Theorem nn0ind-raph 9069

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 8880	. 2 |- ( A e. NN0 <-> ( A e. NN \/ A = 0 ) )
2		dfsbcq2 2881	. . . 4 |- ( z = 1 -> ( [ z / x ] ph <-> [. 1 / x ]. ph ) )
3		nfv 1491	. . . . 5 |- F/ x ch
4		nn0ind-raph.2	. . . . 5 |- ( x = y -> ( ph <-> ch ) )
5	3, 4	sbhypf 2706	. . . 4 |- ( z = y -> ( [ z / x ] ph <-> ch ) )
6		nfv 1491	. . . . 5 |- F/ x th
7		nn0ind-raph.3	. . . . 5 |- ( x = ( y + 1 ) -> ( ph <-> th ) )
8	6, 7	sbhypf 2706	. . . 4 |- ( z = ( y + 1 ) -> ( [ z / x ] ph <-> th ) )
9		nfv 1491	. . . . 5 |- F/ x ta
10		nn0ind-raph.4	. . . . 5 |- ( x = A -> ( ph <-> ta ) )
11	9, 10	sbhypf 2706	. . . 4 |- ( z = A -> ( [ z / x ] ph <-> ta ) )
12		nfsbc1v 2896	. . . . 5 |- F/ x [. 1 / x ]. ph
13		1ex 7682	. . . . 5 |- 1 e. _V
14		c0ex 7681	. . . . . . 7 |- 0 e. _V
15		0nn0 8893	. . . . . . . . . . . 12 |- 0 e. NN0
16		eleq1a 2186	. . . . . . . . . . . 12 |- ( 0 e. NN0 -> ( y = 0 -> y e. NN0 ) )
17	15, 16	ax-mp 7	. . . . . . . . . . 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 2124	. . . . . . . . . . . . . . . 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 2731	. . . . . . . . . . 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 272	. . . . . . . . 9 |- ( ( y = 0 /\ x = 1 ) -> th )
30		oveq1 5735	. . . . . . . . . . . . 13 |- ( y = 0 -> ( y + 1 ) = ( 0 + 1 ) )
31		0p1e1 8741	. . . . . . . . . . . . 13 |- ( 0 + 1 ) = 1
32	30, 31	syl6eq 2163	. . . . . . . . . . . 12 |- ( y = 0 -> ( y + 1 ) = 1 )
33	32	eqeq2d 2126	. . . . . . . . . . 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 2731	. . . . . 6 |- ( x = 1 -> ph )
39		sbceq1a 2887	. . . . . 6 |- ( x = 1 -> ( ph <-> [. 1 / x ]. ph ) )
40	38, 39	mpbid 146	. . . . 5 |- ( x = 1 -> [. 1 / x ]. ph )
41	12, 13, 40	vtoclef 2730	. . . 4 |- [. 1 / x ]. ph
42		nnnn0 8885	. . . . 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 8643	. . 3 |- ( A e. NN -> ta )
45		nfv 1491	. . . . 5 |- F/ x ( 0 = A -> ta )
46		eqeq1 2121	. . . . . 6 |- ( x = 0 -> ( x = A <-> 0 = A ) )
47	19	bicomd 140	. . . . . . . . 9 |- ( x = 0 -> ( ps <-> ph ) )
48	47, 10	sylan9bb 455	. . . . . . . 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 2730	. . . 4 |- ( 0 = A -> ta )
53	52	eqcoms 2118	. . 3 |- ( A = 0 -> ta )
54	44, 53	jaoi 688	. 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 680   = wceq 1314   e. wcel 1463   [wsb 1718   [.wsbc 2878  (class class class)co 5728   0cc0 7544   1c1 7545   + caddc 7547   NNcn 8627   NN0cn0 8878

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-cnex 7633  ax-resscn 7634  ax-1cn 7635  ax-1re 7636  ax-icn 7637  ax-addcl 7638  ax-addrcl 7639  ax-mulcl 7640  ax-addcom 7642  ax-i2m1 7647  ax-0id 7650

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-ss 3050  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-br 3896  df-iota 5046  df-fv 5089  df-ov 5731  df-inn 8628  df-n0 8879

This theorem is referenced by: (None)