Theorem heiborlem4 35899

Description: Lemma for heibor 35906. Using the function 𝑇 constructed in heiborlem3 35898, 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 6756	. . . . 5 ⊢ (𝑥 = 0 → (𝑆‘𝑥) = (𝑆‘0))
2		id 22	. . . . 5 ⊢ (𝑥 = 0 → 𝑥 = 0)
3	1, 2	breq12d 5083	. . . 4 ⊢ (𝑥 = 0 → ((𝑆‘𝑥)𝐺𝑥 ↔ (𝑆‘0)𝐺0))
4	3	imbi2d 340	. . 3 ⊢ (𝑥 = 0 → ((𝜑 → (𝑆‘𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆‘0)𝐺0)))
5		fveq2 6756	. . . . 5 ⊢ (𝑥 = 𝑘 → (𝑆‘𝑥) = (𝑆‘𝑘))
6		id 22	. . . . 5 ⊢ (𝑥 = 𝑘 → 𝑥 = 𝑘)
7	5, 6	breq12d 5083	. . . 4 ⊢ (𝑥 = 𝑘 → ((𝑆‘𝑥)𝐺𝑥 ↔ (𝑆‘𝑘)𝐺𝑘))
8	7	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑘 → ((𝜑 → (𝑆‘𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆‘𝑘)𝐺𝑘)))
9		fveq2 6756	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → (𝑆‘𝑥) = (𝑆‘(𝑘 + 1)))
10		id 22	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → 𝑥 = (𝑘 + 1))
11	9, 10	breq12d 5083	. . . 4 ⊢ (𝑥 = (𝑘 + 1) → ((𝑆‘𝑥)𝐺𝑥 ↔ (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1)))
12	11	imbi2d 340	. . 3 ⊢ (𝑥 = (𝑘 + 1) → ((𝜑 → (𝑆‘𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))))
13		fveq2 6756	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴))
14		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
15	13, 14	breq12d 5083	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥)𝐺𝑥 ↔ (𝑆‘𝐴)𝐺𝐴))
16	15	imbi2d 340	. . 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 6757	. . . . . 6 ⊢ (𝑆‘0) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘0)
19		0z 12260	. . . . . . 7 ⊢ 0 ∈ ℤ
20		seq1 13662	. . . . . . 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 2766	. . . . 5 ⊢ (𝑆‘0) = ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0)
23		0nn0 12178	. . . . . 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 5719	. . . . . . . 8 ⊢ Rel 𝐺
27	26	brrelex1i 5634	. . . . . . 7 ⊢ (𝐶𝐺0 → 𝐶 ∈ V)
28	24, 27	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ V)
29		iftrue 4462	. . . . . . 7 ⊢ (𝑚 = 0 → if(𝑚 = 0, 𝐶, (𝑚 − 1)) = 𝐶)
30		eqid 2738	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))
31	29, 30	fvmptg 6855	. . . . . 6 ⊢ ((0 ∈ ℕ0 ∧ 𝐶 ∈ V) → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0) = 𝐶)
32	23, 28, 31	sylancr 586	. . . . 5 ⊢ (𝜑 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0) = 𝐶)
33	22, 32	syl5eq 2791	. . . 4 ⊢ (𝜑 → (𝑆‘0) = 𝐶)
34	33, 24	eqbrtrd 5092	. . 3 ⊢ (𝜑 → (𝑆‘0)𝐺0)
35		df-br 5071	. . . . . 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 6756	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝑇‘𝑥) = (𝑇‘⟨(𝑆‘𝑘), 𝑘⟩))
38		df-ov 7258	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑘)𝑇𝑘) = (𝑇‘⟨(𝑆‘𝑘), 𝑘⟩)
39	37, 38	eqtr4di 2797	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝑇‘𝑥) = ((𝑆‘𝑘)𝑇𝑘))
40		fvex 6769	. . . . . . . . . . . 12 ⊢ (𝑆‘𝑘) ∈ V
41		vex 3426	. . . . . . . . . . . 12 ⊢ 𝑘 ∈ V
42	40, 41	op2ndd 7815	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (2nd ‘𝑥) = 𝑘)
43	42	oveq1d 7270	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((2nd ‘𝑥) + 1) = (𝑘 + 1))
44	39, 43	breq12d 5083	. . . . . . . . 9 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ↔ ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1)))
45		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝐵‘𝑥) = (𝐵‘⟨(𝑆‘𝑘), 𝑘⟩))
46		df-ov 7258	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑘)𝐵𝑘) = (𝐵‘⟨(𝑆‘𝑘), 𝑘⟩)
47	45, 46	eqtr4di 2797	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝐵‘𝑥) = ((𝑆‘𝑘)𝐵𝑘))
48	39, 43	oveq12d 7273	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1)) = (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)))
49	47, 48	ineq12d 4144	. . . . . . . . . 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 2823	. . . . . . . . 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 630	. . . . . . . 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 3549	. . . . . . 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 17	. . . . . 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	syl5bi 241	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑘)𝐺𝑘 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
55		seqp1 13664	. . . . . . . . . . 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 12549	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘0)
57	55, 56	eleq2s 2857	. . . . . . . . . 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 6757	. . . . . . . . . 10 ⊢ (𝑆‘(𝑘 + 1)) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1))
59	17	fveq1i 6757	. . . . . . . . . . 11 ⊢ (𝑆‘𝑘) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)
60	59	oveq1i 7265	. . . . . . . . . 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 2804	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
62		eqeq1 2742	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → (𝑚 = 0 ↔ (𝑘 + 1) = 0))
63		oveq1 7262	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → (𝑚 − 1) = ((𝑘 + 1) − 1))
64	62, 63	ifbieq2d 4482	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑘 + 1) → if(𝑚 = 0, 𝐶, (𝑚 − 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
65		peano2nn0 12203	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
66		nn0p1nn 12202	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
67		nnne0 11937	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ ℕ → (𝑘 + 1) ≠ 0)
68	67	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑘 + 1) ∈ ℕ → ¬ (𝑘 + 1) = 0)
69		iffalse 4465	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑘 + 1) = 0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) = ((𝑘 + 1) − 1))
70	66, 68, 69	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) = ((𝑘 + 1) − 1))
71		ovex 7288	. . . . . . . . . . . . 13 ⊢ ((𝑘 + 1) − 1) ∈ V
72	70, 71	eqeltrdi 2847	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) ∈ V)
73	30, 64, 65, 72	fvmptd3 6880	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
74		nn0cn 12173	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
75		ax-1cn 10860	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
76		pncan 11157	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
77	74, 75, 76	sylancl 585	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → ((𝑘 + 1) − 1) = 𝑘)
78	73, 70, 77	3eqtrd 2782	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = 𝑘)
79	78	oveq2d 7271	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) = ((𝑆‘𝑘)𝑇𝑘))
80	61, 79	eqtrd 2778	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇𝑘))
81	80	breq1d 5080	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((𝑆‘(𝑘 + 1))𝐺(𝑘 + 1) ↔ ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1)))
82	81	biimprd 247	. . . . . 6 ⊢ (𝑘 ∈ ℕ0 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1)))
83	82	adantrd 491	. . . . 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 78	. . . 4 ⊢ (𝑘 ∈ ℕ0 → (𝜑 → ((𝑆‘𝑘)𝐺𝑘 → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))))
85	84	a2d 29	. . 3 ⊢ (𝑘 ∈ ℕ0 → ((𝜑 → (𝑆‘𝑘)𝐺𝑘) → (𝜑 → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))))
86	4, 8, 12, 16, 34, 85	nn0ind 12345	. 2 ⊢ (𝐴 ∈ ℕ0 → (𝜑 → (𝑆‘𝐴)𝐺𝐴))
87	86	impcom 407	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ0) → (𝑆‘𝐴)𝐺𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ifcif 4456  𝒫 cpw 4530  ⟨cop 4564  ∪ cuni 4836  ∪ ciun 4921   class class class wbr 5070  {copab 5132   ↦ cmpt 5153  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  2^nd c2nd 7803  Fincfn 8691  ℂcc 10800  0cc0 10802  1c1 10803   + caddc 10805   − cmin 11135   / cdiv 11562  ℕcn 11903  2c2 11958  ℕ₀cn0 12163  ℤcz 12249  ℤ_≥cuz 12511  seqcseq 13649  ↑cexp 13710  ballcbl 20497  MetOpencmopn 20500  CMetccmet 24323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-seq 13650

This theorem is referenced by:  heiborlem5  35900  heiborlem6  35901  heiborlem8  35903