Theorem nn0ind-raph 9383

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 9191	. 2 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2		dfsbcq2 2977	. . . 4 ⊢ (𝑧 = 1 → ([𝑧 / 𝑥]𝜑 ↔ [1 / 𝑥]𝜑))
3		nfv 1538	. . . . 5 ⊢ Ⅎ𝑥𝜒
4		nn0ind-raph.2	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
5	3, 4	sbhypf 2798	. . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜒))
6		nfv 1538	. . . . 5 ⊢ Ⅎ𝑥𝜃
7		nn0ind-raph.3	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))
8	6, 7	sbhypf 2798	. . . 4 ⊢ (𝑧 = (𝑦 + 1) → ([𝑧 / 𝑥]𝜑 ↔ 𝜃))
9		nfv 1538	. . . . 5 ⊢ Ⅎ𝑥𝜏
10		nn0ind-raph.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
11	9, 10	sbhypf 2798	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ 𝜏))
12		nfsbc1v 2993	. . . . 5 ⊢ Ⅎ𝑥[1 / 𝑥]𝜑
13		1ex 7965	. . . . 5 ⊢ 1 ∈ V
14		c0ex 7964	. . . . . . 7 ⊢ 0 ∈ V
15		0nn0 9204	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ0
16		eleq1a 2259	. . . . . . . . . . . 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 168	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → 𝜑)
21		eqeq2 2197	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 0 → (𝑥 = 𝑦 ↔ 𝑥 = 0))
22	21, 4	syl6bir 164	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 0 → (𝑥 = 0 → (𝜑 ↔ 𝜒)))
23	22	pm5.74d 182	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 0 → ((𝑥 = 0 → 𝜑) ↔ (𝑥 = 0 → 𝜒)))
24	20, 23	mpbii 148	. . . . . . . . . . . . 13 ⊢ (𝑦 = 0 → (𝑥 = 0 → 𝜒))
25	24	com12 30	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝑦 = 0 → 𝜒))
26	14, 25	vtocle 2823	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → 𝜒)
27		nn0ind-raph.6	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃))
28	17, 26, 27	sylc 62	. . . . . . . . . 10 ⊢ (𝑦 = 0 → 𝜃)
29	28	adantr 276	. . . . . . . . 9 ⊢ ((𝑦 = 0 ∧ 𝑥 = 1) → 𝜃)
30		oveq1 5895	. . . . . . . . . . . . 13 ⊢ (𝑦 = 0 → (𝑦 + 1) = (0 + 1))
31		0p1e1 9046	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
32	30, 31	eqtrdi 2236	. . . . . . . . . . . 12 ⊢ (𝑦 = 0 → (𝑦 + 1) = 1)
33	32	eqeq2d 2199	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → (𝑥 = (𝑦 + 1) ↔ 𝑥 = 1))
34	33, 7	syl6bir 164	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (𝑥 = 1 → (𝜑 ↔ 𝜃)))
35	34	imp 124	. . . . . . . . 9 ⊢ ((𝑦 = 0 ∧ 𝑥 = 1) → (𝜑 ↔ 𝜃))
36	29, 35	mpbird 167	. . . . . . . 8 ⊢ ((𝑦 = 0 ∧ 𝑥 = 1) → 𝜑)
37	36	ex 115	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑥 = 1 → 𝜑))
38	14, 37	vtocle 2823	. . . . . 6 ⊢ (𝑥 = 1 → 𝜑)
39		sbceq1a 2984	. . . . . 6 ⊢ (𝑥 = 1 → (𝜑 ↔ [1 / 𝑥]𝜑))
40	38, 39	mpbid 147	. . . . 5 ⊢ (𝑥 = 1 → [1 / 𝑥]𝜑)
41	12, 13, 40	vtoclef 2822	. . . 4 ⊢ [1 / 𝑥]𝜑
42		nnnn0 9196	. . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
43	42, 27	syl 14	. . . 4 ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃))
44	2, 5, 8, 11, 41, 43	nnind 8948	. . 3 ⊢ (𝐴 ∈ ℕ → 𝜏)
45		nfv 1538	. . . . 5 ⊢ Ⅎ𝑥(0 = 𝐴 → 𝜏)
46		eqeq1 2194	. . . . . 6 ⊢ (𝑥 = 0 → (𝑥 = 𝐴 ↔ 0 = 𝐴))
47	19	bicomd 141	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝜓 ↔ 𝜑))
48	47, 10	sylan9bb 462	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜏))
49	18, 48	mpbii 148	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑥 = 𝐴) → 𝜏)
50	49	ex 115	. . . . . 6 ⊢ (𝑥 = 0 → (𝑥 = 𝐴 → 𝜏))
51	46, 50	sylbird 170	. . . . 5 ⊢ (𝑥 = 0 → (0 = 𝐴 → 𝜏))
52	45, 14, 51	vtoclef 2822	. . . 4 ⊢ (0 = 𝐴 → 𝜏)
53	52	eqcoms 2190	. . 3 ⊢ (𝐴 = 0 → 𝜏)
54	44, 53	jaoi 717	. 2 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → 𝜏)
55	1, 54	sylbi 121	1 ⊢ (𝐴 ∈ ℕ0 → 𝜏)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   = wceq 1363  [wsb 1772   ∈ wcel 2158  [wsbc 2974  (class class class)co 5888  0cc0 7824  1c1 7825   + caddc 7827  ℕcn 8932  ℕ₀cn0 9189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-sep 4133  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-i2m1 7929  ax-0id 7932

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-iota 5190  df-fv 5236  df-ov 5891  df-inn 8933  df-n0 9190

This theorem is referenced by: (None)