Theorem nn0ind-raph 12826

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 12635	. 2 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2		dfsbcq2 3745	. . . 4 ⊢ (𝑧 = 1 → ([𝑧 / 𝑥]𝜑 ↔ [1 / 𝑥]𝜑))
3		nfv 1949	. . . . 5 ⊢ Ⅎ𝑥𝜒
4		nn0ind-raph.2	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
5	3, 4	sbhypf 3513	. . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜒))
6		nfv 1949	. . . . 5 ⊢ Ⅎ𝑥𝜃
7		nn0ind-raph.3	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))
8	6, 7	sbhypf 3513	. . . 4 ⊢ (𝑧 = (𝑦 + 1) → ([𝑧 / 𝑥]𝜑 ↔ 𝜃))
9		nfv 1949	. . . . 5 ⊢ Ⅎ𝑥𝜏
10		nn0ind-raph.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
11	9, 10	sbhypf 3513	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ 𝜏))
12		nfsbc1v 3762	. . . . 5 ⊢ Ⅎ𝑥[1 / 𝑥]𝜑
13		1ex 11330	. . . . 5 ⊢ 1 ∈ V
14		c0ex 11327	. . . . . . 7 ⊢ 0 ∈ V
15		0nn0 12648	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ0
16		eleq1a 2859	. . . . . . . . . . . 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 261	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → 𝜑)
21		eqeq2 2775	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 0 → (𝑥 = 𝑦 ↔ 𝑥 = 0))
22	21, 4	biimtrrdi 257	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 0 → (𝑥 = 0 → (𝜑 ↔ 𝜒)))
23	22	pm5.74d 276	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 0 → ((𝑥 = 0 → 𝜑) ↔ (𝑥 = 0 → 𝜒)))
24	20, 23	mpbii 236	. . . . . . . . . . . . 13 ⊢ (𝑦 = 0 → (𝑥 = 0 → 𝜒))
25	24	com12 33	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝑦 = 0 → 𝜒))
26	14, 25	vtocle 3522	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → 𝜒)
27		nn0ind-raph.6	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃))
28	17, 26, 27	sylc 66	. . . . . . . . . 10 ⊢ (𝑦 = 0 → 𝜃)
29	28	adantr 486	. . . . . . . . 9 ⊢ ((𝑦 = 0 ∧ 𝑥 = 1) → 𝜃)
30		oveq1 7450	. . . . . . . . . . . . 13 ⊢ (𝑦 = 0 → (𝑦 + 1) = (0 + 1))
31		0p1e1 12490	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
32	30, 31	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ (𝑦 = 0 → (𝑦 + 1) = 1)
33	32	eqeq2d 2774	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → (𝑥 = (𝑦 + 1) ↔ 𝑥 = 1))
34	33, 7	biimtrrdi 257	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (𝑥 = 1 → (𝜑 ↔ 𝜃)))
35	34	imp 412	. . . . . . . . 9 ⊢ ((𝑦 = 0 ∧ 𝑥 = 1) → (𝜑 ↔ 𝜃))
36	29, 35	mpbird 260	. . . . . . . 8 ⊢ ((𝑦 = 0 ∧ 𝑥 = 1) → 𝜑)
37	36	ex 418	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑥 = 1 → 𝜑))
38	14, 37	vtocle 3522	. . . . . 6 ⊢ (𝑥 = 1 → 𝜑)
39		sbceq1a 3753	. . . . . 6 ⊢ (𝑥 = 1 → (𝜑 ↔ [1 / 𝑥]𝜑))
40	38, 39	mpbid 235	. . . . 5 ⊢ (𝑥 = 1 → [1 / 𝑥]𝜑)
41	12, 13, 40	vtoclef 3528	. . . 4 ⊢ [1 / 𝑥]𝜑
42		nnnn0 12640	. . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
43	42, 27	syl 18	. . . 4 ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃))
44	2, 5, 8, 11, 41, 43	nnind 12380	. . 3 ⊢ (𝐴 ∈ ℕ → 𝜏)
45		nfv 1949	. . . . 5 ⊢ Ⅎ𝑥(0 = 𝐴 → 𝜏)
46		eqeq1 2767	. . . . . 6 ⊢ (𝑥 = 0 → (𝑥 = 𝐴 ↔ 0 = 𝐴))
47	19	bicomd 226	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝜓 ↔ 𝜑))
48	47, 10	sylan9bb 519	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜏))
49	18, 48	mpbii 236	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑥 = 𝐴) → 𝜏)
50	49	ex 418	. . . . . 6 ⊢ (𝑥 = 0 → (𝑥 = 𝐴 → 𝜏))
51	46, 50	sylbird 263	. . . . 5 ⊢ (𝑥 = 0 → (0 = 𝐴 → 𝜏))
52	45, 14, 51	vtoclef 3528	. . . 4 ⊢ (0 = 𝐴 → 𝜏)
53	52	eqcoms 2771	. . 3 ⊢ (𝐴 = 0 → 𝜏)
54	44, 53	jaoi 871	. 2 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → 𝜏)
55	1, 54	sylbi 220	1 ⊢ (𝐴 ∈ ℕ0 → 𝜏)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570  [wsb 2101   ∈ wcel 2147  [wsbc 3742  (class class class)co 7443  0cc0 11227  1c1 11228   + caddc 11230  ℕcn 12362  ℕ₀cn0 12633

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-pnf 11372  df-mnf 11373  df-ltxr 11375  df-nn 12363  df-n0 12634

This theorem is used by: (None)