cvmliftlem7 - Mathbox for Mario Carneiro

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Theorem cvmliftlem7 33972

Description: Lemma for cvmlift 33980. Prove by induction that every 𝑄 function is well-defined (we can immediately follow this theorem with cvmliftlem6 33971 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 13494	. . . 4 ⊢ (0...(𝑁 − 1)) ⊆ (0...((𝑁 − 1) + 1))
2		cvmliftlem.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3	2	nncnd 12178	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
4	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℂ)
5		ax-1cn 11118	. . . . . 6 ⊢ 1 ∈ ℂ
6		npcan 11419	. . . . . 6 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
7	4, 5, 6	sylancl 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑁 − 1) + 1) = 𝑁)
8	7	oveq2d 7378	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (0...((𝑁 − 1) + 1)) = (0...𝑁))
9	1, 8	sseqtrid 3999	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
10		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑀 ∈ (1...𝑁))
11		elfzelz 13451	. . . . 5 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℤ)
12	2	nnzd 12535	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
13		elfzm1b 13529	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1))))
14	11, 12, 13	syl2anr 597	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1))))
15	10, 14	mpbid 231	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ (0...(𝑁 − 1)))
16	9, 15	sseldd 3948	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ (0...𝑁))
17		elfznn0 13544	. . . 4 ⊢ ((𝑀 − 1) ∈ (0...𝑁) → (𝑀 − 1) ∈ ℕ₀)
18	17	adantl 482	. . 3 ⊢ ((𝜑 ∧ (𝑀 − 1) ∈ (0...𝑁)) → (𝑀 − 1) ∈ ℕ₀)
19		eleq1 2820	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑦 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
20		fveq2 6847	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝑄‘𝑦) = (𝑄‘0))
21		oveq1 7369	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝑦 / 𝑁) = (0 / 𝑁))
22	20, 21	fveq12d 6854	. . . . . . . 8 ⊢ (𝑦 = 0 → ((𝑄‘𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘0)‘(0 / 𝑁)))
23		fvoveq1 7385	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘(0 / 𝑁)))
24	23	sneqd 4603	. . . . . . . . 9 ⊢ (𝑦 = 0 → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘(0 / 𝑁))})
25	24	imaeq2d 6018	. . . . . . . 8 ⊢ (𝑦 = 0 → (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (^◡𝐹 “ {(𝐺‘(0 / 𝑁))}))
26	22, 25	eleq12d 2826	. . . . . . 7 ⊢ (𝑦 = 0 → (((𝑄‘𝑦)‘(𝑦 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) ↔ ((𝑄‘0)‘(0 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(0 / 𝑁))})))
27	19, 26	imbi12d 344	. . . . . 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 340	. . . . 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 2820	. . . . . . 7 ⊢ (𝑦 = 𝑛 → (𝑦 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...𝑁)))
30		fveq2 6847	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → (𝑄‘𝑦) = (𝑄‘𝑛))
31		oveq1 7369	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → (𝑦 / 𝑁) = (𝑛 / 𝑁))
32	30, 31	fveq12d 6854	. . . . . . . 8 ⊢ (𝑦 = 𝑛 → ((𝑄‘𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘𝑛)‘(𝑛 / 𝑁)))
33		fvoveq1 7385	. . . . . . . . . 10 ⊢ (𝑦 = 𝑛 → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘(𝑛 / 𝑁)))
34	33	sneqd 4603	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘(𝑛 / 𝑁))})
35	34	imaeq2d 6018	. . . . . . . 8 ⊢ (𝑦 = 𝑛 → (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))
36	32, 35	eleq12d 2826	. . . . . . 7 ⊢ (𝑦 = 𝑛 → (((𝑄‘𝑦)‘(𝑦 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) ↔ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))})))
37	29, 36	imbi12d 344	. . . . . 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 340	. . . . 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 2820	. . . . . . 7 ⊢ (𝑦 = (𝑛 + 1) → (𝑦 ∈ (0...𝑁) ↔ (𝑛 + 1) ∈ (0...𝑁)))
40		fveq2 6847	. . . . . . . . 9 ⊢ (𝑦 = (𝑛 + 1) → (𝑄‘𝑦) = (𝑄‘(𝑛 + 1)))
41		oveq1 7369	. . . . . . . . 9 ⊢ (𝑦 = (𝑛 + 1) → (𝑦 / 𝑁) = ((𝑛 + 1) / 𝑁))
42	40, 41	fveq12d 6854	. . . . . . . 8 ⊢ (𝑦 = (𝑛 + 1) → ((𝑄‘𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)))
43		fvoveq1 7385	. . . . . . . . . 10 ⊢ (𝑦 = (𝑛 + 1) → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘((𝑛 + 1) / 𝑁)))
44	43	sneqd 4603	. . . . . . . . 9 ⊢ (𝑦 = (𝑛 + 1) → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘((𝑛 + 1) / 𝑁))})
45	44	imaeq2d 6018	. . . . . . . 8 ⊢ (𝑦 = (𝑛 + 1) → (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (^◡𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))}))
46	42, 45	eleq12d 2826	. . . . . . 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 344	. . . . . 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 340	. . . . 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 2820	. . . . . . 7 ⊢ (𝑦 = (𝑀 − 1) → (𝑦 ∈ (0...𝑁) ↔ (𝑀 − 1) ∈ (0...𝑁)))
50		fveq2 6847	. . . . . . . . 9 ⊢ (𝑦 = (𝑀 − 1) → (𝑄‘𝑦) = (𝑄‘(𝑀 − 1)))
51		oveq1 7369	. . . . . . . . 9 ⊢ (𝑦 = (𝑀 − 1) → (𝑦 / 𝑁) = ((𝑀 − 1) / 𝑁))
52	50, 51	fveq12d 6854	. . . . . . . 8 ⊢ (𝑦 = (𝑀 − 1) → ((𝑄‘𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)))
53		fvoveq1 7385	. . . . . . . . . 10 ⊢ (𝑦 = (𝑀 − 1) → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘((𝑀 − 1) / 𝑁)))
54	53	sneqd 4603	. . . . . . . . 9 ⊢ (𝑦 = (𝑀 − 1) → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘((𝑀 − 1) / 𝑁))})
55	54	imaeq2d 6018	. . . . . . . 8 ⊢ (𝑦 = (𝑀 − 1) → (^◡𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (^◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
56	52, 55	eleq12d 2826	. . . . . . 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 344	. . . . . 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 340	. . . . 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 33969	. . . . . . . . . 10 ⊢ (𝑄‘0) = {⟨0, 𝑃⟩}
71	70	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘0) = {⟨0, 𝑃⟩})
72	2	nnne0d 12212	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ≠ 0)
73	3, 72	div0d 11939	. . . . . . . . 9 ⊢ (𝜑 → (0 / 𝑁) = 0)
74	71, 73	fveq12d 6854	. . . . . . . 8 ⊢ (𝜑 → ((𝑄‘0)‘(0 / 𝑁)) = ({⟨0, 𝑃⟩}‘0))
75		0nn0 12437	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
76		fvsng 7131	. . . . . . . . 9 ⊢ ((0 ∈ ℕ₀ ∧ 𝑃 ∈ 𝐵) → ({⟨0, 𝑃⟩}‘0) = 𝑃)
77	75, 64, 76	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → ({⟨0, 𝑃⟩}‘0) = 𝑃)
78	74, 77	eqtrd 2771	. . . . . . 7 ⊢ (𝜑 → ((𝑄‘0)‘(0 / 𝑁)) = 𝑃)
79	73	fveq2d 6851	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘(0 / 𝑁)) = (𝐺‘0))
80	65, 79	eqtr4d 2774	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘(0 / 𝑁)))
81		cvmcn 33943	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
82	62, 81	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Cn 𝐽))
83	60, 61	cnf 22634	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶𝑋)
84		ffn 6673	. . . . . . . . . 10 ⊢ (𝐹:𝐵⟶𝑋 → 𝐹 Fn 𝐵)
85	82, 83, 84	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝐵)
86		fniniseg 7015	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐵 → (𝑃 ∈ (^◡𝐹 “ {(𝐺‘(0 / 𝑁))}) ↔ (𝑃 ∈ 𝐵 ∧ (𝐹‘𝑃) = (𝐺‘(0 / 𝑁)))))
87	85, 86	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∈ (^◡𝐹 “ {(𝐺‘(0 / 𝑁))}) ↔ (𝑃 ∈ 𝐵 ∧ (𝐹‘𝑃) = (𝐺‘(0 / 𝑁)))))
88	64, 80, 87	mpbir2and 711	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ (^◡𝐹 “ {(𝐺‘(0 / 𝑁))}))
89	78, 88	eqeltrd 2832	. . . . . 6 ⊢ (𝜑 → ((𝑄‘0)‘(0 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(0 / 𝑁))}))
90	89	a1d 25	. . . . 5 ⊢ (𝜑 → (0 ∈ (0...𝑁) → ((𝑄‘0)‘(0 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(0 / 𝑁))})))
91		id 22	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℕ₀)
92		nn0uz 12814	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
93	91, 92	eleqtrdi 2842	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ (ℤ_≥‘0))
94	93	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ (ℤ_≥‘0))
95		peano2fzr 13464	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ_≥‘0) ∧ (𝑛 + 1) ∈ (0...𝑁)) → 𝑛 ∈ (0...𝑁))
96	95	ex 413	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ_≥‘0) → ((𝑛 + 1) ∈ (0...𝑁) → 𝑛 ∈ (0...𝑁)))
97	94, 96	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝑛 + 1) ∈ (0...𝑁) → 𝑛 ∈ (0...𝑁)))
98	97	imim1d 82	. . . . . 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 2731	. . . . . . . . . . 11 ⊢ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)) = ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))
100		simprlr 778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ (0...𝑁))
101		elfzle2 13455	. . . . . . . . . . . . 13 ⊢ ((𝑛 + 1) ∈ (0...𝑁) → (𝑛 + 1) ≤ 𝑁)
102	100, 101	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ≤ 𝑁)
103		simprll 777	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℕ₀)
104		nn0p1nn 12461	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
105	103, 104	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ ℕ)
106		nnuz 12815	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘1)
107	105, 106	eleqtrdi 2842	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ (ℤ_≥‘1))
108	12	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℤ)
109		elfz5 13443	. . . . . . . . . . . . 13 ⊢ (((𝑛 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ ℤ) → ((𝑛 + 1) ∈ (1...𝑁) ↔ (𝑛 + 1) ≤ 𝑁))
110	107, 108, 109	syl2anc 584	. . . . . . . . . . . 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 256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ (1...𝑁))
112		simprr 771	. . . . . . . . . . . 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 12484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℂ)
114		pncan 11416	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
115	113, 5, 114	sylancl 586	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) − 1) = 𝑛)
116	115	fveq2d 6851	. . . . . . . . . . . . 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 7377	. . . . . . . . . . . . 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 6854	. . . . . . . . . . . 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 6851	. . . . . . . . . . . . . 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 4603	. . . . . . . . . . . . 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 6018	. . . . . . . . . . . 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 2847	. . . . . . . . . . 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 33971	. . . . . . . . . 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 495	. . . . . . . . 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 12483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℝ)
126	2	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℕ)
127	125, 126	nndivred 12216	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ∈ ℝ)
128	127	rexrd 11214	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ∈ ℝ^*)
129		peano2re 11337	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
130	125, 129	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ ℝ)
131	130, 126	nndivred 12216	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ℝ)
132	131	rexrd 11214	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ℝ^*)
133	125	ltp1d 12094	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 < (𝑛 + 1))
134	126	nnred 12177	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℝ)
135	126	nngt0d 12211	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 0 < 𝑁)
136		ltdiv1 12028	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 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 11312	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁))
140		ubicc2 13392	. . . . . . . . . . 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 1371	. . . . . . . . . 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 7377	. . . . . . . . . 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 2835	. . . . . . . . 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 7041	. . . . . . . 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 496	. . . . . . . . . . 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 5942	. . . . . . . . . . 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 2771	. . . . . . . . . 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 6849	. . . . . . . . 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 6659	. . . . . . . . . . 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 231	. . . . . . . . . 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 6945	. . . . . . . . . 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 584	. . . . . . . . 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 6866	. . . . . . . . . 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 17	. . . . . . . . 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 2779	. . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄‘𝑛)‘(𝑛 / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝐹 Fn 𝐵)
157		fniniseg 7015	. . . . . . . . 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 17	. . . . . . . 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 711	. . . . . . 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 457	. . . . . 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 844	. . . . 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 12607	. . . 4 ⊢ ((𝑀 − 1) ∈ ℕ₀ → (𝜑 → ((𝑀 − 1) ∈ (0...𝑁) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))))
163	162	impd 411	. . 3 ⊢ ((𝑀 − 1) ∈ ℕ₀ → ((𝜑 ∧ (𝑀 − 1) ∈ (0...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))})))
164	18, 163	mpcom 38	. 2 ⊢ ((𝜑 ∧ (𝑀 − 1) ∈ (0...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
165	16, 164	syldan 591	1 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3060 {crab 3405 Vcvv 3446 ∖ cdif 3910 ∪ cun 3911 ∩ cin 3912 ⊆ wss 3913 ∅c0 4287 𝒫 cpw 4565 {csn 4591 ⟨cop 4597 ∪ cuni 4870 ∪ ciun 4959 class class class wbr 5110 ↦ cmpt 5193 I cid 5535 × cxp 5636 ^◡ccnv 5637 ran crn 5639 ↾ cres 5640 “ cima 5641 ∘ ccom 5642 Fn wfn 6496 ⟶wf 6497 ‘cfv 6501 ℩crio 7317 (class class class)co 7362 ∈ cmpo 7364 1^st c1st 7924 2^nd c2nd 7925 ℂcc 11058 ℝcr 11059 0cc0 11060 1c1 11061 + caddc 11063 ℝ^*cxr 11197 < clt 11198 ≤ cle 11199 − cmin 11394 / cdiv 11821 ℕcn 12162 ℕ₀cn0 12422 ℤcz 12508 ℤ_≥cuz 12772 (,)cioo 13274 [,]cicc 13277 ...cfz 13434 seqcseq 13916 ↾_t crest 17316 topGenctg 17333 Cn ccn 22612 Homeochmeo 23141 IIcii 24275 CovMap ccvm 33936

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137 ax-pre-sup 11138

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fi 9356 df-sup 9387 df-inf 9388 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-div 11822 df-nn 12163 df-2 12225 df-3 12226 df-n0 12423 df-z 12509 df-uz 12773 df-q 12883 df-rp 12925 df-xneg 13042 df-xadd 13043 df-xmul 13044 df-icc 13281 df-fz 13435 df-seq 13917 df-exp 13978 df-cj 14996 df-re 14997 df-im 14998 df-sqrt 15132 df-abs 15133 df-rest 17318 df-topgen 17339 df-psmet 20825 df-xmet 20826 df-met 20827 df-bl 20828 df-mopn 20829 df-top 22280 df-topon 22297 df-bases 22333 df-cn 22615 df-hmeo 23143 df-ii 24277 df-cvm 33937

This theorem is referenced by: cvmliftlem8 33973 cvmliftlem9 33974 cvmliftlem10 33975 cvmliftlem13 33977

W3C validator