Theorem ovolicc2lem5 23725

Description: Lemma for ovolicc2 23726. (Contributed by Mario Carneiro, 14-Jun-2014.)

Hypotheses

Ref	Expression
ovolicc.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
ovolicc.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
ovolicc.3	⊢ (𝜑 → 𝐴 ≤ 𝐵)
ovolicc2.4	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ovolicc2.5	⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ovolicc2.6	⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
ovolicc2.7	⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)
ovolicc2.8	⊢ (𝜑 → 𝐺:𝑈⟶ℕ)
ovolicc2.9	⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
ovolicc2.10	⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}

Assertion

Ref	Expression
ovolicc2lem5	⊢ (𝜑 → (𝐵 − 𝐴) ≤ 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 ℎ 𝑚 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovolicc2.7	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)
2		ovolicc.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	2	rexrd 10426	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
4		ovolicc.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5	4	rexrd 10426	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
6		ovolicc.3	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
7		lbicc2 12602	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
8	3, 5, 6, 7	syl3anc 1439	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
9	1, 8	sseldd 3822	. . 3 ⊢ (𝜑 → 𝐴 ∈ ∪ 𝑈)
10		eluni2 4675	. . 3 ⊢ (𝐴 ∈ ∪ 𝑈 ↔ ∃𝑧 ∈ 𝑈 𝐴 ∈ 𝑧)
11	9, 10	sylib 210	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝑈 𝐴 ∈ 𝑧)
12		ovolicc2.6	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
13		elin 4019	. . . . . . . 8 ⊢ (𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin) ↔ (𝑈 ∈ 𝒫 ran ((,) ∘ 𝐹) ∧ 𝑈 ∈ Fin))
14	12, 13	sylib 210	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∈ 𝒫 ran ((,) ∘ 𝐹) ∧ 𝑈 ∈ Fin))
15	14	simprd 491	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Fin)
16		ovolicc2.10	. . . . . . 7 ⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
17		ssrab2 3908	. . . . . . 7 ⊢ {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} ⊆ 𝑈
18	16, 17	eqsstri 3854	. . . . . 6 ⊢ 𝑇 ⊆ 𝑈
19		ssfi 8468	. . . . . 6 ⊢ ((𝑈 ∈ Fin ∧ 𝑇 ⊆ 𝑈) → 𝑇 ∈ Fin)
20	15, 18, 19	sylancl 580	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ Fin)
21	1	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐴[,]𝐵) ⊆ ∪ 𝑈)
22		inss2 4054	. . . . . . . . . . . . 13 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
23		ovolicc2.8	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺:𝑈⟶ℕ)
24		ineq1 4030	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑡 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑡 ∩ (𝐴[,]𝐵)))
25	24	neeq1d 3028	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑡 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅))
26	25, 16	elrab2 3576	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ 𝑇 ↔ (𝑡 ∈ 𝑈 ∧ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅))
27	26	simplbi 493	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ 𝑇 → 𝑡 ∈ 𝑈)
28		ffvelrn 6621	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:𝑈⟶ℕ ∧ 𝑡 ∈ 𝑈) → (𝐺‘𝑡) ∈ ℕ)
29	23, 27, 28	syl2an 589	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℕ)
30		ovolicc2.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
31	30	ffvelrnda 6623	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐺‘𝑡) ∈ ℕ) → (𝐹‘(𝐺‘𝑡)) ∈ ( ≤ ∩ (ℝ × ℝ)))
32	29, 31	syldan 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘(𝐺‘𝑡)) ∈ ( ≤ ∩ (ℝ × ℝ)))
33	22, 32	sseldi 3819	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ))
34		xp2nd 7478	. . . . . . . . . . . 12 ⊢ ((𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ)
35	33, 34	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ)
36	4	adantr 474	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐵 ∈ ℝ)
37	35, 36	ifcld 4352	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ℝ)
38	26	simprbi 492	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑇 → (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅)
39	38	adantl 475	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅)
40		n0 4159	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
41	39, 40	sylib 210	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
42	2	adantr 474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ∈ ℝ)
43		simprr 763	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
44		elin 4019	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) ↔ (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ (𝐴[,]𝐵)))
45	43, 44	sylib 210	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ (𝐴[,]𝐵)))
46	45	simprd 491	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝐴[,]𝐵))
47	4	adantr 474	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐵 ∈ ℝ)
48		elicc2 12550	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
49	42, 47, 48	syl2anc 579	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
50	46, 49	mpbid 224	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))
51	50	simp1d 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ℝ)
52	33	adantrr 707	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ))
53	52, 34	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ)
54	50	simp2d 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ≤ 𝑦)
55	45	simpld 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ 𝑡)
56	29	adantrr 707	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐺‘𝑡) ∈ ℕ)
57		fvco3 6535	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐺‘𝑡) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = ((,)‘(𝐹‘(𝐺‘𝑡))))
58	30, 57	sylan 575	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝐺‘𝑡) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = ((,)‘(𝐹‘(𝐺‘𝑡))))
59	56, 58	syldan 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = ((,)‘(𝐹‘(𝐺‘𝑡))))
60		ovolicc2.9	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
61	27, 60	sylan2 586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
62	61	adantrr 707	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
63		1st2nd2 7484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ) → (𝐹‘(𝐺‘𝑡)) = ⟨(1st ‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))⟩)
64	52, 63	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐹‘(𝐺‘𝑡)) = ⟨(1st ‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))⟩)
65	64	fveq2d 6450	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺‘𝑡))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))⟩))
66		df-ov 6925	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))⟩)
67	65, 66	syl6eqr 2832	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺‘𝑡))) = ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))))
68	59, 62, 67	3eqtr3d 2822	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑡 = ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))))
69	55, 68	eleqtrd 2861	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))))
70		xp1st 7477	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ)
71	52, 70	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ)
72		rexr 10422	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ → (1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ*)
73		rexr 10422	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ → (2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ*)
74		elioo2 12528	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ* ∧ (2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ*) → (𝑦 ∈ ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘𝑡))) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡))))))
75	72, 73, 74	syl2an 589	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ ∧ (2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) → (𝑦 ∈ ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘𝑡))) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡))))))
76	71, 53, 75	syl2anc 579	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘𝑡))) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡))))))
77	69, 76	mpbid 224	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘𝑡))) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡)))))
78	77	simp3d 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡))))
79	51, 53, 78	ltled 10524	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))))
80	42, 51, 53, 54, 79	letrd 10533	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))))
81	80	expr 450	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡)))))
82	81	exlimdv 1976	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡)))))
83	41, 82	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))))
84	6	adantr 474	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐴 ≤ 𝐵)
85		breq2 4890	. . . . . . . . . . . 12 ⊢ ((2nd ‘(𝐹‘(𝐺‘𝑡))) = if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) → (𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))) ↔ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵)))
86		breq2 4890	. . . . . . . . . . . 12 ⊢ (𝐵 = if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) → (𝐴 ≤ 𝐵 ↔ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵)))
87	85, 86	ifboth 4345	. . . . . . . . . . 11 ⊢ ((𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵))
88	83, 84, 87	syl2anc 579	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵))
89		min2 12333	. . . . . . . . . . 11 ⊢ (((2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵)
90	35, 36, 89	syl2anc 579	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵)
91		elicc2 12550	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵) ↔ (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ℝ ∧ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵)))
92	2, 4, 91	syl2anc 579	. . . . . . . . . . 11 ⊢ (𝜑 → (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵) ↔ (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ℝ ∧ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵)))
93	92	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵) ↔ (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ℝ ∧ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵)))
94	37, 88, 90, 93	mpbir3and 1399	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵))
95	21, 94	sseldd 3822	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ∪ 𝑈)
96		eluni2 4675	. . . . . . . 8 ⊢ (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ∪ 𝑈 ↔ ∃𝑥 ∈ 𝑈 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)
97	95, 96	sylib 210	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∃𝑥 ∈ 𝑈 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)
98		simprl 761	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → 𝑥 ∈ 𝑈)
99		simprr 763	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)
100	94	adantr 474	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵))
101		inelcm 4257	. . . . . . . . 9 ⊢ ((if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵)) → (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅)
102	99, 100, 101	syl2anc 579	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅)
103		ineq1 4030	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑥 ∩ (𝐴[,]𝐵)))
104	103	neeq1d 3028	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅))
105	104, 16	elrab2 3576	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑇 ↔ (𝑥 ∈ 𝑈 ∧ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅))
106	98, 102, 105	sylanbrc 578	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → 𝑥 ∈ 𝑇)
107	97, 106, 99	reximssdv 3200	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∃𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)
108	107	ralrimiva 3148	. . . . 5 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 ∃𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)
109		eleq2 2848	. . . . . 6 ⊢ (𝑥 = (ℎ‘𝑡) → (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥 ↔ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)))
110	109	ac6sfi 8492	. . . . 5 ⊢ ((𝑇 ∈ Fin ∧ ∀𝑡 ∈ 𝑇 ∃𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) → ∃ℎ(ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)))
111	20, 108, 110	syl2anc 579	. . . 4 ⊢ (𝜑 → ∃ℎ(ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)))
112	111	adantr 474	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) → ∃ℎ(ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)))
113		2fveq3 6451	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → (𝐹‘(𝐺‘𝑥)) = (𝐹‘(𝐺‘𝑡)))
114	113	fveq2d 6450	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → (2nd ‘(𝐹‘(𝐺‘𝑥))) = (2nd ‘(𝐹‘(𝐺‘𝑡))))
115	114	breq1d 4896	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → ((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵 ↔ (2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵))
116	115, 114	ifbieq1d 4330	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) = if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵))
117		fveq2 6446	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (ℎ‘𝑥) = (ℎ‘𝑡))
118	116, 117	eleq12d 2853	. . . . . . 7 ⊢ (𝑥 = 𝑡 → (if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥) ↔ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)))
119	118	cbvralv 3367	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥) ↔ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡))
120	2	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐴 ∈ ℝ)
121	4	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐵 ∈ ℝ)
122	6	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐴 ≤ 𝐵)
123		ovolicc2.4	. . . . . . . . 9 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
124	30	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
125	12	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
126	1	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → (𝐴[,]𝐵) ⊆ ∪ 𝑈)
127	23	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐺:𝑈⟶ℕ)
128	60	adantlr 705	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
129		simprrl 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → ℎ:𝑇⟶𝑇)
130		simprrr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥))
131	118	rspccva 3510	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥) ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡))
132	130, 131	sylan 575	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡))
133		simprlr 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐴 ∈ 𝑧)
134		simprll 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝑧 ∈ 𝑈)
135	8	adantr 474	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐴 ∈ (𝐴[,]𝐵))
136		inelcm 4257	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑧 ∧ 𝐴 ∈ (𝐴[,]𝐵)) → (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)
137	133, 135, 136	syl2anc 579	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)
138		ineq1 4030	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑧 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑧 ∩ (𝐴[,]𝐵)))
139	138	neeq1d 3028	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑧 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅))
140	139, 16	elrab2 3576	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑇 ↔ (𝑧 ∈ 𝑈 ∧ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅))
141	134, 137, 140	sylanbrc 578	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝑧 ∈ 𝑇)
142		eqid 2778	. . . . . . . . 9 ⊢ seq1((ℎ ∘ 1st ), (ℕ × {𝑧})) = seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))
143		fveq2 6446	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))‘𝑚) = (seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))‘𝑛))
144	143	eleq2d 2845	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))‘𝑚) ↔ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))‘𝑛)))
145	144	cbvrabv 3396	. . . . . . . . 9 ⊢ {𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))‘𝑚)} = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))‘𝑛)}
146		eqid 2778	. . . . . . . . 9 ⊢ inf({𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))‘𝑚)}, ℝ, < ) = inf({𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))‘𝑚)}, ℝ, < )
147	120, 121, 122, 123, 124, 125, 126, 127, 128, 16, 129, 132, 133, 141, 142, 145, 146	ovolicc2lem4 23724	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
148	147	anassrs 461	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥))) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
149	148	expr 450	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) ∧ ℎ:𝑇⟶𝑇) → (∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < )))
150	119, 149	syl5bir 235	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) ∧ ℎ:𝑇⟶𝑇) → (∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < )))
151	150	expimpd 447	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) → ((ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < )))
152	151	exlimdv 1976	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) → (∃ℎ(ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < )))
153	112, 152	mpd 15	. 2 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
154	11, 153	rexlimddv 3218	1 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601  ∃wex 1823   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090  ∃wrex 3091  {crab 3094   ∩ cin 3791   ⊆ wss 3792  ∅c0 4141  ifcif 4307  𝒫 cpw 4379  {csn 4398  ⟨cop 4404  ∪ cuni 4671   class class class wbr 4886   × cxp 5353  ran crn 5356   ∘ ccom 5359  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922  1^st c1st 7443  2^nd c2nd 7444  Fincfn 8241  supcsup 8634  infcinf 8635  ℝcr 10271  1c1 10273   + caddc 10275  ℝ^*cxr 10410   < clt 10411   ≤ cle 10412   − cmin 10606  ℕcn 11374  (,)cioo 12487  [,]cicc 12490  seqcseq 13119  abscabs 14381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-z 11729  df-uz 11993  df-rp 12138  df-ioo 12491  df-ico 12493  df-icc 12494  df-fz 12644  df-fzo 12785  df-seq 13120  df-exp 13179  df-hash 13436  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-clim 14627  df-sum 14825

This theorem is referenced by:  ovolicc2  23726