Theorem nn0ind-raph 8355

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 8183	. 2 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2		dfsbcq2 2767	. . . 4 ⊢ (𝑧 = 1 → ([𝑧 / 𝑥]𝜑 ↔ [1 / 𝑥]𝜑))
3		nfv 1421	. . . . 5 ⊢ Ⅎ𝑥𝜒
4		nn0ind-raph.2	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
5	3, 4	sbhypf 2603	. . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜒))
6		nfv 1421	. . . . 5 ⊢ Ⅎ𝑥𝜃
7		nn0ind-raph.3	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))
8	6, 7	sbhypf 2603	. . . 4 ⊢ (𝑧 = (𝑦 + 1) → ([𝑧 / 𝑥]𝜑 ↔ 𝜃))
9		nfv 1421	. . . . 5 ⊢ Ⅎ𝑥𝜏
10		nn0ind-raph.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
11	9, 10	sbhypf 2603	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ 𝜏))
12		nfsbc1v 2782	. . . . 5 ⊢ Ⅎ𝑥[1 / 𝑥]𝜑
13		1ex 7022	. . . . 5 ⊢ 1 ∈ V
14		c0ex 7021	. . . . . . 7 ⊢ 0 ∈ V
15		0nn0 8196	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ0
16		eleq1a 2109	. . . . . . . . . . . 12 ⊢ (0 ∈ ℕ0 → (𝑦 = 0 → 𝑦 ∈ ℕ0))
17	15, 16	ax-mp 7	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → 𝑦 ∈ ℕ0)
18		nn0ind-raph.5	. . . . . . . . . . . . . . 15 ⊢ 𝜓
19		nn0ind-raph.1	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓))
20	18, 19	mpbiri 157	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → 𝜑)
21		eqeq2 2049	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 0 → (𝑥 = 𝑦 ↔ 𝑥 = 0))
22	21, 4	syl6bir 153	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 0 → (𝑥 = 0 → (𝜑 ↔ 𝜒)))
23	22	pm5.74d 171	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 0 → ((𝑥 = 0 → 𝜑) ↔ (𝑥 = 0 → 𝜒)))
24	20, 23	mpbii 136	. . . . . . . . . . . . 13 ⊢ (𝑦 = 0 → (𝑥 = 0 → 𝜒))
25	24	com12 27	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝑦 = 0 → 𝜒))
26	14, 25	vtocle 2627	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → 𝜒)
27		nn0ind-raph.6	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃))
28	17, 26, 27	sylc 56	. . . . . . . . . 10 ⊢ (𝑦 = 0 → 𝜃)
29	28	adantr 261	. . . . . . . . 9 ⊢ ((𝑦 = 0 ∧ 𝑥 = 1) → 𝜃)
30		oveq1 5519	. . . . . . . . . . . . 13 ⊢ (𝑦 = 0 → (𝑦 + 1) = (0 + 1))
31		0p1e1 8031	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
32	30, 31	syl6eq 2088	. . . . . . . . . . . 12 ⊢ (𝑦 = 0 → (𝑦 + 1) = 1)
33	32	eqeq2d 2051	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → (𝑥 = (𝑦 + 1) ↔ 𝑥 = 1))
34	33, 7	syl6bir 153	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (𝑥 = 1 → (𝜑 ↔ 𝜃)))
35	34	imp 115	. . . . . . . . 9 ⊢ ((𝑦 = 0 ∧ 𝑥 = 1) → (𝜑 ↔ 𝜃))
36	29, 35	mpbird 156	. . . . . . . 8 ⊢ ((𝑦 = 0 ∧ 𝑥 = 1) → 𝜑)
37	36	ex 108	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑥 = 1 → 𝜑))
38	14, 37	vtocle 2627	. . . . . 6 ⊢ (𝑥 = 1 → 𝜑)
39		sbceq1a 2773	. . . . . 6 ⊢ (𝑥 = 1 → (𝜑 ↔ [1 / 𝑥]𝜑))
40	38, 39	mpbid 135	. . . . 5 ⊢ (𝑥 = 1 → [1 / 𝑥]𝜑)
41	12, 13, 40	vtoclef 2626	. . . 4 ⊢ [1 / 𝑥]𝜑
42		nnnn0 8188	. . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
43	42, 27	syl 14	. . . 4 ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃))
44	2, 5, 8, 11, 41, 43	nnind 7930	. . 3 ⊢ (𝐴 ∈ ℕ → 𝜏)
45		nfv 1421	. . . . 5 ⊢ Ⅎ𝑥(0 = 𝐴 → 𝜏)
46		eqeq1 2046	. . . . . 6 ⊢ (𝑥 = 0 → (𝑥 = 𝐴 ↔ 0 = 𝐴))
47	19	bicomd 129	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝜓 ↔ 𝜑))
48	47, 10	sylan9bb 435	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜏))
49	18, 48	mpbii 136	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑥 = 𝐴) → 𝜏)
50	49	ex 108	. . . . . 6 ⊢ (𝑥 = 0 → (𝑥 = 𝐴 → 𝜏))
51	46, 50	sylbird 159	. . . . 5 ⊢ (𝑥 = 0 → (0 = 𝐴 → 𝜏))
52	45, 14, 51	vtoclef 2626	. . . 4 ⊢ (0 = 𝐴 → 𝜏)
53	52	eqcoms 2043	. . 3 ⊢ (𝐴 = 0 → 𝜏)
54	44, 53	jaoi 636	. 2 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → 𝜏)
55	1, 54	sylbi 114	1 ⊢ (𝐴 ∈ ℕ0 → 𝜏)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   = wceq 1243   ∈ wcel 1393  [wsb 1645  [wsbc 2764  (class class class)co 5512  0cc0 6889  1c1 6890   + caddc 6892  ℕcn 7914  ℕ₀cn0 8181

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-i2m1 6989  ax-0id 6992

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515  df-inn 7915  df-n0 8182

This theorem is referenced by: (None)