Theorem heiborlem4 37815

Description: Lemma for heibor 37822. Using the function 𝑇 constructed in heiborlem3 37814, 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 6861	. . . . 5 ⊢ (𝑥 = 0 → (𝑆‘𝑥) = (𝑆‘0))
2		id 22	. . . . 5 ⊢ (𝑥 = 0 → 𝑥 = 0)
3	1, 2	breq12d 5123	. . . 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 6861	. . . . 5 ⊢ (𝑥 = 𝑘 → (𝑆‘𝑥) = (𝑆‘𝑘))
6		id 22	. . . . 5 ⊢ (𝑥 = 𝑘 → 𝑥 = 𝑘)
7	5, 6	breq12d 5123	. . . 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 6861	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → (𝑆‘𝑥) = (𝑆‘(𝑘 + 1)))
10		id 22	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → 𝑥 = (𝑘 + 1))
11	9, 10	breq12d 5123	. . . 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 6861	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴))
14		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
15	13, 14	breq12d 5123	. . . 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 6862	. . . . . 6 ⊢ (𝑆‘0) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘0)
19		0z 12547	. . . . . . 7 ⊢ 0 ∈ ℤ
20		seq1 13986	. . . . . . 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 2753	. . . . 5 ⊢ (𝑆‘0) = ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0)
23		0nn0 12464	. . . . . 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 5786	. . . . . . . 8 ⊢ Rel 𝐺
27	26	brrelex1i 5697	. . . . . . 7 ⊢ (𝐶𝐺0 → 𝐶 ∈ V)
28	24, 27	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ V)
29		iftrue 4497	. . . . . . 7 ⊢ (𝑚 = 0 → if(𝑚 = 0, 𝐶, (𝑚 − 1)) = 𝐶)
30		eqid 2730	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))
31	29, 30	fvmptg 6969	. . . . . 6 ⊢ ((0 ∈ ℕ0 ∧ 𝐶 ∈ V) → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0) = 𝐶)
32	23, 28, 31	sylancr 587	. . . . 5 ⊢ (𝜑 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0) = 𝐶)
33	22, 32	eqtrid 2777	. . . 4 ⊢ (𝜑 → (𝑆‘0) = 𝐶)
34	33, 24	eqbrtrd 5132	. . 3 ⊢ (𝜑 → (𝑆‘0)𝐺0)
35		df-br 5111	. . . . . 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 6861	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝑇‘𝑥) = (𝑇‘⟨(𝑆‘𝑘), 𝑘⟩))
38		df-ov 7393	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑘)𝑇𝑘) = (𝑇‘⟨(𝑆‘𝑘), 𝑘⟩)
39	37, 38	eqtr4di 2783	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝑇‘𝑥) = ((𝑆‘𝑘)𝑇𝑘))
40		fvex 6874	. . . . . . . . . . . 12 ⊢ (𝑆‘𝑘) ∈ V
41		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑘 ∈ V
42	40, 41	op2ndd 7982	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (2nd ‘𝑥) = 𝑘)
43	42	oveq1d 7405	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((2nd ‘𝑥) + 1) = (𝑘 + 1))
44	39, 43	breq12d 5123	. . . . . . . . 9 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ↔ ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1)))
45		fveq2 6861	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝐵‘𝑥) = (𝐵‘⟨(𝑆‘𝑘), 𝑘⟩))
46		df-ov 7393	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑘)𝐵𝑘) = (𝐵‘⟨(𝑆‘𝑘), 𝑘⟩)
47	45, 46	eqtr4di 2783	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝐵‘𝑥) = ((𝑆‘𝑘)𝐵𝑘))
48	39, 43	oveq12d 7408	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1)) = (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)))
49	47, 48	ineq12d 4187	. . . . . . . . . 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 2814	. . . . . . . . 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 632	. . . . . . . 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 3588	. . . . . . 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	biimtrid 242	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑘)𝐺𝑘 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
55		seqp1 13988	. . . . . . . . . . 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 12842	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘0)
57	55, 56	eleq2s 2847	. . . . . . . . . 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 6862	. . . . . . . . . 10 ⊢ (𝑆‘(𝑘 + 1)) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1))
59	17	fveq1i 6862	. . . . . . . . . . 11 ⊢ (𝑆‘𝑘) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)
60	59	oveq1i 7400	. . . . . . . . . 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 2790	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
62		eqeq1 2734	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → (𝑚 = 0 ↔ (𝑘 + 1) = 0))
63		oveq1 7397	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → (𝑚 − 1) = ((𝑘 + 1) − 1))
64	62, 63	ifbieq2d 4518	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑘 + 1) → if(𝑚 = 0, 𝐶, (𝑚 − 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
65		peano2nn0 12489	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
66		nn0p1nn 12488	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
67		nnne0 12227	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ ℕ → (𝑘 + 1) ≠ 0)
68	67	neneqd 2931	. . . . . . . . . . . . . 14 ⊢ ((𝑘 + 1) ∈ ℕ → ¬ (𝑘 + 1) = 0)
69		iffalse 4500	. . . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 13 ⊢ ((𝑘 + 1) − 1) ∈ V
72	70, 71	eqeltrdi 2837	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) ∈ V)
73	30, 64, 65, 72	fvmptd3 6994	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
74		nn0cn 12459	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
75		ax-1cn 11133	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
76		pncan 11434	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
77	74, 75, 76	sylancl 586	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → ((𝑘 + 1) − 1) = 𝑘)
78	73, 70, 77	3eqtrd 2769	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = 𝑘)
79	78	oveq2d 7406	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) = ((𝑆‘𝑘)𝑇𝑘))
80	61, 79	eqtrd 2765	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇𝑘))
81	80	breq1d 5120	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((𝑆‘(𝑘 + 1))𝐺(𝑘 + 1) ↔ ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1)))
82	81	biimprd 248	. . . . . 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 12636	. 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 1086   = wceq 1540   ∈ wcel 2109  {cab 2708  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ∩ cin 3916   ⊆ wss 3917  ifcif 4491  𝒫 cpw 4566  ⟨cop 4598  ∪ cuni 4874  ∪ ciun 4958   class class class wbr 5110  {copab 5172   ↦ cmpt 5191  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  2^nd c2nd 7970  Fincfn 8921  ℂcc 11073  0cc0 11075  1c1 11076   + caddc 11078   − cmin 11412   / cdiv 11842  ℕcn 12193  2c2 12248  ℕ₀cn0 12449  ℤcz 12536  ℤ_≥cuz 12800  seqcseq 13973  ↑cexp 14033  ballcbl 21258  MetOpencmopn 21261  CMetccmet 25161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-n0 12450  df-z 12537  df-uz 12801  df-seq 13974

This theorem is referenced by:  heiborlem5  37816  heiborlem6  37817  heiborlem8  37819