Theorem ovolicc2lem3 25731

Description: Lemma for ovolicc2 25734. (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)) ≠ ∅}
ovolicc2.11	⊢ (𝜑 → 𝐻:𝑇⟶𝑇)
ovolicc2.12	⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡))
ovolicc2.13	⊢ (𝜑 → 𝐴 ∈ 𝐶)
ovolicc2.14	⊢ (𝜑 → 𝐶 ∈ 𝑇)
ovolicc2.15	⊢ 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))
ovolicc2.16	⊢ 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾‘𝑛)}

Assertion

Ref	Expression
ovolicc2lem3	⊢ ((𝜑 ∧ (𝑁 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ∧ 𝑃 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})) → (𝑁 = 𝑃 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑁)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑃))))))

Distinct variable groups:   𝑚,𝑛,𝑡,𝑢,𝐴   𝐵,𝑚,𝑛,𝑡,𝑢   𝑡,𝐻   𝐶,𝑚,𝑛,𝑡   𝑛,𝐹,𝑡   𝑛,𝐾,𝑡,𝑢   𝑛,𝐺,𝑡   𝑚,𝑊,𝑛   𝜑,𝑚,𝑛,𝑡   𝑇,𝑛,𝑡   𝑛,𝑁,𝑡,𝑢   𝑈,𝑛,𝑡,𝑢

Allowed substitution hints:   𝜑(𝑢)   𝐶(𝑢)   𝑃(𝑢, 𝑡, 𝑚, 𝑛)   𝑆(𝑢, 𝑡, 𝑚, 𝑛)   𝑇(𝑢, 𝑚)   𝑈(𝑚)   𝐹(𝑢, 𝑚)   𝐺(𝑢, 𝑚)   𝐻(𝑢, 𝑚, 𝑛)   𝐾(𝑚)   𝑁(𝑚)   𝑊(𝑢, 𝑡)

Proof of Theorem ovolicc2lem3

Dummy variables 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2fveq3 6890	. . . 4 ⊢ (𝑦 = 𝑘 → (𝐺‘(𝐾‘𝑦)) = (𝐺‘(𝐾‘𝑘)))
2	1	fveq2d 6889	. . 3 ⊢ (𝑦 = 𝑘 → (𝐹‘(𝐺‘(𝐾‘𝑦))) = (𝐹‘(𝐺‘(𝐾‘𝑘))))
3	2	fveq2d 6889	. 2 ⊢ (𝑦 = 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))))
4		2fveq3 6890	. . . 4 ⊢ (𝑦 = 𝑁 → (𝐺‘(𝐾‘𝑦)) = (𝐺‘(𝐾‘𝑁)))
5	4	fveq2d 6889	. . 3 ⊢ (𝑦 = 𝑁 → (𝐹‘(𝐺‘(𝐾‘𝑦))) = (𝐹‘(𝐺‘(𝐾‘𝑁))))
6	5	fveq2d 6889	. 2 ⊢ (𝑦 = 𝑁 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑁)))))
7		2fveq3 6890	. . . 4 ⊢ (𝑦 = 𝑃 → (𝐺‘(𝐾‘𝑦)) = (𝐺‘(𝐾‘𝑃)))
8	7	fveq2d 6889	. . 3 ⊢ (𝑦 = 𝑃 → (𝐹‘(𝐺‘(𝐾‘𝑦))) = (𝐹‘(𝐺‘(𝐾‘𝑃))))
9	8	fveq2d 6889	. 2 ⊢ (𝑦 = 𝑃 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑃)))))
10		ssrab2 4035	. . 3 ⊢ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ⊆ ℕ
11		nnssre 12254	. . 3 ⊢ ℕ ⊆ ℝ
12	10, 11	sstri 3947	. 2 ⊢ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ⊆ ℝ
13	10	sseli 3934	. . 3 ⊢ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} → 𝑦 ∈ ℕ)
14		ovolicc2.5	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
15		inss2 4190	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
16		fss 6726	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ × ℝ))
17	14, 15, 16	sylancl 598	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ))
18	17	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
19		ovolicc2.8	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝑈⟶ℕ)
20	19	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → 𝐺:𝑈⟶ℕ)
21		nnuz 12919	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
22		ovolicc2.15	. . . . . . . . . 10 ⊢ 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))
23		1zzd 12642	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℤ)
24		ovolicc2.14	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ 𝑇)
25		ovolicc2.11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻:𝑇⟶𝑇)
26	21, 22, 23, 24, 25	algrf 16655	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:ℕ⟶𝑇)
27	26	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → 𝐾:ℕ⟶𝑇)
28		ovolicc2.10	. . . . . . . . 9 ⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
29	28	ssrab3 4037	. . . . . . . 8 ⊢ 𝑇 ⊆ 𝑈
30		fss 6726	. . . . . . . 8 ⊢ ((𝐾:ℕ⟶𝑇 ∧ 𝑇 ⊆ 𝑈) → 𝐾:ℕ⟶𝑈)
31	27, 29, 30	sylancl 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → 𝐾:ℕ⟶𝑈)
32		ffvelcdm 7080	. . . . . . 7 ⊢ ((𝐾:ℕ⟶𝑈 ∧ 𝑦 ∈ ℕ) → (𝐾‘𝑦) ∈ 𝑈)
33	31, 32	sylancom 600	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → (𝐾‘𝑦) ∈ 𝑈)
34	20, 33	ffvelcdmd 7084	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → (𝐺‘(𝐾‘𝑦)) ∈ ℕ)
35	18, 34	ffvelcdmd 7084	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → (𝐹‘(𝐺‘(𝐾‘𝑦))) ∈ (ℝ × ℝ))
36		xp2nd 8025	. . . 4 ⊢ ((𝐹‘(𝐺‘(𝐾‘𝑦))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) ∈ ℝ)
37	35, 36	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) ∈ ℝ)
38	13, 37	sylan2 605	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚}) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) ∈ ℝ)
39	10	sseli 3934	. . . 4 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} → 𝑘 ∈ ℕ)
40	39	ad2antll 742	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})) → 𝑘 ∈ ℕ)
41	13	anim2i 629	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚}) → (𝜑 ∧ 𝑦 ∈ ℕ))
42	41	adantrr 730	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})) → (𝜑 ∧ 𝑦 ∈ ℕ))
43		breq1 5114	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 ≤ 𝑚 ↔ 𝑘 ≤ 𝑚))
44	43	ralbidv 3190	. . . . . 6 ⊢ (𝑛 = 𝑘 → (∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚 ↔ ∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚))
45	44	elrab 3652	. . . . 5 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ↔ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚))
46	45	simprbi 503	. . . 4 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} → ∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚)
47	46	ad2antll 742	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})) → ∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚)
48		breq1 5114	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑥 ≤ 𝑚 ↔ 1 ≤ 𝑚))
49	48	ralbidv 3190	. . . . . 6 ⊢ (𝑥 = 1 → (∀𝑚 ∈ 𝑊 𝑥 ≤ 𝑚 ↔ ∀𝑚 ∈ 𝑊 1 ≤ 𝑚))
50		breq2 5115	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑦 < 𝑥 ↔ 𝑦 < 1))
51		2fveq3 6890	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝐺‘(𝐾‘𝑥)) = (𝐺‘(𝐾‘1)))
52	51	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑥 = 1 → (𝐹‘(𝐺‘(𝐾‘𝑥))) = (𝐹‘(𝐺‘(𝐾‘1))))
53	52	fveq2d 6889	. . . . . . . 8 ⊢ (𝑥 = 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘1)))))
54	53	breq2d 5123	. . . . . . 7 ⊢ (𝑥 = 1 → ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥)))) ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1))))))
55	50, 54	imbi12d 347	. . . . . 6 ⊢ (𝑥 = 1 → ((𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥))))) ↔ (𝑦 < 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1)))))))
56	49, 55	imbi12d 347	. . . . 5 ⊢ (𝑥 = 1 → ((∀𝑚 ∈ 𝑊 𝑥 ≤ 𝑚 → (𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥)))))) ↔ (∀𝑚 ∈ 𝑊 1 ≤ 𝑚 → (𝑦 < 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1))))))))
57	56	imbi2d 343	. . . 4 ⊢ (𝑥 = 1 → (((𝜑 ∧ 𝑦 ∈ ℕ) → (∀𝑚 ∈ 𝑊 𝑥 ≤ 𝑚 → (𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥))))))) ↔ ((𝜑 ∧ 𝑦 ∈ ℕ) → (∀𝑚 ∈ 𝑊 1 ≤ 𝑚 → (𝑦 < 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1)))))))))
58		breq1 5114	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝑥 ≤ 𝑚 ↔ 𝑘 ≤ 𝑚))
59	58	ralbidv 3190	. . . . . 6 ⊢ (𝑥 = 𝑘 → (∀𝑚 ∈ 𝑊 𝑥 ≤ 𝑚 ↔ ∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚))
60		breq2 5115	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝑦 < 𝑥 ↔ 𝑦 < 𝑘))
61		2fveq3 6890	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → (𝐺‘(𝐾‘𝑥)) = (𝐺‘(𝐾‘𝑘)))
62	61	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝐹‘(𝐺‘(𝐾‘𝑥))) = (𝐹‘(𝐺‘(𝐾‘𝑘))))
63	62	fveq2d 6889	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))))
64	63	breq2d 5123	. . . . . . 7 ⊢ (𝑥 = 𝑘 → ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥)))) ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘))))))
65	60, 64	imbi12d 347	. . . . . 6 ⊢ (𝑥 = 𝑘 → ((𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥))))) ↔ (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))))))
66	59, 65	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑘 → ((∀𝑚 ∈ 𝑊 𝑥 ≤ 𝑚 → (𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥)))))) ↔ (∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘))))))))
67	66	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑘 → (((𝜑 ∧ 𝑦 ∈ ℕ) → (∀𝑚 ∈ 𝑊 𝑥 ≤ 𝑚 → (𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥))))))) ↔ ((𝜑 ∧ 𝑦 ∈ ℕ) → (∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))))))))
68		breq1 5114	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝑥 ≤ 𝑚 ↔ (𝑘 + 1) ≤ 𝑚))
69	68	ralbidv 3190	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (∀𝑚 ∈ 𝑊 𝑥 ≤ 𝑚 ↔ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚))
70		breq2 5115	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝑦 < 𝑥 ↔ 𝑦 < (𝑘 + 1)))
71		2fveq3 6890	. . . . . . . . . 10 ⊢ (𝑥 = (𝑘 + 1) → (𝐺‘(𝐾‘𝑥)) = (𝐺‘(𝐾‘(𝑘 + 1))))
72	71	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑥 = (𝑘 + 1) → (𝐹‘(𝐺‘(𝐾‘𝑥))) = (𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))
73	72	fveq2d 6889	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))
74	73	breq2d 5123	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥)))) ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
75	70, 74	imbi12d 347	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → ((𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥))))) ↔ (𝑦 < (𝑘 + 1) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
76	69, 75	imbi12d 347	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → ((∀𝑚 ∈ 𝑊 𝑥 ≤ 𝑚 → (𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥)))))) ↔ (∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚 → (𝑦 < (𝑘 + 1) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))))
77	76	imbi2d 343	. . . 4 ⊢ (𝑥 = (𝑘 + 1) → (((𝜑 ∧ 𝑦 ∈ ℕ) → (∀𝑚 ∈ 𝑊 𝑥 ≤ 𝑚 → (𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥))))))) ↔ ((𝜑 ∧ 𝑦 ∈ ℕ) → (∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚 → (𝑦 < (𝑘 + 1) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))))
78		nnnlt1 12285	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → ¬ 𝑦 < 1)
79	78	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → ¬ 𝑦 < 1)
80	79	pm2.21d 122	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → (𝑦 < 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1))))))
81	80	a1d 26	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → (∀𝑚 ∈ 𝑊 1 ≤ 𝑚 → (𝑦 < 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1)))))))
82		nnre 12257	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
83	82	adantr 486	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑊) → 𝑘 ∈ ℝ)
84	83	lep1d 12163	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑊) → 𝑘 ≤ (𝑘 + 1))
85		peano2re 11400	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
86	83, 85	syl 18	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑊) → (𝑘 + 1) ∈ ℝ)
87		ovolicc2.16	. . . . . . . . . . . . . 14 ⊢ 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾‘𝑛)}
88	87	ssrab3 4037	. . . . . . . . . . . . 13 ⊢ 𝑊 ⊆ ℕ
89	88, 11	sstri 3947	. . . . . . . . . . . 12 ⊢ 𝑊 ⊆ ℝ
90	89	sseli 3934	. . . . . . . . . . 11 ⊢ (𝑚 ∈ 𝑊 → 𝑚 ∈ ℝ)
91	90	adantl 487	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑊) → 𝑚 ∈ ℝ)
92		letr 11321	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ 𝑚 ∈ ℝ) → ((𝑘 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≤ 𝑚) → 𝑘 ≤ 𝑚))
93	83, 86, 91, 92	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑊) → ((𝑘 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≤ 𝑚) → 𝑘 ≤ 𝑚))
94	84, 93	mpand 708	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑊) → ((𝑘 + 1) ≤ 𝑚 → 𝑘 ≤ 𝑚))
95	94	ralimdva 3179	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚 → ∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚))
96	95	imim1d 83	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ((∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))))) → (∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘))))))))
97	96	adantl 487	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))))) → (∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘))))))))
98		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝑦 ∈ ℕ)
99		simprl 783	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝑘 ∈ ℕ)
100		nnleltp1 12669	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑦 ≤ 𝑘 ↔ 𝑦 < (𝑘 + 1)))
101	98, 99, 100	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦 ≤ 𝑘 ↔ 𝑦 < (𝑘 + 1)))
102	98	nnred 12265	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝑦 ∈ ℝ)
103	99	nnred 12265	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝑘 ∈ ℝ)
104	102, 103	leloed 11370	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦 ≤ 𝑘 ↔ (𝑦 < 𝑘 ∨ 𝑦 = 𝑘)))
105	101, 104	bitr3d 284	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦 < (𝑘 + 1) ↔ (𝑦 < 𝑘 ∨ 𝑦 = 𝑘)))
106		simpll 779	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝜑)
107		ltp1 12072	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℝ → 𝑘 < (𝑘 + 1))
108		ltnle 11306	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
109	85, 108	mpdan 700	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℝ → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
110	107, 109	mpbid 235	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℝ → ¬ (𝑘 + 1) ≤ 𝑘)
111	103, 110	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → ¬ (𝑘 + 1) ≤ 𝑘)
112		breq2 5115	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑘 → ((𝑘 + 1) ≤ 𝑚 ↔ (𝑘 + 1) ≤ 𝑘))
113	112	rspccv 3580	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚 → (𝑘 ∈ 𝑊 → (𝑘 + 1) ≤ 𝑘))
114	113	ad2antll 742	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑘 ∈ 𝑊 → (𝑘 + 1) ≤ 𝑘))
115	111, 114	mtod 201	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → ¬ 𝑘 ∈ 𝑊)
116		ovolicc.1	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℝ)
117		ovolicc.2	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ)
118		ovolicc.3	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
119		ovolicc2.4	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
120		ovolicc2.6	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
121		ovolicc2.7	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)
122		ovolicc2.9	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
123		ovolicc2.12	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡))
124		ovolicc2.13	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
125	116, 117, 118, 119, 14, 120, 121, 19, 122, 28, 25, 123, 124, 24, 22, 87	ovolicc2lem2 25730	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ ¬ 𝑘 ∈ 𝑊)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ≤ 𝐵)
126	106, 99, 115, 125	syl12anc 850	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ≤ 𝐵)
127	126	iftrued 4497	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → if((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))), 𝐵) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))))
128		2fveq3 6890	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = (𝐾‘𝑘) → (𝐹‘(𝐺‘𝑡)) = (𝐹‘(𝐺‘(𝐾‘𝑘))))
129	128	fveq2d 6889	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = (𝐾‘𝑘) → (2nd ‘(𝐹‘(𝐺‘𝑡))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))))
130	129	breq1d 5121	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = (𝐾‘𝑘) → ((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ≤ 𝐵))
131	130, 129	ifbieq1d 4514	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (𝐾‘𝑘) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) = if((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))), 𝐵))
132		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (𝐾‘𝑘) → (𝐻‘𝑡) = (𝐻‘(𝐾‘𝑘)))
133	131, 132	eleq12d 2859	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝐾‘𝑘) → (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡) ↔ if((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))), 𝐵) ∈ (𝐻‘(𝐾‘𝑘))))
134	123	ralrimiva 3159	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡))
135	134	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡))
136	26	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝐾:ℕ⟶𝑇)
137	136, 99	ffvelcdmd 7084	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐾‘𝑘) ∈ 𝑇)
138	133, 135, 137	rspcdva 3584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → if((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))), 𝐵) ∈ (𝐻‘(𝐾‘𝑘)))
139	127, 138	eqeltrrd 2866	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∈ (𝐻‘(𝐾‘𝑘)))
140	21, 22, 23, 24, 25	algrp1 16656	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾‘(𝑘 + 1)) = (𝐻‘(𝐾‘𝑘)))
141	140	ad2ant2r 760	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐾‘(𝑘 + 1)) = (𝐻‘(𝐾‘𝑘)))
142	139, 141	eleqtrrd 2868	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∈ (𝐾‘(𝑘 + 1)))
143	136, 29, 30	sylancl 598	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝐾:ℕ⟶𝑈)
144	99	peano2nnd 12267	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑘 + 1) ∈ ℕ)
145	143, 144	ffvelcdmd 7084	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐾‘(𝑘 + 1)) ∈ 𝑈)
146	116, 117, 118, 119, 14, 120, 121, 19, 122	ovolicc2lem1 25729	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐾‘(𝑘 + 1)) ∈ 𝑈) → ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∈ (𝐾‘(𝑘 + 1)) ↔ ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
147	106, 145, 146	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∈ (𝐾‘(𝑘 + 1)) ↔ ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
148	142, 147	mpbid 235	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
149	148	simp3d 1162	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))
150	37	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) ∈ ℝ)
151	17	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝐹:ℕ⟶(ℝ × ℝ))
152	19	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝐺:𝑈⟶ℕ)
153	143, 99	ffvelcdmd 7084	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐾‘𝑘) ∈ 𝑈)
154	152, 153	ffvelcdmd 7084	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐺‘(𝐾‘𝑘)) ∈ ℕ)
155	151, 154	ffvelcdmd 7084	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐹‘(𝐺‘(𝐾‘𝑘))) ∈ (ℝ × ℝ))
156		xp2nd 8025	. . . . . . . . . . . . 13 ⊢ ((𝐹‘(𝐺‘(𝐾‘𝑘))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∈ ℝ)
157	155, 156	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∈ ℝ)
158	152, 145	ffvelcdmd 7084	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐺‘(𝐾‘(𝑘 + 1))) ∈ ℕ)
159	151, 158	ffvelcdmd 7084	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))) ∈ (ℝ × ℝ))
160		xp2nd 8025	. . . . . . . . . . . . 13 ⊢ ((𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) ∈ ℝ)
161	159, 160	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) ∈ ℝ)
162		lttr 11303	. . . . . . . . . . . 12 ⊢ (((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) ∈ ℝ ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∈ ℝ ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) ∈ ℝ) → (((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
163	150, 157, 161, 162	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
164	149, 163	mpan2d 707	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
165	164	imim2d 58	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → ((𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘))))) → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
166	165	com23 87	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦 < 𝑘 → ((𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘))))) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
167	3	breq1d 5121	. . . . . . . . . 10 ⊢ (𝑦 = 𝑘 → ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
168	149, 167	syl5ibrcom 250	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦 = 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
169	168	a1dd 51	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦 = 𝑘 → ((𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘))))) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
170	166, 169	jaod 873	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → ((𝑦 < 𝑘 ∨ 𝑦 = 𝑘) → ((𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘))))) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
171	105, 170	sylbid 243	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦 < (𝑘 + 1) → ((𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘))))) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
172	171	com23 87	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → ((𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘))))) → (𝑦 < (𝑘 + 1) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
173	97, 172	animpimp2impd 860	. . . 4 ⊢ (𝑘 ∈ ℕ → (((𝜑 ∧ 𝑦 ∈ ℕ) → (∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘))))))) → ((𝜑 ∧ 𝑦 ∈ ℕ) → (∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚 → (𝑦 < (𝑘 + 1) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))))
174	57, 67, 77, 67, 81, 173	nnind 12268	. . 3 ⊢ (𝑘 ∈ ℕ → ((𝜑 ∧ 𝑦 ∈ ℕ) → (∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘))))))))
175	40, 42, 47, 174	syl3c 67	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})) → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘))))))
176	3, 6, 9, 12, 38, 175	eqord1 11759	1 ⊢ ((𝜑 ∧ (𝑁 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ∧ 𝑃 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})) → (𝑁 = 𝑃 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑁)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑃))))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  {crab 3418   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  ifcif 4489  𝒫 cpw 4564  {csn 4591  ∪ cuni 4874   class class class wbr 5111   × cxp 5661  ran crn 5664   ∘ ccom 5667  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  1^st c1st 7990  2^nd c2nd 7991  Fincfn 8949  ℝcr 11116  1c1 11118   + caddc 11120   < clt 11260   ≤ cle 11261   − cmin 11458  ℕcn 12250  (,)cioo 13390  [,]cicc 13393  seqcseq 14057  abscabs 15311

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-nn 12251  df-n0 12522  df-z 12609  df-uz 12881  df-ioo 13394  df-icc 13397  df-fz 13554  df-seq 14058

This theorem is used by:  ovolicc2lem4  25732