Theorem nn0ind-raph 9161

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 8972	. 2 |- ( A e. NN0 <-> ( A e. NN \/ A = 0 ) )
2		dfsbcq2 2907	. . . 4 |- ( z = 1 -> ( [ z / x ] ph <-> [. 1 / x ]. ph ) )
3		nfv 1508	. . . . 5 |- F/ x ch
4		nn0ind-raph.2	. . . . 5 |- ( x = y -> ( ph <-> ch ) )
5	3, 4	sbhypf 2730	. . . 4 |- ( z = y -> ( [ z / x ] ph <-> ch ) )
6		nfv 1508	. . . . 5 |- F/ x th
7		nn0ind-raph.3	. . . . 5 |- ( x = ( y + 1 ) -> ( ph <-> th ) )
8	6, 7	sbhypf 2730	. . . 4 |- ( z = ( y + 1 ) -> ( [ z / x ] ph <-> th ) )
9		nfv 1508	. . . . 5 |- F/ x ta
10		nn0ind-raph.4	. . . . 5 |- ( x = A -> ( ph <-> ta ) )
11	9, 10	sbhypf 2730	. . . 4 |- ( z = A -> ( [ z / x ] ph <-> ta ) )
12		nfsbc1v 2922	. . . . 5 |- F/ x [. 1 / x ]. ph
13		1ex 7754	. . . . 5 |- 1 e. _V
14		c0ex 7753	. . . . . . 7 |- 0 e. _V
15		0nn0 8985	. . . . . . . . . . . 12 |- 0 e. NN0
16		eleq1a 2209	. . . . . . . . . . . 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 167	. . . . . . . . . . . . . 14 |- ( x = 0 -> ph )
21		eqeq2 2147	. . . . . . . . . . . . . . . 16 |- ( y = 0 -> ( x = y <-> x = 0 ) )
22	21, 4	syl6bir 163	. . . . . . . . . . . . . . 15 |- ( y = 0 -> ( x = 0 -> ( ph <-> ch ) ) )
23	22	pm5.74d 181	. . . . . . . . . . . . . 14 |- ( y = 0 -> ( ( x = 0 -> ph ) <-> ( x = 0 -> ch ) ) )
24	20, 23	mpbii 147	. . . . . . . . . . . . 13 |- ( y = 0 -> ( x = 0 -> ch ) )
25	24	com12 30	. . . . . . . . . . . 12 |- ( x = 0 -> ( y = 0 -> ch ) )
26	14, 25	vtocle 2755	. . . . . . . . . . 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 274	. . . . . . . . 9 |- ( ( y = 0 /\ x = 1 ) -> th )
30		oveq1 5774	. . . . . . . . . . . . 13 |- ( y = 0 -> ( y + 1 ) = ( 0 + 1 ) )
31		0p1e1 8827	. . . . . . . . . . . . 13 |- ( 0 + 1 ) = 1
32	30, 31	syl6eq 2186	. . . . . . . . . . . 12 |- ( y = 0 -> ( y + 1 ) = 1 )
33	32	eqeq2d 2149	. . . . . . . . . . 11 |- ( y = 0 -> ( x = ( y + 1 ) <-> x = 1 ) )
34	33, 7	syl6bir 163	. . . . . . . . . 10 |- ( y = 0 -> ( x = 1 -> ( ph <-> th ) ) )
35	34	imp 123	. . . . . . . . 9 |- ( ( y = 0 /\ x = 1 ) -> ( ph <-> th ) )
36	29, 35	mpbird 166	. . . . . . . 8 |- ( ( y = 0 /\ x = 1 ) -> ph )
37	36	ex 114	. . . . . . 7 |- ( y = 0 -> ( x = 1 -> ph ) )
38	14, 37	vtocle 2755	. . . . . 6 |- ( x = 1 -> ph )
39		sbceq1a 2913	. . . . . 6 |- ( x = 1 -> ( ph <-> [. 1 / x ]. ph ) )
40	38, 39	mpbid 146	. . . . 5 |- ( x = 1 -> [. 1 / x ]. ph )
41	12, 13, 40	vtoclef 2754	. . . 4 |- [. 1 / x ]. ph
42		nnnn0 8977	. . . . 5 |- ( y e. NN -> y e. NN0 )
43	42, 27	syl 14	. . . 4 |- ( y e. NN -> ( ch -> th ) )
44	2, 5, 8, 11, 41, 43	nnind 8729	. . 3 |- ( A e. NN -> ta )
45		nfv 1508	. . . . 5 |- F/ x ( 0 = A -> ta )
46		eqeq1 2144	. . . . . 6 |- ( x = 0 -> ( x = A <-> 0 = A ) )
47	19	bicomd 140	. . . . . . . . 9 |- ( x = 0 -> ( ps <-> ph ) )
48	47, 10	sylan9bb 457	. . . . . . . 8 |- ( ( x = 0 /\ x = A ) -> ( ps <-> ta ) )
49	18, 48	mpbii 147	. . . . . . 7 |- ( ( x = 0 /\ x = A ) -> ta )
50	49	ex 114	. . . . . 6 |- ( x = 0 -> ( x = A -> ta ) )
51	46, 50	sylbird 169	. . . . 5 |- ( x = 0 -> ( 0 = A -> ta ) )
52	45, 14, 51	vtoclef 2754	. . . 4 |- ( 0 = A -> ta )
53	52	eqcoms 2140	. . 3 |- ( A = 0 -> ta )
54	44, 53	jaoi 705	. 2 |- ( ( A e. NN \/ A = 0 ) -> ta )
55	1, 54	sylbi 120	1 |- ( A e. NN0 -> ta )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   = wceq 1331   e. wcel 1480   [wsb 1735   [.wsbc 2904  (class class class)co 5767   0cc0 7613   1c1 7614   + caddc 7616   NNcn 8713   NN0cn0 8970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-i2m1 7718  ax-0id 7721

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770  df-inn 8714  df-n0 8971

This theorem is referenced by: (None)