Theorem ovolicc2lem3 24388

Description: Lemma for ovolicc2 24391. (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 6711	. . . 4 ⊢ (𝑦 = 𝑘 → (𝐺‘(𝐾‘𝑦)) = (𝐺‘(𝐾‘𝑘)))
2	1	fveq2d 6710	. . 3 ⊢ (𝑦 = 𝑘 → (𝐹‘(𝐺‘(𝐾‘𝑦))) = (𝐹‘(𝐺‘(𝐾‘𝑘))))
3	2	fveq2d 6710	. 2 ⊢ (𝑦 = 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))))
4		2fveq3 6711	. . . 4 ⊢ (𝑦 = 𝑁 → (𝐺‘(𝐾‘𝑦)) = (𝐺‘(𝐾‘𝑁)))
5	4	fveq2d 6710	. . 3 ⊢ (𝑦 = 𝑁 → (𝐹‘(𝐺‘(𝐾‘𝑦))) = (𝐹‘(𝐺‘(𝐾‘𝑁))))
6	5	fveq2d 6710	. 2 ⊢ (𝑦 = 𝑁 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑁)))))
7		2fveq3 6711	. . . 4 ⊢ (𝑦 = 𝑃 → (𝐺‘(𝐾‘𝑦)) = (𝐺‘(𝐾‘𝑃)))
8	7	fveq2d 6710	. . 3 ⊢ (𝑦 = 𝑃 → (𝐹‘(𝐺‘(𝐾‘𝑦))) = (𝐹‘(𝐺‘(𝐾‘𝑃))))
9	8	fveq2d 6710	. 2 ⊢ (𝑦 = 𝑃 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑃)))))
10		ssrab2 3983	. . 3 ⊢ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ⊆ ℕ
11		nnssre 11817	. . 3 ⊢ ℕ ⊆ ℝ
12	10, 11	sstri 3900	. 2 ⊢ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ⊆ ℝ
13	10	sseli 3887	. . 3 ⊢ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} → 𝑦 ∈ ℕ)
14		ovolicc2.5	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
15		inss2 4134	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
16		fss 6551	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ × ℝ))
17	14, 15, 16	sylancl 589	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ))
18	17	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
19		ovolicc2.8	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝑈⟶ℕ)
20	19	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → 𝐺:𝑈⟶ℕ)
21		nnuz 12460	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
22		ovolicc2.15	. . . . . . . . . 10 ⊢ 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))
23		1zzd 12191	. . . . . . . . . 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 16111	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:ℕ⟶𝑇)
27	26	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → 𝐾:ℕ⟶𝑇)
28		ovolicc2.10	. . . . . . . . 9 ⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
29	28	ssrab3 3985	. . . . . . . 8 ⊢ 𝑇 ⊆ 𝑈
30		fss 6551	. . . . . . . 8 ⊢ ((𝐾:ℕ⟶𝑇 ∧ 𝑇 ⊆ 𝑈) → 𝐾:ℕ⟶𝑈)
31	27, 29, 30	sylancl 589	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → 𝐾:ℕ⟶𝑈)
32		ffvelrn 6891	. . . . . . 7 ⊢ ((𝐾:ℕ⟶𝑈 ∧ 𝑦 ∈ ℕ) → (𝐾‘𝑦) ∈ 𝑈)
33	31, 32	sylancom 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → (𝐾‘𝑦) ∈ 𝑈)
34	20, 33	ffvelrnd 6894	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → (𝐺‘(𝐾‘𝑦)) ∈ ℕ)
35	18, 34	ffvelrnd 6894	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → (𝐹‘(𝐺‘(𝐾‘𝑦))) ∈ (ℝ × ℝ))
36		xp2nd 7783	. . . 4 ⊢ ((𝐹‘(𝐺‘(𝐾‘𝑦))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) ∈ ℝ)
37	35, 36	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) ∈ ℝ)
38	13, 37	sylan2 596	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚}) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) ∈ ℝ)
39	10	sseli 3887	. . . 4 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} → 𝑘 ∈ ℕ)
40	39	ad2antll 729	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})) → 𝑘 ∈ ℕ)
41	13	anim2i 620	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚}) → (𝜑 ∧ 𝑦 ∈ ℕ))
42	41	adantrr 717	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})) → (𝜑 ∧ 𝑦 ∈ ℕ))
43		breq1 5046	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 ≤ 𝑚 ↔ 𝑘 ≤ 𝑚))
44	43	ralbidv 3111	. . . . . 6 ⊢ (𝑛 = 𝑘 → (∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚 ↔ ∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚))
45	44	elrab 3595	. . . . 5 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ↔ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚))
46	45	simprbi 500	. . . 4 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} → ∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚)
47	46	ad2antll 729	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})) → ∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚)
48		breq1 5046	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑥 ≤ 𝑚 ↔ 1 ≤ 𝑚))
49	48	ralbidv 3111	. . . . . 6 ⊢ (𝑥 = 1 → (∀𝑚 ∈ 𝑊 𝑥 ≤ 𝑚 ↔ ∀𝑚 ∈ 𝑊 1 ≤ 𝑚))
50		breq2 5047	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑦 < 𝑥 ↔ 𝑦 < 1))
51		2fveq3 6711	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝐺‘(𝐾‘𝑥)) = (𝐺‘(𝐾‘1)))
52	51	fveq2d 6710	. . . . . . . . 9 ⊢ (𝑥 = 1 → (𝐹‘(𝐺‘(𝐾‘𝑥))) = (𝐹‘(𝐺‘(𝐾‘1))))
53	52	fveq2d 6710	. . . . . . . 8 ⊢ (𝑥 = 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘1)))))
54	53	breq2d 5055	. . . . . . 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 348	. . . . . 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 348	. . . . 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 344	. . . 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 5046	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝑥 ≤ 𝑚 ↔ 𝑘 ≤ 𝑚))
59	58	ralbidv 3111	. . . . . 6 ⊢ (𝑥 = 𝑘 → (∀𝑚 ∈ 𝑊 𝑥 ≤ 𝑚 ↔ ∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚))
60		breq2 5047	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝑦 < 𝑥 ↔ 𝑦 < 𝑘))
61		2fveq3 6711	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → (𝐺‘(𝐾‘𝑥)) = (𝐺‘(𝐾‘𝑘)))
62	61	fveq2d 6710	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝐹‘(𝐺‘(𝐾‘𝑥))) = (𝐹‘(𝐺‘(𝐾‘𝑘))))
63	62	fveq2d 6710	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))))
64	63	breq2d 5055	. . . . . . 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 348	. . . . . 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 348	. . . . 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 344	. . . 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 5046	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝑥 ≤ 𝑚 ↔ (𝑘 + 1) ≤ 𝑚))
69	68	ralbidv 3111	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (∀𝑚 ∈ 𝑊 𝑥 ≤ 𝑚 ↔ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚))
70		breq2 5047	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝑦 < 𝑥 ↔ 𝑦 < (𝑘 + 1)))
71		2fveq3 6711	. . . . . . . . . 10 ⊢ (𝑥 = (𝑘 + 1) → (𝐺‘(𝐾‘𝑥)) = (𝐺‘(𝐾‘(𝑘 + 1))))
72	71	fveq2d 6710	. . . . . . . . 9 ⊢ (𝑥 = (𝑘 + 1) → (𝐹‘(𝐺‘(𝐾‘𝑥))) = (𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))
73	72	fveq2d 6710	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑥)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))
74	73	breq2d 5055	. . . . . . 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 348	. . . . . 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 348	. . . . 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 344	. . . 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 11845	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → ¬ 𝑦 < 1)
79	78	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → ¬ 𝑦 < 1)
80	79	pm2.21d 121	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → (𝑦 < 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1))))))
81	80	a1d 25	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → (∀𝑚 ∈ 𝑊 1 ≤ 𝑚 → (𝑦 < 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1)))))))
82		nnre 11820	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
83	82	adantr 484	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑊) → 𝑘 ∈ ℝ)
84	83	lep1d 11746	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑊) → 𝑘 ≤ (𝑘 + 1))
85		peano2re 10988	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
86	83, 85	syl 17	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑊) → (𝑘 + 1) ∈ ℝ)
87		ovolicc2.16	. . . . . . . . . . . . . 14 ⊢ 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾‘𝑛)}
88	87	ssrab3 3985	. . . . . . . . . . . . 13 ⊢ 𝑊 ⊆ ℕ
89	88, 11	sstri 3900	. . . . . . . . . . . 12 ⊢ 𝑊 ⊆ ℝ
90	89	sseli 3887	. . . . . . . . . . 11 ⊢ (𝑚 ∈ 𝑊 → 𝑚 ∈ ℝ)
91	90	adantl 485	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑊) → 𝑚 ∈ ℝ)
92		letr 10909	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ 𝑚 ∈ ℝ) → ((𝑘 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≤ 𝑚) → 𝑘 ≤ 𝑚))
93	83, 86, 91, 92	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑊) → ((𝑘 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≤ 𝑚) → 𝑘 ≤ 𝑚))
94	84, 93	mpand 695	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑊) → ((𝑘 + 1) ≤ 𝑚 → 𝑘 ≤ 𝑚))
95	94	ralimdva 3093	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚 → ∀𝑚 ∈ 𝑊 𝑘 ≤ 𝑚))
96	95	imim1d 82	. . . . . 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 485	. . . . 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 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝑦 ∈ ℕ)
99		simprl 771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝑘 ∈ ℕ)
100		nnleltp1 12215	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑦 ≤ 𝑘 ↔ 𝑦 < (𝑘 + 1)))
101	98, 99, 100	syl2anc 587	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦 ≤ 𝑘 ↔ 𝑦 < (𝑘 + 1)))
102	98	nnred 11828	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝑦 ∈ ℝ)
103	99	nnred 11828	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝑘 ∈ ℝ)
104	102, 103	leloed 10958	. . . . . . . 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 767	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝜑)
107		ltp1 11655	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℝ → 𝑘 < (𝑘 + 1))
108		ltnle 10895	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
109	85, 108	mpdan 687	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℝ → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
110	107, 109	mpbid 235	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℝ → ¬ (𝑘 + 1) ≤ 𝑘)
111	103, 110	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → ¬ (𝑘 + 1) ≤ 𝑘)
112		breq2 5047	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑘 → ((𝑘 + 1) ≤ 𝑚 ↔ (𝑘 + 1) ≤ 𝑘))
113	112	rspccv 3527	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚 → (𝑘 ∈ 𝑊 → (𝑘 + 1) ≤ 𝑘))
114	113	ad2antll 729	. . . . . . . . . . . . . . . . . 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 24387	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ ¬ 𝑘 ∈ 𝑊)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ≤ 𝐵)
126	106, 99, 115, 125	syl12anc 837	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ≤ 𝐵)
127	126	iftrued 4437	. . . . . . . . . . . . . . 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 6711	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = (𝐾‘𝑘) → (𝐹‘(𝐺‘𝑡)) = (𝐹‘(𝐺‘(𝐾‘𝑘))))
129	128	fveq2d 6710	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = (𝐾‘𝑘) → (2nd ‘(𝐹‘(𝐺‘𝑡))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))))
130	129	breq1d 5053	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = (𝐾‘𝑘) → ((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ≤ 𝐵))
131	130, 129	ifbieq1d 4453	. . . . . . . . . . . . . . . . 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 6706	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (𝐾‘𝑘) → (𝐻‘𝑡) = (𝐻‘(𝐾‘𝑘)))
133	131, 132	eleq12d 2828	. . . . . . . . . . . . . . . 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 3098	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡))
135	134	ad2antrr 726	. . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝐾:ℕ⟶𝑇)
137	136, 99	ffvelrnd 6894	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐾‘𝑘) ∈ 𝑇)
138	133, 135, 137	rspcdva 3532	. . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . 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 16112	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾‘(𝑘 + 1)) = (𝐻‘(𝐾‘𝑘)))
141	140	ad2ant2r 747	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐾‘(𝑘 + 1)) = (𝐻‘(𝐾‘𝑘)))
142	139, 141	eleqtrrd 2837	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∈ (𝐾‘(𝑘 + 1)))
143	136, 29, 30	sylancl 589	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝐾:ℕ⟶𝑈)
144	99	peano2nnd 11830	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑘 + 1) ∈ ℕ)
145	143, 144	ffvelrnd 6894	. . . . . . . . . . . . . 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 24386	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 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 1146	. . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑦)))) ∈ ℝ)
151	17	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝐹:ℕ⟶(ℝ × ℝ))
152	19	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝐺:𝑈⟶ℕ)
153	143, 99	ffvelrnd 6894	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐾‘𝑘) ∈ 𝑈)
154	152, 153	ffvelrnd 6894	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐺‘(𝐾‘𝑘)) ∈ ℕ)
155	151, 154	ffvelrnd 6894	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐹‘(𝐺‘(𝐾‘𝑘))) ∈ (ℝ × ℝ))
156		xp2nd 7783	. . . . . . . . . . . . 13 ⊢ ((𝐹‘(𝐺‘(𝐾‘𝑘))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∈ ℝ)
157	155, 156	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑘)))) ∈ ℝ)
158	152, 145	ffvelrnd 6894	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐺‘(𝐾‘(𝑘 + 1))) ∈ ℕ)
159	151, 158	ffvelrnd 6894	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))) ∈ (ℝ × ℝ))
160		xp2nd 7783	. . . . . . . . . . . . 13 ⊢ ((𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) ∈ ℝ)
161	159, 160	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) ∈ ℝ)
162		lttr 10892	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 694	. . . . . . . . . 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 57	. . . . . . . . 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 86	. . . . . . . 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 5053	. . . . . . . . . 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 50	. . . . . . . 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 859	. . . . . . 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 86	. . . . 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 846	. . . 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 11831	. . 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 66	. 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 11343	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))))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ≠ wne 2935  ∀wral 3054  {crab 3058   ∩ cin 3856   ⊆ wss 3857  ∅c0 4227  ifcif 4429  𝒫 cpw 4503  {csn 4531  ∪ cuni 4809   class class class wbr 5043   × cxp 5538  ran crn 5541   ∘ ccom 5544  ⟶wf 6365  ‘cfv 6369  (class class class)co 7202  1^st c1st 7748  2^nd c2nd 7749  Fincfn 8615  ℝcr 10711  1c1 10713   + caddc 10715   < clt 10850   ≤ cle 10851   − cmin 11045  ℕcn 11813  (,)cioo 12918  [,]cicc 12921  seqcseq 13557  abscabs 14780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-1st 7750  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-er 8380  df-en 8616  df-dom 8617  df-sdom 8618  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-nn 11814  df-n0 12074  df-z 12160  df-uz 12422  df-ioo 12922  df-icc 12925  df-fz 13079  df-seq 13558

This theorem is referenced by:  ovolicc2lem4  24389