cvmliftlem7 - Mathbox for Mario Carneiro

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Theorem cvmliftlem7 36223

Description: Lemma for cvmlift 36231. Prove by induction that every 𝑄 function is well-defined (we can immediately follow this theorem with cvmliftlem6 36222 to show functionality and lifting of 𝑄). (Contributed by Mario Carneiro, 14-Feb-2015.)

**Hypotheses**
Ref	Expression
cvmliftlem.1	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (^◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾_t 𝑢)Homeo(𝐽 ↾_t 𝑘))))})
cvmliftlem.b	⊢ 𝐵 = ∪ 𝐶
cvmliftlem.x	⊢ 𝑋 = ∪ 𝐽
cvmliftlem.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftlem.g	⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
cvmliftlem.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmliftlem.e	⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))
cvmliftlem.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
cvmliftlem.t	⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
cvmliftlem.a	⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1^st ‘(𝑇‘𝑘)))
cvmliftlem.l	⊢ 𝐿 = (topGen‘ran (,))
cvmliftlem.q	⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))
cvmliftlem5.3	⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))

**Assertion**
Ref	Expression
cvmliftlem7	⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))

Distinct variable groups: 𝑣,𝑏,𝑧,𝐵 𝑗,𝑏,𝑘,𝑚,𝑠,𝑢,𝑥,𝐹,𝑣,𝑧 𝑧,𝐿 𝑀,𝑏,𝑗,𝑘,𝑚,𝑠,𝑢,𝑣,𝑥,𝑧 𝑃,𝑏,𝑘,𝑚,𝑢,𝑣,𝑥,𝑧 𝐶,𝑏,𝑗,𝑘,𝑠,𝑢,𝑣,𝑧 𝜑,𝑗,𝑠,𝑥,𝑧 𝑁,𝑏,𝑘,𝑚,𝑢,𝑣,𝑥,𝑧 𝑆,𝑏,𝑗,𝑘,𝑠,𝑢,𝑣,𝑥,𝑧 𝑗,𝑋 𝐺,𝑏,𝑗,𝑘,𝑚,𝑠,𝑢,𝑣,𝑥,𝑧 𝑇,𝑏,𝑗,𝑘,𝑚,𝑠,𝑢,𝑣,𝑥,𝑧 𝐽,𝑏,𝑗,𝑘,𝑠,𝑢,𝑣,𝑥,𝑧 𝑄,𝑏,𝑘,𝑚,𝑢,𝑣,𝑥,𝑧 𝑘,𝑊,𝑚,𝑥,𝑧 Allowed substitution hints: 𝜑(𝑣, 𝑢, 𝑘, 𝑚, 𝑏) 𝐵(𝑥, 𝑢, 𝑗, 𝑘, 𝑚, 𝑠) 𝐶(𝑥, 𝑚) 𝑃(𝑗, 𝑠) 𝑄(𝑗, 𝑠) 𝑆(𝑚) 𝐽(𝑚) 𝐿(𝑥, 𝑣, 𝑢, 𝑗, 𝑘, 𝑚, 𝑠, 𝑏) 𝑁(𝑗, 𝑠) 𝑊(𝑣, 𝑢, 𝑗, 𝑠, 𝑏) 𝑋(𝑥, 𝑧, 𝑣, 𝑢, 𝑘, 𝑚, 𝑠, 𝑏)

Proof of Theorem cvmliftlem7 Dummy variables 𝑦 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzssp1 13728	. . . 4 ⊢ (0...(𝑁 − 1)) ⊆ (0...((𝑁 − 1) + 1))
2		cvmliftlem.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3	2	nncnd 12378	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
4	3	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℂ)
5		ax-1cn 11285	. . . . . 6 ⊢ 1 ∈ ℂ
6		npcan 11593	. . . . . 6 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
7	4, 5, 6	sylancl 598	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑁 − 1) + 1) = 𝑁)
8	7	oveq2d 7459	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (0...((𝑁 − 1) + 1)) = (0...𝑁))
9	1, 8	sseqtrid 3955	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
10		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑀 ∈ (1...𝑁))
11		elfzelz 13683	. . . . 5 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℤ)
12	2	nnzd 12746	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
13		elfzm1b 13763	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1))))
14	11, 12, 13	syl2anr 609	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1))))
15	10, 14	mpbid 235	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ (0...(𝑁 − 1)))
16	9, 15	sseldd 3914	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ (0...𝑁))
17		elfznn0 13781	. . . 4 ⊢ ((𝑀 − 1) ∈ (0...𝑁) → (𝑀 − 1) ∈ ℕ₀)
18	17	adantl 487	. . 3 ⊢ ((𝜑 ∧ (𝑀 − 1) ∈ (0...𝑁)) → (𝑀 − 1) ∈ ℕ₀)
19		eleq1 2852	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑦 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
20		fveq2 6908	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝑄‘𝑦) = (𝑄‘0))
21		oveq1 7450	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝑦 / 𝑁) = (0 / 𝑁))
22	20, 21	fveq12d 6915	. . . . . . . 8 ⊢ (𝑦 = 0 → ((𝑄‘𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘0)‘(0 / 𝑁)))
23		fvoveq1 7466	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘(0 / 𝑁)))
24	23	sneqd 4596	. . . . . . . . 9 ⊢ (𝑦 = 0 → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘(0 / 𝑁))})
25	24	imaeq2d 6064	. . . . . . . 8 ⊢ (𝑦 = 0 → (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (^◡𝐹 “ {(𝐺‘(0 / 𝑁))}))
26	22, 25	eleq12d 2858	. . . . . . 7 ⊢ (𝑦 = 0 → (((𝑄‘𝑦)‘(𝑦 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) ↔ ((𝑄‘0)‘(0 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(0 / 𝑁))})))
27	19, 26	imbi12d 347	. . . . . 6 ⊢ (𝑦 = 0 → ((𝑦 ∈ (0...𝑁) → ((𝑄‘𝑦)‘(𝑦 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))})) ↔ (0 ∈ (0...𝑁) → ((𝑄‘0)‘(0 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(0 / 𝑁))}))))
28	27	imbi2d 343	. . . . 5 ⊢ (𝑦 = 0 → ((𝜑 → (𝑦 ∈ (0...𝑁) → ((𝑄‘𝑦)‘(𝑦 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}))) ↔ (𝜑 → (0 ∈ (0...𝑁) → ((𝑄‘0)‘(0 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(0 / 𝑁))})))))
29		eleq1 2852	. . . . . . 7 ⊢ (𝑦 = 𝑛 → (𝑦 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...𝑁)))
30		fveq2 6908	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → (𝑄‘𝑦) = (𝑄‘𝑛))
31		oveq1 7450	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → (𝑦 / 𝑁) = (𝑛 / 𝑁))
32	30, 31	fveq12d 6915	. . . . . . . 8 ⊢ (𝑦 = 𝑛 → ((𝑄‘𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘𝑛)‘(𝑛 / 𝑁)))
33		fvoveq1 7466	. . . . . . . . . 10 ⊢ (𝑦 = 𝑛 → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘(𝑛 / 𝑁)))
34	33	sneqd 4596	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘(𝑛 / 𝑁))})
35	34	imaeq2d 6064	. . . . . . . 8 ⊢ (𝑦 = 𝑛 → (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))
36	32, 35	eleq12d 2858	. . . . . . 7 ⊢ (𝑦 = 𝑛 → (((𝑄‘𝑦)‘(𝑦 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) ↔ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))})))
37	29, 36	imbi12d 347	. . . . . 6 ⊢ (𝑦 = 𝑛 → ((𝑦 ∈ (0...𝑁) → ((𝑄‘𝑦)‘(𝑦 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))})) ↔ (𝑛 ∈ (0...𝑁) → ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))))
38	37	imbi2d 343	. . . . 5 ⊢ (𝑦 = 𝑛 → ((𝜑 → (𝑦 ∈ (0...𝑁) → ((𝑄‘𝑦)‘(𝑦 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}))) ↔ (𝜑 → (𝑛 ∈ (0...𝑁) → ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))})))))
39		eleq1 2852	. . . . . . 7 ⊢ (𝑦 = (𝑛 + 1) → (𝑦 ∈ (0...𝑁) ↔ (𝑛 + 1) ∈ (0...𝑁)))
40		fveq2 6908	. . . . . . . . 9 ⊢ (𝑦 = (𝑛 + 1) → (𝑄‘𝑦) = (𝑄‘(𝑛 + 1)))
41		oveq1 7450	. . . . . . . . 9 ⊢ (𝑦 = (𝑛 + 1) → (𝑦 / 𝑁) = ((𝑛 + 1) / 𝑁))
42	40, 41	fveq12d 6915	. . . . . . . 8 ⊢ (𝑦 = (𝑛 + 1) → ((𝑄‘𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)))
43		fvoveq1 7466	. . . . . . . . . 10 ⊢ (𝑦 = (𝑛 + 1) → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘((𝑛 + 1) / 𝑁)))
44	43	sneqd 4596	. . . . . . . . 9 ⊢ (𝑦 = (𝑛 + 1) → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘((𝑛 + 1) / 𝑁))})
45	44	imaeq2d 6064	. . . . . . . 8 ⊢ (𝑦 = (𝑛 + 1) → (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (^◡𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))}))
46	42, 45	eleq12d 2858	. . . . . . 7 ⊢ (𝑦 = (𝑛 + 1) → (((𝑄‘𝑦)‘(𝑦 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) ↔ ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))})))
47	39, 46	imbi12d 347	. . . . . 6 ⊢ (𝑦 = (𝑛 + 1) → ((𝑦 ∈ (0...𝑁) → ((𝑄‘𝑦)‘(𝑦 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))})) ↔ ((𝑛 + 1) ∈ (0...𝑁) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))}))))
48	47	imbi2d 343	. . . . 5 ⊢ (𝑦 = (𝑛 + 1) → ((𝜑 → (𝑦 ∈ (0...𝑁) → ((𝑄‘𝑦)‘(𝑦 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}))) ↔ (𝜑 → ((𝑛 + 1) ∈ (0...𝑁) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))})))))
49		eleq1 2852	. . . . . . 7 ⊢ (𝑦 = (𝑀 − 1) → (𝑦 ∈ (0...𝑁) ↔ (𝑀 − 1) ∈ (0...𝑁)))
50		fveq2 6908	. . . . . . . . 9 ⊢ (𝑦 = (𝑀 − 1) → (𝑄‘𝑦) = (𝑄‘(𝑀 − 1)))
51		oveq1 7450	. . . . . . . . 9 ⊢ (𝑦 = (𝑀 − 1) → (𝑦 / 𝑁) = ((𝑀 − 1) / 𝑁))
52	50, 51	fveq12d 6915	. . . . . . . 8 ⊢ (𝑦 = (𝑀 − 1) → ((𝑄‘𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)))
53		fvoveq1 7466	. . . . . . . . . 10 ⊢ (𝑦 = (𝑀 − 1) → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘((𝑀 − 1) / 𝑁)))
54	53	sneqd 4596	. . . . . . . . 9 ⊢ (𝑦 = (𝑀 − 1) → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘((𝑀 − 1) / 𝑁))})
55	54	imaeq2d 6064	. . . . . . . 8 ⊢ (𝑦 = (𝑀 − 1) → (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (^◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
56	52, 55	eleq12d 2858	. . . . . . 7 ⊢ (𝑦 = (𝑀 − 1) → (((𝑄‘𝑦)‘(𝑦 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) ↔ ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))})))
57	49, 56	imbi12d 347	. . . . . 6 ⊢ (𝑦 = (𝑀 − 1) → ((𝑦 ∈ (0...𝑁) → ((𝑄‘𝑦)‘(𝑦 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))})) ↔ ((𝑀 − 1) ∈ (0...𝑁) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))))
58	57	imbi2d 343	. . . . 5 ⊢ (𝑦 = (𝑀 − 1) → ((𝜑 → (𝑦 ∈ (0...𝑁) → ((𝑄‘𝑦)‘(𝑦 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}))) ↔ (𝜑 → ((𝑀 − 1) ∈ (0...𝑁) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))})))))
59		cvmliftlem.1	. . . . . . . . . . 11 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (^◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾_t 𝑢)Homeo(𝐽 ↾_t 𝑘))))})
60		cvmliftlem.b	. . . . . . . . . . 11 ⊢ 𝐵 = ∪ 𝐶
61		cvmliftlem.x	. . . . . . . . . . 11 ⊢ 𝑋 = ∪ 𝐽
62		cvmliftlem.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
63		cvmliftlem.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
64		cvmliftlem.p	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
65		cvmliftlem.e	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))
66		cvmliftlem.t	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
67		cvmliftlem.a	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1^st ‘(𝑇‘𝑘)))
68		cvmliftlem.l	. . . . . . . . . . 11 ⊢ 𝐿 = (topGen‘ran (,))
69		cvmliftlem.q	. . . . . . . . . . 11 ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))
70	59, 60, 61, 62, 63, 64, 65, 2, 66, 67, 68, 69	cvmliftlem4 36220	. . . . . . . . . 10 ⊢ (𝑄‘0) = {⟨0, 𝑃⟩}
71	70	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘0) = {⟨0, 𝑃⟩})
72	2	nnne0d 12415	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ≠ 0)
73	3, 72	div0d 12119	. . . . . . . . 9 ⊢ (𝜑 → (0 / 𝑁) = 0)
74	71, 73	fveq12d 6915	. . . . . . . 8 ⊢ (𝜑 → ((𝑄‘0)‘(0 / 𝑁)) = ({⟨0, 𝑃⟩}‘0))
75		0nn0 12648	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
76		fvsng 7208	. . . . . . . . 9 ⊢ ((0 ∈ ℕ₀ ∧ 𝑃 ∈ 𝐵) → ({⟨0, 𝑃⟩}‘0) = 𝑃)
77	75, 64, 76	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → ({⟨0, 𝑃⟩}‘0) = 𝑃)
78	74, 77	eqtrd 2798	. . . . . . 7 ⊢ (𝜑 → ((𝑄‘0)‘(0 / 𝑁)) = 𝑃)
79	73	fveq2d 6912	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘(0 / 𝑁)) = (𝐺‘0))
80	65, 79	eqtr4d 2801	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘(0 / 𝑁)))
81		cvmcn 36194	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
82	62, 81	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Cn 𝐽))
83	60, 61	cnf 23594	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶𝑋)
84		ffn 6731	. . . . . . . . . 10 ⊢ (𝐹:𝐵⟶𝑋 → 𝐹 Fn 𝐵)
85	82, 83, 84	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝐵)
86		fniniseg 7083	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐵 → (𝑃 ∈ (^◡𝐹 “ {(𝐺‘(0 / 𝑁))}) ↔ (𝑃 ∈ 𝐵 ∧ (𝐹‘𝑃) = (𝐺‘(0 / 𝑁)))))
87	85, 86	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∈ (^◡𝐹 “ {(𝐺‘(0 / 𝑁))}) ↔ (𝑃 ∈ 𝐵 ∧ (𝐹‘𝑃) = (𝐺‘(0 / 𝑁)))))
88	64, 80, 87	mpbir2and 726	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ (^◡𝐹 “ {(𝐺‘(0 / 𝑁))}))
89	78, 88	eqeltrd 2864	. . . . . 6 ⊢ (𝜑 → ((𝑄‘0)‘(0 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(0 / 𝑁))}))
90	89	a1d 26	. . . . 5 ⊢ (𝜑 → (0 ∈ (0...𝑁) → ((𝑄‘0)‘(0 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(0 / 𝑁))})))
91		id 23	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℕ₀)
92		nn0uz 13030	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
93	91, 92	eleqtrdi 2874	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ (ℤ_≥‘0))
94	93	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ (ℤ_≥‘0))
95		peano2fzr 13697	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ_≥‘0) ∧ (𝑛 + 1) ∈ (0...𝑁)) → 𝑛 ∈ (0...𝑁))
96	95	ex 418	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ_≥‘0) → ((𝑛 + 1) ∈ (0...𝑁) → 𝑛 ∈ (0...𝑁)))
97	94, 96	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝑛 + 1) ∈ (0...𝑁) → 𝑛 ∈ (0...𝑁)))
98	97	imim1d 83	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝑛 ∈ (0...𝑁) → ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))})) → ((𝑛 + 1) ∈ (0...𝑁) → ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))))
99		eqid 2763	. . . . . . . . . . 11 ⊢ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)) = ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))
100		simprlr 792	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ (0...𝑁))
101		elfzle2 13688	. . . . . . . . . . . . 13 ⊢ ((𝑛 + 1) ∈ (0...𝑁) → (𝑛 + 1) ≤ 𝑁)
102	100, 101	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ≤ 𝑁)
103		simprll 791	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℕ₀)
104		nn0p1nn 12672	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
105	103, 104	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ ℕ)
106		nnuz 13031	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘1)
107	105, 106	eleqtrdi 2874	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ (ℤ_≥‘1))
108	12	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℤ)
109		elfz5 13675	. . . . . . . . . . . . 13 ⊢ (((𝑛 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ ℤ) → ((𝑛 + 1) ∈ (1...𝑁) ↔ (𝑛 + 1) ≤ 𝑁))
110	107, 108, 109	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) ∈ (1...𝑁) ↔ (𝑛 + 1) ≤ 𝑁))
111	102, 110	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ (1...𝑁))
112		simprr 785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))
113	103	nn0cnd 12696	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℂ)
114		pncan 11590	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
115	113, 5, 114	sylancl 598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) − 1) = 𝑛)
116	115	fveq2d 6912	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑄‘((𝑛 + 1) − 1)) = (𝑄‘𝑛))
117	115	oveq1d 7458	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (((𝑛 + 1) − 1) / 𝑁) = (𝑛 / 𝑁))
118	116, 117	fveq12d 6915	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄‘𝑛)‘(𝑛 / 𝑁)))
119	117	fveq2d 6912	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝐺‘(((𝑛 + 1) − 1) / 𝑁)) = (𝐺‘(𝑛 / 𝑁)))
120	119	sneqd 4596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → {(𝐺‘(((𝑛 + 1) − 1) / 𝑁))} = {(𝐺‘(𝑛 / 𝑁))})
121	120	imaeq2d 6064	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (^◡𝐹 “ {(𝐺‘(((𝑛 + 1) − 1) / 𝑁))}) = (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))
122	112, 118, 121	3eltr4d 2879	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(((𝑛 + 1) − 1) / 𝑁))}))
123	59, 60, 61, 62, 63, 64, 65, 2, 66, 67, 68, 69, 99, 111, 122	cvmliftlem6 36222	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄‘(𝑛 + 1)):((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
124	123	simpld 500	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑄‘(𝑛 + 1)):((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵)
125	103	nn0red 12695	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℝ)
126	2	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℕ)
127	125, 126	nndivred 12419	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ∈ ℝ)
128	127	rexrd 11386	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ∈ ℝ^*)
129		peano2re 11510	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
130	125, 129	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ ℝ)
131	130, 126	nndivred 12419	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ℝ)
132	131	rexrd 11386	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ℝ^*)
133	125	ltp1d 12274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 < (𝑛 + 1))
134	126	nnred 12377	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℝ)
135	126	nngt0d 12414	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 0 < 𝑁)
136		ltdiv1 12208	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (𝑛 < (𝑛 + 1) ↔ (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁)))
137	125, 130, 134, 135, 136	syl112anc 1402	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 < (𝑛 + 1) ↔ (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁)))
138	133, 137	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁))
139	127, 131, 138	ltled 11485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁))
140		ubicc2 13623	. . . . . . . . . . 11 ⊢ (((𝑛 / 𝑁) ∈ ℝ^* ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ^* ∧ (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁)) → ((𝑛 + 1) / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
141	128, 132, 139, 140	syl3anc 1399	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
142	117	oveq1d 7458	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)) = ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
143	141, 142	eleqtrrd 2867	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)))
144	124, 143	ffvelcdmd 7109	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ 𝐵)
145	123	simprd 501	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))))
146	142	reseq2d 5981	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
147	145, 146	eqtrd 2798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
148	147	fveq1d 6910	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝐹 ∘ (𝑄‘(𝑛 + 1)))‘((𝑛 + 1) / 𝑁)) = ((𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))‘((𝑛 + 1) / 𝑁)))
149	142	feq2d 6715	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄‘(𝑛 + 1)):((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ↔ (𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵))
150	124, 149	mpbid 235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵)
151		fvco3 7008	. . . . . . . . . 10 ⊢ (((𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ∧ ((𝑛 + 1) / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) → ((𝐹 ∘ (𝑄‘(𝑛 + 1)))‘((𝑛 + 1) / 𝑁)) = (𝐹‘((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁))))
152	150, 141, 151	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝐹 ∘ (𝑄‘(𝑛 + 1)))‘((𝑛 + 1) / 𝑁)) = (𝐹‘((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁))))
153		fvres 6927	. . . . . . . . . 10 ⊢ (((𝑛 + 1) / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) → ((𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))‘((𝑛 + 1) / 𝑁)) = (𝐺‘((𝑛 + 1) / 𝑁)))
154	141, 153	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))‘((𝑛 + 1) / 𝑁)) = (𝐺‘((𝑛 + 1) / 𝑁)))
155	148, 152, 154	3eqtr3d 2806	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝐹‘((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁))) = (𝐺‘((𝑛 + 1) / 𝑁)))
156	85	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝐹 Fn 𝐵)
157		fniniseg 7083	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐵 → (((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))}) ↔ (((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ 𝐵 ∧ (𝐹‘((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁))) = (𝐺‘((𝑛 + 1) / 𝑁)))))
158	156, 157	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))}) ↔ (((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ 𝐵 ∧ (𝐹‘((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁))) = (𝐺‘((𝑛 + 1) / 𝑁)))))
159	144, 155, 158	mpbir2and 726	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))}))
160	159	expr 462	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁))) → (((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))})))
161	98, 160	animpimp2impd 860	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((𝜑 → (𝑛 ∈ (0...𝑁) → ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝜑 → ((𝑛 + 1) ∈ (0...𝑁) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))})))))
162	28, 38, 48, 58, 90, 161	nn0ind 12821	. . . 4 ⊢ ((𝑀 − 1) ∈ ℕ₀ → (𝜑 → ((𝑀 − 1) ∈ (0...𝑁) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))))
163	162	impd 416	. . 3 ⊢ ((𝑀 − 1) ∈ ℕ₀ → ((𝜑 ∧ (𝑀 − 1) ∈ (0...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))})))
164	18, 163	mpcom 39	. 2 ⊢ ((𝜑 ∧ (𝑀 − 1) ∈ (0...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
165	16, 164	syldan 603	1 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∀wral 3080 {crab 3416 Vcvv 3454 ⊆ wss 3900 ∖ cdif 4048 ∩ cin 4092 ∪ cun 4152 ∅c0 4281 𝒫 cpw 4560 {csn 4588 ⟨cop 4838 ∪ cuni 4870 ∪ ciun 4954 class class class wbr 5106 ↦ cmpt 5189 I cid 5549 × cxp 5653 ^◡ccnv 5654 ran crn 5656 ↾ cres 5657 “ cima 5658 ∘ ccom 5659 Fn wfn 6553 ⟶wf 6554 ‘cfv 6558 ℩crio 7399 (class class class)co 7443 ∈ cmpo 7445 1^st c1st 8023 2^nd c2nd 8024 ℂcc 11225 ℝcr 11226 0cc0 11227 1c1 11228 + caddc 11230 ℝ^*cxr 11369 < clt 11370 ≤ cle 11371 − cmin 11568 / cdiv 12000 ℕcn 12362 ℕ₀cn0 12633 ℤcz 12720 ℤ_≥cuz 12992 (,)cioo 13503 [,]cicc 13506 ...cfz 13666 seqcseq 14171 ↾_t crest 17620 topGenctg 17637 Cn ccn 23572 Homeochmeo 24102 IIcii 25226 CovMap ccvm 36187

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-er 8745 df-map 8877 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fi 9430 df-sup 9461 df-inf 9462 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-n0 12634 df-z 12721 df-uz 12993 df-q 13103 df-rp 13148 df-xneg 13268 df-xadd 13269 df-xmul 13270 df-icc 13510 df-fz 13667 df-seq 14172 df-exp 14232 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432 df-rest 17622 df-topgen 17643 df-psmet 21700 df-xmet 21701 df-met 21702 df-bl 21703 df-mopn 21704 df-top 23242 df-topon 23259 df-bases 23294 df-cn 23575 df-hmeo 24104 df-ii 25228 df-cvm 36188

This theorem is used by: cvmliftlem8 36224 cvmliftlem9 36225 cvmliftlem10 36226 cvmliftlem13 36228

W3C validator