Theorem nn0ind-raph 6216

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-2004. Raph says: "This seems a bit painful. I wonder if an explicit substitution version would be easier.")

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

Proof of Theorem nn0ind-raph

Step	Hyp	Ref	Expression
1		elnn0 6103	. 2 ⊢ (A ∈ ℕ0 ↔ (A ∈ ℕ ⋁ A = 0))
2		dfsbcq 1946	. . . 4 ⊢ (z = 1 → ([z / x]φ ↔ [1 / x]φ))
3		nn0ind-raph.2	. . . . 5 ⊢ (x = y → (φ ↔ χ))
4	3	sbhyp 1943	. . . 4 ⊢ (z = y → ([z / x]φ ↔ χ))
5		nn0ind-raph.3	. . . . 5 ⊢ (x = (y + 1) → (φ ↔ θ))
6	5	sbhyp 1943	. . . 4 ⊢ (z = (y + 1) → ([z / x]φ ↔ θ))
7		nn0ind-raph.4	. . . . 5 ⊢ (x = A → (φ ↔ τ))
8	7	sbhyp 1943	. . . 4 ⊢ (z = A → ([z / x]φ ↔ τ))
9		1re 5447	. . . . . . 7 ⊢ 1 ∈ ℝ
10	9	elisseti 1821	. . . . . 6 ⊢ 1 ∈ V
11	10	hbsbc1v 1953	. . . . 5 ⊢ ([1 / x]φ → ∀x[1 / x]φ)
12		0nn0 6115	. . . . . . . 8 ⊢ 0 ∈ ℕ0
13	12	elisseti 1821	. . . . . . 7 ⊢ 0 ∈ V
14		nn0ind-raph.6	. . . . . . . . . . 11 ⊢ (y ∈ ℕ0 → (χ → θ))
15		eleq1a 1546	. . . . . . . . . . . 12 ⊢ (0 ∈ ℕ0 → (y = 0 → y ∈ ℕ0))
16	12, 15	ax-mp 7	. . . . . . . . . . 11 ⊢ (y = 0 → y ∈ ℕ0)
17		nn0ind-raph.5	. . . . . . . . . . . . . . 15 ⊢ ψ
18		nn0ind-raph.1	. . . . . . . . . . . . . . 15 ⊢ (x = 0 → (φ ↔ ψ))
19	17, 18	mpbiri 194	. . . . . . . . . . . . . 14 ⊢ (x = 0 → φ)
20		eqeq2 1487	. . . . . . . . . . . . . . . 16 ⊢ (y = 0 → (x = y ↔ x = 0))
21	20, 3	syl6bir 215	. . . . . . . . . . . . . . 15 ⊢ (y = 0 → (x = 0 → (φ ↔ χ)))
22	21	pm5.74d 587	. . . . . . . . . . . . . 14 ⊢ (y = 0 → ((x = 0 → φ) ↔ (x = 0 → χ)))
23	19, 22	mpbii 193	. . . . . . . . . . . . 13 ⊢ (y = 0 → (x = 0 → χ))
24	23	com12 11	. . . . . . . . . . . 12 ⊢ (x = 0 → (y = 0 → χ))
25	13, 24	vtocle 1861	. . . . . . . . . . 11 ⊢ (y = 0 → χ)
26	14, 16, 25	sylc 68	. . . . . . . . . 10 ⊢ (y = 0 → θ)
27	26	adantr 391	. . . . . . . . 9 ⊢ ((y = 0 ⋀ x = 1) → θ)
28		opreq1 3974	. . . . . . . . . . . . 13 ⊢ (y = 0 → (y + 1) = (0 + 1))
29		ax1cn 5281	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
30	29	addid2 5343	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
31	28, 30	syl6eq 1526	. . . . . . . . . . . 12 ⊢ (y = 0 → (y + 1) = 1)
32	31	eqeq2d 1489	. . . . . . . . . . 11 ⊢ (y = 0 → (x = (y + 1) ↔ x = 1))
33	32, 5	syl6bir 215	. . . . . . . . . 10 ⊢ (y = 0 → (x = 1 → (φ ↔ θ)))
34	33	imp 350	. . . . . . . . 9 ⊢ ((y = 0 ⋀ x = 1) → (φ ↔ θ))
35	27, 34	mpbird 196	. . . . . . . 8 ⊢ ((y = 0 ⋀ x = 1) → φ)
36	35	ex 373	. . . . . . 7 ⊢ (y = 0 → (x = 1 → φ))
37	13, 36	vtocle 1861	. . . . . 6 ⊢ (x = 1 → φ)
38		sbceq1a 1947	. . . . . 6 ⊢ (x = 1 → (φ ↔ [1 / x]φ))
39	37, 38	mpbid 195	. . . . 5 ⊢ (x = 1 → [1 / x]φ)
40	11, 10, 39	vtoclef 1860	. . . 4 ⊢ [1 / x]φ
41		nnnn0t 6108	. . . . 5 ⊢ (y ∈ ℕ → y ∈ ℕ0)
42	41, 14	syl 10	. . . 4 ⊢ (y ∈ ℕ → (χ → θ))
43	2, 4, 6, 8, 40, 42	nnind 5939	. . 3 ⊢ (A ∈ ℕ → τ)
44		ax-17 973	. . . . . 6 ⊢ (0 = A → ∀x0 = A)
45		ax-17 973	. . . . . 6 ⊢ (τ → ∀xτ)
46	44, 45	hbim 1009	. . . . 5 ⊢ ((0 = A → τ) → ∀x(0 = A → τ))
47		eqeq1 1484	. . . . . 6 ⊢ (x = 0 → (x = A ↔ 0 = A))
48	18	bicomd 523	. . . . . . . . 9 ⊢ (x = 0 → (ψ ↔ φ))
49	48, 7	sylan9bb 542	. . . . . . . 8 ⊢ ((x = 0 ⋀ x = A) → (ψ ↔ τ))
50	17, 49	mpbii 193	. . . . . . 7 ⊢ ((x = 0 ⋀ x = A) → τ)
51	50	ex 373	. . . . . 6 ⊢ (x = 0 → (x = A → τ))
52	47, 51	sylbird 205	. . . . 5 ⊢ (x = 0 → (0 = A → τ))
53	46, 13, 52	vtoclef 1860	. . . 4 ⊢ (0 = A → τ)
54	53	eqcoms 1481	. . 3 ⊢ (A = 0 → τ)
55	43, 54	jaoi 341	. 2 ⊢ ((A ∈ ℕ ⋁ A = 0) → τ)
56	1, 55	sylbi 199	1 ⊢ (A ∈ ℕ0 → τ)

Syntax hints:   → wi 3   ↔ wb 146   ⋁ wo 222   ⋀ wa 223   = wceq 958   ∈ wcel 960  [wsbc 1172  (class class class)co 3969  ℝcr 5245  0cc0 5246  1c1 5247   + caddc 5249  ℕcn 5308  ℕ₀cn0 5309

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-sub 5368  df-neg 5370  df-n 5927  df-n0 6102

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