Theorem nn0ind-raph 11477

Description: Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Raph Levien remarks: "This seems a bit painful. I wonder if an explicit substitution version would be easier." (Contributed by Raph Levien, 10-Apr-2004.)

Hypotheses

Ref	Expression
nn0ind-raph.1	⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓))
nn0ind-raph.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
nn0ind-raph.3	⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))
nn0ind-raph.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
nn0ind-raph.5	⊢ 𝜓
nn0ind-raph.6	⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃))

Assertion

Ref	Expression
nn0ind-raph	⊢ (𝐴 ∈ ℕ0 → 𝜏)

Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝜃(𝑦)   𝜏(𝑦)   𝐴(𝑦)

Proof of Theorem nn0ind-raph

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnn0 11294	. 2 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2		dfsbcq2 3438	. . . 4 ⊢ (𝑧 = 1 → ([𝑧 / 𝑥]𝜑 ↔ [1 / 𝑥]𝜑))
3		nfv 1843	. . . . 5 ⊢ Ⅎ𝑥𝜒
4		nn0ind-raph.2	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
5	3, 4	sbhypf 3253	. . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜒))
6		nfv 1843	. . . . 5 ⊢ Ⅎ𝑥𝜃
7		nn0ind-raph.3	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))
8	6, 7	sbhypf 3253	. . . 4 ⊢ (𝑧 = (𝑦 + 1) → ([𝑧 / 𝑥]𝜑 ↔ 𝜃))
9		nfv 1843	. . . . 5 ⊢ Ⅎ𝑥𝜏
10		nn0ind-raph.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
11	9, 10	sbhypf 3253	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ 𝜏))
12		nfsbc1v 3455	. . . . 5 ⊢ Ⅎ𝑥[1 / 𝑥]𝜑
13		1ex 10035	. . . . 5 ⊢ 1 ∈ V
14		c0ex 10034	. . . . . . 7 ⊢ 0 ∈ V
15		0nn0 11307	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ0
16		eleq1a 2696	. . . . . . . . . . . 12 ⊢ (0 ∈ ℕ0 → (𝑦 = 0 → 𝑦 ∈ ℕ0))
17	15, 16	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → 𝑦 ∈ ℕ0)
18		nn0ind-raph.5	. . . . . . . . . . . . . . 15 ⊢ 𝜓
19		nn0ind-raph.1	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓))
20	18, 19	mpbiri 248	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → 𝜑)
21		eqeq2 2633	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 0 → (𝑥 = 𝑦 ↔ 𝑥 = 0))
22	21, 4	syl6bir 244	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 0 → (𝑥 = 0 → (𝜑 ↔ 𝜒)))
23	22	pm5.74d 262	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 0 → ((𝑥 = 0 → 𝜑) ↔ (𝑥 = 0 → 𝜒)))
24	20, 23	mpbii 223	. . . . . . . . . . . . 13 ⊢ (𝑦 = 0 → (𝑥 = 0 → 𝜒))
25	24	com12 32	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝑦 = 0 → 𝜒))
26	14, 25	vtocle 3282	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → 𝜒)
27		nn0ind-raph.6	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃))
28	17, 26, 27	sylc 65	. . . . . . . . . 10 ⊢ (𝑦 = 0 → 𝜃)
29	28	adantr 481	. . . . . . . . 9 ⊢ ((𝑦 = 0 ∧ 𝑥 = 1) → 𝜃)
30		oveq1 6657	. . . . . . . . . . . . 13 ⊢ (𝑦 = 0 → (𝑦 + 1) = (0 + 1))
31		0p1e1 11132	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
32	30, 31	syl6eq 2672	. . . . . . . . . . . 12 ⊢ (𝑦 = 0 → (𝑦 + 1) = 1)
33	32	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → (𝑥 = (𝑦 + 1) ↔ 𝑥 = 1))
34	33, 7	syl6bir 244	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (𝑥 = 1 → (𝜑 ↔ 𝜃)))
35	34	imp 445	. . . . . . . . 9 ⊢ ((𝑦 = 0 ∧ 𝑥 = 1) → (𝜑 ↔ 𝜃))
36	29, 35	mpbird 247	. . . . . . . 8 ⊢ ((𝑦 = 0 ∧ 𝑥 = 1) → 𝜑)
37	36	ex 450	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑥 = 1 → 𝜑))
38	14, 37	vtocle 3282	. . . . . 6 ⊢ (𝑥 = 1 → 𝜑)
39		sbceq1a 3446	. . . . . 6 ⊢ (𝑥 = 1 → (𝜑 ↔ [1 / 𝑥]𝜑))
40	38, 39	mpbid 222	. . . . 5 ⊢ (𝑥 = 1 → [1 / 𝑥]𝜑)
41	12, 13, 40	vtoclef 3281	. . . 4 ⊢ [1 / 𝑥]𝜑
42		nnnn0 11299	. . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
43	42, 27	syl 17	. . . 4 ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃))
44	2, 5, 8, 11, 41, 43	nnind 11038	. . 3 ⊢ (𝐴 ∈ ℕ → 𝜏)
45		nfv 1843	. . . . 5 ⊢ Ⅎ𝑥(0 = 𝐴 → 𝜏)
46		eqeq1 2626	. . . . . 6 ⊢ (𝑥 = 0 → (𝑥 = 𝐴 ↔ 0 = 𝐴))
47	19	bicomd 213	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝜓 ↔ 𝜑))
48	47, 10	sylan9bb 736	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜏))
49	18, 48	mpbii 223	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑥 = 𝐴) → 𝜏)
50	49	ex 450	. . . . . 6 ⊢ (𝑥 = 0 → (𝑥 = 𝐴 → 𝜏))
51	46, 50	sylbird 250	. . . . 5 ⊢ (𝑥 = 0 → (0 = 𝐴 → 𝜏))
52	45, 14, 51	vtoclef 3281	. . . 4 ⊢ (0 = 𝐴 → 𝜏)
53	52	eqcoms 2630	. . 3 ⊢ (𝐴 = 0 → 𝜏)
54	44, 53	jaoi 394	. 2 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → 𝜏)
55	1, 54	sylbi 207	1 ⊢ (𝐴 ∈ ℕ0 → 𝜏)

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483  [wsb 1880   ∈ wcel 1990  [wsbc 3435  (class class class)co 6650  0cc0 9936  1c1 9937   + caddc 9939  ℕcn 11020  ℕ₀cn0 11292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-nn 11021  df-n0 11293

This theorem is referenced by: (None)