Theorem nn0ind 4612

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 hypothesis. (Contributed by Raph Levien, 10-Apr-04. Raph says: "This seems a bit painful. I wonder if an explicit substitution version would be easier.")

Hypotheses

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

Assertion

Ref	Expression
nn0ind	⊢ (A ∈ ℕ0 → τ)

Distinct variable group(s):   x,y   x,A   ψ,x   χ,x   θ,x   τ,x   φ,y

Proof of Theorem nn0ind

Step	Hyp	Ref	Expression
1		elnn0 4536	. 2 ⊢ (A ∈ ℕ0 ↔ (A ∈ ℕ ∨ A = 0))
2		dfsbcq 1442	. . . 4 ⊢ (z = 1 → ([z / x]φ ↔ [1 / x]φ))
3		nn0ind.2	. . . . 5 ⊢ (x = y → (φ ↔ χ))
4	3	vsbcint 1438	. . . 4 ⊢ (z = y → ([z / x]φ ↔ χ))
5		nn0ind.3	. . . . 5 ⊢ (x = (y + 1) → (φ ↔ θ))
6	5	vsbcint 1438	. . . 4 ⊢ (z = (y + 1) → ([z / x]φ ↔ θ))
7		nn0ind.4	. . . . 5 ⊢ (x = A → (φ ↔ τ))
8	7	vsbcint 1438	. . . 4 ⊢ (z = A → ([z / x]φ ↔ τ))
9		ax1re 4064	. . . . . . 7 ⊢ 1 ∈ ℝ
10	9	elisseti 1355	. . . . . 6 ⊢ 1 ∈ V
11	10	hbsbcv 1447	. . . . 5 ⊢ ([1 / x]φ → ∀x[1 / x]φ)
12		0nn0 4546	. . . . . . . 8 ⊢ 0 ∈ ℕ0
13	12	elisseti 1355	. . . . . . 7 ⊢ 0 ∈ V
14		nn0ind.6	. . . . . . . . . . 11 ⊢ (y ∈ ℕ0 → (χ → θ))
15		eleq1a 1158	. . . . . . . . . . . 12 ⊢ (0 ∈ ℕ0 → (y = 0 → y ∈ ℕ0))
16	12, 15	ax-mp 6	. . . . . . . . . . 11 ⊢ (y = 0 → y ∈ ℕ0)
17		nn0ind.5	. . . . . . . . . . . . . . 15 ⊢ ψ
18		nn0ind.1	. . . . . . . . . . . . . . 15 ⊢ (x = 0 → (φ ↔ ψ))
19	17, 18	mpbiri 169	. . . . . . . . . . . . . 14 ⊢ (x = 0 → φ)
20		cleq2 1110	. . . . . . . . . . . . . . . 16 ⊢ (y = 0 → (x = y ↔ x = 0))
21	20, 3	syl6bir 188	. . . . . . . . . . . . . . 15 ⊢ (y = 0 → (x = 0 → (φ ↔ χ)))
22	21	pm5.74d 444	. . . . . . . . . . . . . 14 ⊢ (y = 0 → ((x = 0 → φ) ↔ (x = 0 → χ)))
23	19, 22	mpbii 168	. . . . . . . . . . . . 13 ⊢ (y = 0 → (x = 0 → χ))
24	23	com12 13	. . . . . . . . . . . 12 ⊢ (x = 0 → (y = 0 → χ))
25	13, 24	vtocle 1391	. . . . . . . . . . 11 ⊢ (y = 0 → χ)
26	14, 16, 25	sylc 62	. . . . . . . . . 10 ⊢ (y = 0 → θ)
27	26	adantr 306	. . . . . . . . 9 ⊢ ((y = 0 ∧ x = 1) → θ)
28		opreq1 3006	. . . . . . . . . . . . 13 ⊢ (y = 0 → (y + 1) = (0 + 1))
29		1cn 4101	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
30	29	addid2 4113	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
31	28, 30	syl6eq 1140	. . . . . . . . . . . 12 ⊢ (y = 0 → (y + 1) = 1)
32	31	cleq2d 1112	. . . . . . . . . . 11 ⊢ (y = 0 → (x = (y + 1) ↔ x = 1))
33	32, 5	syl6bir 188	. . . . . . . . . 10 ⊢ (y = 0 → (x = 1 → (φ ↔ θ)))
34	33	imp 277	. . . . . . . . 9 ⊢ ((y = 0 ∧ x = 1) → (φ ↔ θ))
35	27, 34	mpbird 171	. . . . . . . 8 ⊢ ((y = 0 ∧ x = 1) → φ)
36	35	exp 291	. . . . . . 7 ⊢ (y = 0 → (x = 1 → φ))
37	13, 36	vtocle 1391	. . . . . 6 ⊢ (x = 1 → φ)
38		sbceq1 1443	. . . . . 6 ⊢ (x = 1 → (φ ↔ [1 / x]φ))
39	37, 38	mpbid 170	. . . . 5 ⊢ (x = 1 → [1 / x]φ)
40	11, 10, 39	vtoclef 1392	. . . 4 ⊢ [1 / x]φ
41		nnnn0t 4541	. . . . 5 ⊢ (y ∈ ℕ → y ∈ ℕ0)
42	41, 14	syl 12	. . . 4 ⊢ (y ∈ ℕ → (χ → θ))
43	2, 4, 6, 8, 40, 42	nnind 4434	. . 3 ⊢ (A ∈ ℕ → τ)
44		ax-17 925	. . . . . 6 ⊢ (0 = A → ∀x0 = A)
45		ax-17 925	. . . . . 6 ⊢ (τ → ∀xτ)
46	44, 45	hbim 702	. . . . 5 ⊢ ((0 = A → τ) → ∀x(0 = A → τ))
47		cleq1 1107	. . . . . 6 ⊢ (x = 0 → (x = A ↔ 0 = A))
48	18	bicomd 399	. . . . . . . . 9 ⊢ (x = 0 → (ψ ↔ φ))
49	48, 7	sylan9bb 418	. . . . . . . 8 ⊢ ((x = 0 ∧ x = A) → (ψ ↔ τ))
50	17, 49	mpbii 168	. . . . . . 7 ⊢ ((x = 0 ∧ x = A) → τ)
51	50	exp 291	. . . . . 6 ⊢ (x = 0 → (x = A → τ))
52	47, 51	sylbird 180	. . . . 5 ⊢ (x = 0 → (0 = A → τ))
53	46, 13, 52	vtoclef 1392	. . . 4 ⊢ (0 = A → τ)
54	53	cleqcoms 1104	. . 3 ⊢ (A = 0 → τ)
55	43, 54	jaoi 275	. 2 ⊢ ((A ∈ ℕ ∨ A = 0) → τ)
56	1, 55	sylbi 174	1 ⊢ (A ∈ ℕ0 → τ)

Syntax hints:   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = weq 797  [wsb 852   = wceq 1091   ∈ wcel 1092  [wsbc 1440  (class class class)co 3001  ℝcr 4027  0cc0 4028  1c1 4029   + caddc 4031  ℕcn 4093  ℕ₀cn0 4094

This theorem is referenced by:  facclt 4874

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-ltp 3884  df-plpr 3958  df-enr 3960  df-nr 3961  df-plr 3962  df-0r 3965  df-1r 3966  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-n 4423  df-n0 4535

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