 eqid 2770 . . . . . . . . 9 seq1(( ∘ 1st ), (ℕ × {𝑧})) = seq1(( ∘ 1st ), (ℕ × {𝑧}))
147 fveq2 6332 . . . . . . . . . . 11 (𝑚 = 𝑛 → (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑚) = (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑛))
148147eleq2d 2835 . . . . . . . . . 10 (𝑚 = 𝑛 → (𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑚) ↔ 𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑛)))
149148cbvrabv 3348 . . . . . . . . 9 {𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑚)} = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑛)}
150 eqid 2770 . . . . . . . . 9 inf({𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑚)}, ℝ, < ) = inf({𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑚)}, ℝ, < )
151124, 125, 126, 127, 128, 129, 130, 131, 132, 16, 133, 136, 137, 145, 146, 149, 150ovolicc2lem4 23507 . . . . . . . 8 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
152151anassrs 458 . . . . . . 7 (((𝜑 ∧ (𝑧𝑈𝐴𝑧)) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥))) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
153152expr 444 . . . . . 6 (((𝜑 ∧ (𝑧𝑈𝐴𝑧)) ∧ :𝑇𝑇) → (∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < )))
154123, 153syl5bir 233 . . . . 5 (((𝜑 ∧ (𝑧𝑈𝐴𝑧)) ∧ :𝑇𝑇) → (∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < )))
155154expimpd 441 . . . 4 ((𝜑 ∧ (𝑧𝑈𝐴𝑧)) → ((:𝑇𝑇 ∧ ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < )))
156155exlimdv 2012 . . 3 ((𝜑 ∧ (𝑧𝑈𝐴𝑧)) → (∃(:𝑇𝑇 ∧ ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < )))
157115, 156mpd 15 . 2 ((𝜑 ∧ (𝑧𝑈𝐴𝑧)) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
15811, 157rexlimddv 3182 1 (𝜑 → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1070   = wceq 1630  ∃wex 1851   ∈ wcel 2144   ≠ wne 2942  ∀wral 3060  ∃wrex 3061  {crab 3064   ∩ cin 3720   ⊆ wss 3721  ∅c0 4061  ifcif 4223  𝒫 cpw 4295  {csn 4314  ⟨cop 4320  ∪ cuni 4572   class class class wbr 4784   × cxp 5247  ran crn 5250   ∘ ccom 5253  ⟶wf 6027  ‘cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313  Fincfn 8108  supcsup 8501  infcinf 8502  ℝcr 10136  1c1 10138   + caddc 10140  ℝ*cxr 10274   < clt 10275   ≤ cle 10276   − cmin 10467  ℕcn 11221  (,)cioo 12379  [,]cicc 12382  seqcseq 13007  abscabs 14181 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-pre-sup 10215 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-sup 8503  df-inf 8504  df-oi 8570  df-card 8964  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-div 10886  df-nn 11222  df-2 11280  df-3 11281  df-n0 11494  df-z 11579  df-uz 11888  df-rp 12035  df-ioo 12383  df-ico 12385  df-icc 12386  df-fz 12533  df-fzo 12673  df-seq 13008  df-exp 13067  df-hash 13321  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-clim 14426  df-sum 14624 This theorem is referenced by:  ovolicc2  23509
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