Theorem ovolicc2lem5 25455

Description: Lemma for ovolicc2 25456. (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 11200	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
4		ovolicc.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5	4	rexrd 11200	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
6		ovolicc.3	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
7		lbicc2 13401	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
8	3, 5, 6, 7	syl3anc 1373	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
9	1, 8	sseldd 3944	. . 3 ⊢ (𝜑 → 𝐴 ∈ ∪ 𝑈)
10		eluni2 4871	. . 3 ⊢ (𝐴 ∈ ∪ 𝑈 ↔ ∃𝑧 ∈ 𝑈 𝐴 ∈ 𝑧)
11	9, 10	sylib 218	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝑈 𝐴 ∈ 𝑧)
12		ovolicc2.6	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
13	12	elin2d 4164	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Fin)
14		ovolicc2.10	. . . . . . 7 ⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
15	14	ssrab3 4041	. . . . . 6 ⊢ 𝑇 ⊆ 𝑈
16		ssfi 9114	. . . . . 6 ⊢ ((𝑈 ∈ Fin ∧ 𝑇 ⊆ 𝑈) → 𝑇 ∈ Fin)
17	13, 15, 16	sylancl 586	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ Fin)
18	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐴[,]𝐵) ⊆ ∪ 𝑈)
19		ovolicc2.8	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺:𝑈⟶ℕ)
20		ineq1 4172	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑡 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑡 ∩ (𝐴[,]𝐵)))
21	20	neeq1d 2984	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑡 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅))
22	21, 14	elrab2 3659	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ 𝑇 ↔ (𝑡 ∈ 𝑈 ∧ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅))
23	22	simplbi 497	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ 𝑇 → 𝑡 ∈ 𝑈)
24		ffvelcdm 7035	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:𝑈⟶ℕ ∧ 𝑡 ∈ 𝑈) → (𝐺‘𝑡) ∈ ℕ)
25	19, 23, 24	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℕ)
26		ovolicc2.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
27	26	ffvelcdmda 7038	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐺‘𝑡) ∈ ℕ) → (𝐹‘(𝐺‘𝑡)) ∈ ( ≤ ∩ (ℝ × ℝ)))
28	25, 27	syldan 591	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘(𝐺‘𝑡)) ∈ ( ≤ ∩ (ℝ × ℝ)))
29	28	elin2d 4164	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ))
30		xp2nd 7980	. . . . . . . . . . . 12 ⊢ ((𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ)
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ)
32	4	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐵 ∈ ℝ)
33	31, 32	ifcld 4531	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ℝ)
34	22	simprbi 496	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑇 → (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅)
35	34	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅)
36		n0 4312	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
37	35, 36	sylib 218	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
38	2	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ∈ ℝ)
39		simprr 772	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
40	39	elin2d 4164	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝐴[,]𝐵))
41	4	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐵 ∈ ℝ)
42		elicc2 13348	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
43	2, 41, 42	syl2an2r 685	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
44	40, 43	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))
45	44	simp1d 1142	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ℝ)
46	29	adantrr 717	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ))
47	46, 30	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ)
48	44	simp2d 1143	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ≤ 𝑦)
49	39	elin1d 4163	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ 𝑡)
50	25	adantrr 717	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐺‘𝑡) ∈ ℕ)
51		fvco3 6942	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐺‘𝑡) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = ((,)‘(𝐹‘(𝐺‘𝑡))))
52	26, 50, 51	syl2an2r 685	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = ((,)‘(𝐹‘(𝐺‘𝑡))))
53		ovolicc2.9	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
54	23, 53	sylan2 593	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
55	54	adantrr 717	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
56		1st2nd2 7986	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ) → (𝐹‘(𝐺‘𝑡)) = ⟨(1st ‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))⟩)
57	46, 56	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐹‘(𝐺‘𝑡)) = ⟨(1st ‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))⟩)
58	57	fveq2d 6844	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺‘𝑡))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))⟩))
59		df-ov 7372	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))⟩)
60	58, 59	eqtr4di 2782	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺‘𝑡))) = ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))))
61	52, 55, 60	3eqtr3d 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑡 = ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))))
62	49, 61	eleqtrd 2830	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))))
63		xp1st 7979	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ)
64	46, 63	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ)
65		rexr 11196	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ → (1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ*)
66		rexr 11196	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ → (2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ*)
67		elioo2 13323	. . . . . . . . . . . . . . . . . . . 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))))))
68	65, 66, 67	syl2an 596	. . . . . . . . . . . . . . . . . . 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))))))
69	64, 47, 68	syl2anc 584	. . . . . . . . . . . . . . . . . 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))))))
70	62, 69	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘𝑡))) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡)))))
71	70	simp3d 1144	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡))))
72	45, 47, 71	ltled 11298	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))))
73	38, 45, 47, 48, 72	letrd 11307	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))))
74	73	expr 456	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡)))))
75	74	exlimdv 1933	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡)))))
76	37, 75	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))))
77	6	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐴 ≤ 𝐵)
78		breq2 5106	. . . . . . . . . . . 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)))
79		breq2 5106	. . . . . . . . . . . 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)))
80	78, 79	ifboth 4524	. . . . . . . . . . 11 ⊢ ((𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵))
81	76, 77, 80	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵))
82		min2 13126	. . . . . . . . . . 11 ⊢ (((2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵)
83	31, 32, 82	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵)
84		elicc2 13348	. . . . . . . . . . . 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)))
85	2, 4, 84	syl2anc 584	. . . . . . . . . . 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)))
86	85	adantr 480	. . . . . . . . . 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)))
87	33, 81, 83, 86	mpbir3and 1343	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵))
88	18, 87	sseldd 3944	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ∪ 𝑈)
89		eluni2 4871	. . . . . . . 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)
90	88, 89	sylib 218	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∃𝑥 ∈ 𝑈 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)
91		simprl 770	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → 𝑥 ∈ 𝑈)
92		simprr 772	. . . . . . . . 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)
93	87	adantr 480	. . . . . . . . 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))
94		inelcm 4424	. . . . . . . . 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)) ≠ ∅)
95	92, 93, 94	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅)
96		ineq1 4172	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑥 ∩ (𝐴[,]𝐵)))
97	96	neeq1d 2984	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅))
98	97, 14	elrab2 3659	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑇 ↔ (𝑥 ∈ 𝑈 ∧ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅))
99	91, 95, 98	sylanbrc 583	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → 𝑥 ∈ 𝑇)
100	90, 99, 92	reximssdv 3151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∃𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)
101	100	ralrimiva 3125	. . . . 5 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 ∃𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)
102		eleq2 2817	. . . . . 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)))
103	102	ac6sfi 9207	. . . . 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)))
104	17, 101, 103	syl2anc 584	. . . 4 ⊢ (𝜑 → ∃ℎ(ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)))
105	104	adantr 480	. . 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)))
106		2fveq3 6845	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → (𝐹‘(𝐺‘𝑥)) = (𝐹‘(𝐺‘𝑡)))
107	106	fveq2d 6844	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → (2nd ‘(𝐹‘(𝐺‘𝑥))) = (2nd ‘(𝐹‘(𝐺‘𝑡))))
108	107	breq1d 5112	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → ((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵 ↔ (2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵))
109	108, 107	ifbieq1d 4509	. . . . . . . 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))
110		fveq2 6840	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (ℎ‘𝑥) = (ℎ‘𝑡))
111	109, 110	eleq12d 2822	. . . . . . 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)))
112	111	cbvralvw 3213	. . . . . 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))
113	2	adantr 480	. . . . . . . . 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)
114	4	adantr 480	. . . . . . . . 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)
115	6	adantr 480	. . . . . . . . 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)
116		ovolicc2.4	. . . . . . . . 9 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
117	26	adantr 480	. . . . . . . . 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)))
118	12	adantr 480	. . . . . . . . 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))
119	1	adantr 480	. . . . . . . . 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)
120	19	adantr 480	. . . . . . . . 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)
121	53	adantlr 715	. . . . . . . . 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)
122		simprrl 780	. . . . . . . . 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)
123		simprrr 781	. . . . . . . . . 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))
124	111	rspccva 3584	. . . . . . . . . 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))
125	123, 124	sylan 580	. . . . . . . . 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))
126		simprlr 779	. . . . . . . . 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)
127		simprll 778	. . . . . . . . . 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)
128	8	adantr 480	. . . . . . . . . . 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))
129		inelcm 4424	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑧 ∧ 𝐴 ∈ (𝐴[,]𝐵)) → (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)
130	126, 128, 129	syl2anc 584	. . . . . . . . . 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)) ≠ ∅)
131		ineq1 4172	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑧 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑧 ∩ (𝐴[,]𝐵)))
132	131	neeq1d 2984	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑧 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅))
133	132, 14	elrab2 3659	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑇 ↔ (𝑧 ∈ 𝑈 ∧ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅))
134	127, 130, 133	sylanbrc 583	. . . . . . . . 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)
135		eqid 2729	. . . . . . . . 9 ⊢ seq1((ℎ ∘ 1st ), (ℕ × {𝑧})) = seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))
136		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))‘𝑚) = (seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))‘𝑛))
137	136	eleq2d 2814	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))‘𝑚) ↔ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))‘𝑛)))
138	137	cbvrabv 3413	. . . . . . . . 9 ⊢ {𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))‘𝑚)} = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))‘𝑛)}
139		eqid 2729	. . . . . . . . 9 ⊢ inf({𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))‘𝑚)}, ℝ, < ) = inf({𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ × {𝑧}))‘𝑚)}, ℝ, < )
140	113, 114, 115, 116, 117, 118, 119, 120, 121, 14, 122, 125, 126, 134, 135, 138, 139	ovolicc2lem4 25454	. . . . . . . 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*, < ))
141	140	anassrs 467	. . . . . . 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*, < ))
142	141	expr 456	. . . . . 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*, < )))
143	112, 142	biimtrrid 243	. . . . 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*, < )))
144	143	expimpd 453	. . . 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*, < )))
145	144	exlimdv 1933	. . 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*, < )))
146	105, 145	mpd 15	. 2 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
147	11, 146	rexlimddv 3140	1 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3402   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  ifcif 4484  𝒫 cpw 4559  {csn 4585  ⟨cop 4591  ∪ cuni 4867   class class class wbr 5102   × cxp 5629  ran crn 5632   ∘ ccom 5635  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946  Fincfn 8895  supcsup 9367  infcinf 9368  ℝcr 11043  1c1 11045   + caddc 11047  ℝ^*cxr 11183   < clt 11184   ≤ cle 11185   − cmin 11381  ℕcn 12162  (,)cioo 13282  [,]cicc 13285  seqcseq 13942  abscabs 15176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-inf 9370  df-oi 9439  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-n0 12419  df-z 12506  df-uz 12770  df-rp 12928  df-ioo 13286  df-ico 13288  df-icc 13289  df-fz 13445  df-fzo 13592  df-seq 13943  df-exp 14003  df-hash 14272  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-clim 15430  df-sum 15629

This theorem is referenced by:  ovolicc2  25456