Theorem heiborlem4 38493

Description: Lemma for heibor 38500. Using the function 𝑇 constructed in heiborlem3 38492, construct an infinite path in 𝐺. (Contributed by Jeff Madsen, 23-Jan-2014.)

Hypotheses

Ref	Expression
heibor.1	⊢ 𝐽 = (MetOpen‘𝐷)
heibor.3	⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
heibor.4	⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
heibor.5	⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
heibor.6	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
heibor.7	⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
heibor.8	⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))
heibor.9	⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
heibor.10	⊢ (𝜑 → 𝐶𝐺0)
heibor.11	⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))

Assertion

Ref	Expression
heiborlem4	⊢ ((𝜑 ∧ 𝐴 ∈ ℕ0) → (𝑆‘𝐴)𝐺𝐴)

Distinct variable groups:   𝑥,𝑛,𝑦,𝐴   𝑢,𝑛,𝐹,𝑥,𝑦   𝑥,𝐺   𝜑,𝑥   𝑚,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧,𝐷   𝑇,𝑚,𝑛,𝑥,𝑦,𝑧   𝐵,𝑛,𝑢,𝑣,𝑦   𝑚,𝐽,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑈,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑆,𝑚,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑚,𝑋,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝐶,𝑚,𝑛,𝑢,𝑣,𝑦   𝑛,𝐾,𝑥,𝑦,𝑧   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑦, 𝑧, 𝑣, 𝑢, 𝑚, 𝑛)   𝐴(𝑧, 𝑣, 𝑢, 𝑚)   𝐵(𝑧, 𝑚)   𝐶(𝑥, 𝑧)   𝑇(𝑣, 𝑢)   𝑈(𝑚)   𝐹(𝑧, 𝑣, 𝑚)   𝐺(𝑦, 𝑧, 𝑣, 𝑢, 𝑚, 𝑛)   𝐾(𝑣, 𝑢, 𝑚)

Proof of Theorem heiborlem4

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . . . 5 ⊢ (𝑥 = 0 → (𝑆‘𝑥) = (𝑆‘0))
2		id 23	. . . . 5 ⊢ (𝑥 = 0 → 𝑥 = 0)
3	1, 2	breq12d 5122	. . . 4 ⊢ (𝑥 = 0 → ((𝑆‘𝑥)𝐺𝑥 ↔ (𝑆‘0)𝐺0))
4	3	imbi2d 343	. . 3 ⊢ (𝑥 = 0 → ((𝜑 → (𝑆‘𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆‘0)𝐺0)))
5		fveq2 6881	. . . . 5 ⊢ (𝑥 = 𝑘 → (𝑆‘𝑥) = (𝑆‘𝑘))
6		id 23	. . . . 5 ⊢ (𝑥 = 𝑘 → 𝑥 = 𝑘)
7	5, 6	breq12d 5122	. . . 4 ⊢ (𝑥 = 𝑘 → ((𝑆‘𝑥)𝐺𝑥 ↔ (𝑆‘𝑘)𝐺𝑘))
8	7	imbi2d 343	. . 3 ⊢ (𝑥 = 𝑘 → ((𝜑 → (𝑆‘𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆‘𝑘)𝐺𝑘)))
9		fveq2 6881	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → (𝑆‘𝑥) = (𝑆‘(𝑘 + 1)))
10		id 23	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → 𝑥 = (𝑘 + 1))
11	9, 10	breq12d 5122	. . . 4 ⊢ (𝑥 = (𝑘 + 1) → ((𝑆‘𝑥)𝐺𝑥 ↔ (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1)))
12	11	imbi2d 343	. . 3 ⊢ (𝑥 = (𝑘 + 1) → ((𝜑 → (𝑆‘𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))))
13		fveq2 6881	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴))
14		id 23	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
15	13, 14	breq12d 5122	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥)𝐺𝑥 ↔ (𝑆‘𝐴)𝐺𝐴))
16	15	imbi2d 343	. . 3 ⊢ (𝑥 = 𝐴 → ((𝜑 → (𝑆‘𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆‘𝐴)𝐺𝐴)))
17		heibor.11	. . . . . . 7 ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))
18	17	fveq1i 6882	. . . . . 6 ⊢ (𝑆‘0) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘0)
19		0z 12606	. . . . . . 7 ⊢ 0 ∈ ℤ
20		seq1 14055	. . . . . . 7 ⊢ (0 ∈ ℤ → (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘0) = ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0))
21	19, 20	ax-mp 5	. . . . . 6 ⊢ (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘0) = ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0)
22	18, 21	eqtri 2786	. . . . 5 ⊢ (𝑆‘0) = ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0)
23		0nn0 12523	. . . . . 6 ⊢ 0 ∈ ℕ0
24		heibor.10	. . . . . . 7 ⊢ (𝜑 → 𝐶𝐺0)
25		heibor.4	. . . . . . . . 9 ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
26	25	relopabiv 5807	. . . . . . . 8 ⊢ Rel 𝐺
27	26	brrelex1i 5717	. . . . . . 7 ⊢ (𝐶𝐺0 → 𝐶 ∈ V)
28	24, 27	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ V)
29		iftrue 4493	. . . . . . 7 ⊢ (𝑚 = 0 → if(𝑚 = 0, 𝐶, (𝑚 − 1)) = 𝐶)
30		eqid 2763	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))
31	29, 30	fvmptg 6987	. . . . . 6 ⊢ ((0 ∈ ℕ0 ∧ 𝐶 ∈ V) → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0) = 𝐶)
32	23, 28, 31	sylancr 598	. . . . 5 ⊢ (𝜑 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0) = 𝐶)
33	22, 32	eqtrid 2810	. . . 4 ⊢ (𝜑 → (𝑆‘0) = 𝐶)
34	33, 24	eqbrtrd 5133	. . 3 ⊢ (𝜑 → (𝑆‘0)𝐺0)
35		df-br 5110	. . . . . 6 ⊢ ((𝑆‘𝑘)𝐺𝑘 ↔ ⟨(𝑆‘𝑘), 𝑘⟩ ∈ 𝐺)
36		heibor.9	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
37		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝑇‘𝑥) = (𝑇‘⟨(𝑆‘𝑘), 𝑘⟩))
38		df-ov 7413	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑘)𝑇𝑘) = (𝑇‘⟨(𝑆‘𝑘), 𝑘⟩)
39	37, 38	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝑇‘𝑥) = ((𝑆‘𝑘)𝑇𝑘))
40		fvex 6894	. . . . . . . . . . . 12 ⊢ (𝑆‘𝑘) ∈ V
41		vex 3459	. . . . . . . . . . . 12 ⊢ 𝑘 ∈ V
42	40, 41	op2ndd 7993	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (2nd ‘𝑥) = 𝑘)
43	42	oveq1d 7425	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((2nd ‘𝑥) + 1) = (𝑘 + 1))
44	39, 43	breq12d 5122	. . . . . . . . 9 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ↔ ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1)))
45		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝐵‘𝑥) = (𝐵‘⟨(𝑆‘𝑘), 𝑘⟩))
46		df-ov 7413	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑘)𝐵𝑘) = (𝐵‘⟨(𝑆‘𝑘), 𝑘⟩)
47	45, 46	eqtr4di 2816	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝐵‘𝑥) = ((𝑆‘𝑘)𝐵𝑘))
48	39, 43	oveq12d 7428	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1)) = (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)))
49	47, 48	ineq12d 4174	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) = (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))))
50	49	eleq1d 2848	. . . . . . . . 9 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾 ↔ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾))
51	44, 50	anbi12d 643	. . . . . . . 8 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾) ↔ (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
52	51	rspccv 3578	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾) → (⟨(𝑆‘𝑘), 𝑘⟩ ∈ 𝐺 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
53	36, 52	syl 18	. . . . . 6 ⊢ (𝜑 → (⟨(𝑆‘𝑘), 𝑘⟩ ∈ 𝐺 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
54	35, 53	biimtrid 245	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑘)𝐺𝑘 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
55		seqp1 14057	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ≥‘0) → (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1)) = ((seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
56		nn0uz 12904	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘0)
57	55, 56	eleq2s 2881	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1)) = ((seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
58	17	fveq1i 6882	. . . . . . . . . 10 ⊢ (𝑆‘(𝑘 + 1)) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1))
59	17	fveq1i 6882	. . . . . . . . . . 11 ⊢ (𝑆‘𝑘) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)
60	59	oveq1i 7420	. . . . . . . . . 10 ⊢ ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) = ((seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)))
61	57, 58, 60	3eqtr4g 2823	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
62		eqeq1 2767	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → (𝑚 = 0 ↔ (𝑘 + 1) = 0))
63		oveq1 7417	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → (𝑚 − 1) = ((𝑘 + 1) − 1))
64	62, 63	ifbieq2d 4514	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑘 + 1) → if(𝑚 = 0, 𝐶, (𝑚 − 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
65		peano2nn0 12548	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
66		nn0p1nn 12547	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
67		nnne0 12274	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ ℕ → (𝑘 + 1) ≠ 0)
68	67	neneqd 2963	. . . . . . . . . . . . . 14 ⊢ ((𝑘 + 1) ∈ ℕ → ¬ (𝑘 + 1) = 0)
69		iffalse 4496	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑘 + 1) = 0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) = ((𝑘 + 1) − 1))
70	66, 68, 69	3syl 19	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) = ((𝑘 + 1) − 1))
71		ovex 7443	. . . . . . . . . . . . 13 ⊢ ((𝑘 + 1) − 1) ∈ V
72	70, 71	eqeltrdi 2871	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) ∈ V)
73	30, 64, 65, 72	fvmptd3 7013	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
74		nn0cn 12518	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
75		ax-1cn 11162	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
76		pncan 11467	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
77	74, 75, 76	sylancl 597	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → ((𝑘 + 1) − 1) = 𝑘)
78	73, 70, 77	3eqtrd 2802	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = 𝑘)
79	78	oveq2d 7426	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) = ((𝑆‘𝑘)𝑇𝑘))
80	61, 79	eqtrd 2798	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇𝑘))
81	80	breq1d 5119	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((𝑆‘(𝑘 + 1))𝐺(𝑘 + 1) ↔ ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1)))
82	81	biimprd 251	. . . . . 6 ⊢ (𝑘 ∈ ℕ0 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1)))
83	82	adantrd 496	. . . . 5 ⊢ (𝑘 ∈ ℕ0 → ((((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾) → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1)))
84	54, 83	syl9r 79	. . . 4 ⊢ (𝑘 ∈ ℕ0 → (𝜑 → ((𝑆‘𝑘)𝐺𝑘 → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))))
85	84	a2d 30	. . 3 ⊢ (𝑘 ∈ ℕ0 → ((𝜑 → (𝑆‘𝑘)𝐺𝑘) → (𝜑 → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))))
86	4, 8, 12, 16, 34, 85	nn0ind 12695	. 2 ⊢ (𝐴 ∈ ℕ0 → (𝜑 → (𝑆‘𝐴)𝐺𝐴))
87	86	impcom 412	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ0) → (𝑆‘𝐴)𝐺𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  {cab 2741  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905  ifcif 4487  𝒫 cpw 4562  ⟨cop 4595  ∪ cuni 4872  ∪ ciun 4956   class class class wbr 5109  {copab 5173   ↦ cmpt 5192  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  2^nd c2nd 7981  Fincfn 8939  ℂcc 11102  0cc0 11104  1c1 11105   + caddc 11107   − cmin 11445   / cdiv 11875  ℕcn 12237  2c2 12299  ℕ₀cn0 12508  ℤcz 12595  ℤ_≥cuz 12866  seqcseq 14042  ↑cexp 14102  ballcbl 21518  MetOpencmopn 21521  CMetccmet 25422

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448  df-nn 12238  df-n0 12509  df-z 12596  df-uz 12867  df-seq 14043

This theorem is used by:  heiborlem5  38494  heiborlem6  38495  heiborlem8  38497