Theorem nn0ind-raph 11063

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 -> ( ph <-> ps ) )
nn0ind-raph.2	|- ( x = y -> ( ph <-> ch ) )
nn0ind-raph.3	|- ( x = ( y + 1 ) -> ( ph <-> th ) )
nn0ind-raph.4	|- ( x = A -> ( ph <-> ta ) )
nn0ind-raph.5	|- ps
nn0ind-raph.6	|- ( y e. NN0 -> ( ch -> th ) )

Assertion

Ref	Expression
nn0ind-raph	|- ( A e. NN0 -> ta )

Distinct variable groups:   x, y   x, A   ps, x   ch, x   th, x   ta, x   ph, y

Allowed substitution hints:   ph( x)   ps( y)   ch( y)   th( y)   ta( y)   A( y)

Proof of Theorem nn0ind-raph

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnn0 10899	. 2 |- ( A e. NN0 <-> ( A e. NN \/ A = 0 ) )
2		dfsbcq2 3281	. . . 4 |- ( z = 1 -> ( [ z / x ] ph <-> [. 1 / x ]. ph ) )
3		nfv 1771	. . . . 5 |- F/ x ch
4		nn0ind-raph.2	. . . . 5 |- ( x = y -> ( ph <-> ch ) )
5	3, 4	sbhypf 3106	. . . 4 |- ( z = y -> ( [ z / x ] ph <-> ch ) )
6		nfv 1771	. . . . 5 |- F/ x th
7		nn0ind-raph.3	. . . . 5 |- ( x = ( y + 1 ) -> ( ph <-> th ) )
8	6, 7	sbhypf 3106	. . . 4 |- ( z = ( y + 1 ) -> ( [ z / x ] ph <-> th ) )
9		nfv 1771	. . . . 5 |- F/ x ta
10		nn0ind-raph.4	. . . . 5 |- ( x = A -> ( ph <-> ta ) )
11	9, 10	sbhypf 3106	. . . 4 |- ( z = A -> ( [ z / x ] ph <-> ta ) )
12		nfsbc1v 3298	. . . . 5 |- F/ x [. 1 / x ]. ph
13		1ex 9663	. . . . 5 |- 1 e. _V
14		c0ex 9662	. . . . . . 7 |- 0 e. _V
15		0nn0 10912	. . . . . . . . . . . 12 |- 0 e. NN0
16		eleq1a 2534	. . . . . . . . . . . 12 |- ( 0 e. NN0 -> ( y = 0 -> y e. NN0 ) )
17	15, 16	ax-mp 5	. . . . . . . . . . 11 |- ( y = 0 -> y e. NN0 )
18		nn0ind-raph.5	. . . . . . . . . . . . . . 15 |- ps
19		nn0ind-raph.1	. . . . . . . . . . . . . . 15 |- ( x = 0 -> ( ph <-> ps ) )
20	18, 19	mpbiri 241	. . . . . . . . . . . . . 14 |- ( x = 0 -> ph )
21		eqeq2 2472	. . . . . . . . . . . . . . . 16 |- ( y = 0 -> ( x = y <-> x = 0 ) )
22	21, 4	syl6bir 237	. . . . . . . . . . . . . . 15 |- ( y = 0 -> ( x = 0 -> ( ph <-> ch ) ) )
23	22	pm5.74d 255	. . . . . . . . . . . . . 14 |- ( y = 0 -> ( ( x = 0 -> ph ) <-> ( x = 0 -> ch ) ) )
24	20, 23	mpbii 216	. . . . . . . . . . . . 13 |- ( y = 0 -> ( x = 0 -> ch ) )
25	24	com12 32	. . . . . . . . . . . 12 |- ( x = 0 -> ( y = 0 -> ch ) )
26	14, 25	vtocle 3134	. . . . . . . . . . 11 |- ( y = 0 -> ch )
27		nn0ind-raph.6	. . . . . . . . . . 11 |- ( y e. NN0 -> ( ch -> th ) )
28	17, 26, 27	sylc 62	. . . . . . . . . 10 |- ( y = 0 -> th )
29	28	adantr 471	. . . . . . . . 9 |- ( ( y = 0 /\ x = 1 ) -> th )
30		oveq1 6321	. . . . . . . . . . . . 13 |- ( y = 0 -> ( y + 1 ) = ( 0 + 1 ) )
31		0p1e1 10748	. . . . . . . . . . . . 13 |- ( 0 + 1 ) = 1
32	30, 31	syl6eq 2511	. . . . . . . . . . . 12 |- ( y = 0 -> ( y + 1 ) = 1 )
33	32	eqeq2d 2471	. . . . . . . . . . 11 |- ( y = 0 -> ( x = ( y + 1 ) <-> x = 1 ) )
34	33, 7	syl6bir 237	. . . . . . . . . 10 |- ( y = 0 -> ( x = 1 -> ( ph <-> th ) ) )
35	34	imp 435	. . . . . . . . 9 |- ( ( y = 0 /\ x = 1 ) -> ( ph <-> th ) )
36	29, 35	mpbird 240	. . . . . . . 8 |- ( ( y = 0 /\ x = 1 ) -> ph )
37	36	ex 440	. . . . . . 7 |- ( y = 0 -> ( x = 1 -> ph ) )
38	14, 37	vtocle 3134	. . . . . 6 |- ( x = 1 -> ph )
39		sbceq1a 3289	. . . . . 6 |- ( x = 1 -> ( ph <-> [. 1 / x ]. ph ) )
40	38, 39	mpbid 215	. . . . 5 |- ( x = 1 -> [. 1 / x ]. ph )
41	12, 13, 40	vtoclef 3133	. . . 4 |- [. 1 / x ]. ph
42		nnnn0 10904	. . . . 5 |- ( y e. NN -> y e. NN0 )
43	42, 27	syl 17	. . . 4 |- ( y e. NN -> ( ch -> th ) )
44	2, 5, 8, 11, 41, 43	nnind 10654	. . 3 |- ( A e. NN -> ta )
45		nfv 1771	. . . . 5 |- F/ x ( 0 = A -> ta )
46		eqeq1 2465	. . . . . 6 |- ( x = 0 -> ( x = A <-> 0 = A ) )
47	19	bicomd 206	. . . . . . . . 9 |- ( x = 0 -> ( ps <-> ph ) )
48	47, 10	sylan9bb 711	. . . . . . . 8 |- ( ( x = 0 /\ x = A ) -> ( ps <-> ta ) )
49	18, 48	mpbii 216	. . . . . . 7 |- ( ( x = 0 /\ x = A ) -> ta )
50	49	ex 440	. . . . . 6 |- ( x = 0 -> ( x = A -> ta ) )
51	46, 50	sylbird 243	. . . . 5 |- ( x = 0 -> ( 0 = A -> ta ) )
52	45, 14, 51	vtoclef 3133	. . . 4 |- ( 0 = A -> ta )
53	52	eqcoms 2469	. . 3 |- ( A = 0 -> ta )
54	44, 53	jaoi 385	. 2 |- ( ( A e. NN \/ A = 0 ) -> ta )
55	1, 54	sylbi 200	1 |- ( A e. NN0 -> ta )

Syntax hints:   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   = wceq 1454   [wsb 1807   e. wcel 1897   [.wsbc 3278  (class class class)co 6314   0cc0 9564   1c1 9565   + caddc 9567   NNcn 10636   NN0cn0 10897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-om 6719  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-pnf 9702  df-mnf 9703  df-ltxr 9705  df-nn 10637  df-n0 10898

This theorem is referenced by: (None)