Theorem nn0ind-raph 10957

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 10793	. 2 |- ( A e. NN0 <-> ( A e. NN \/ A = 0 ) )
2		dfsbcq2 3334	. . . 4 |- ( z = 1 -> ( [ z / x ] ph <-> [. 1 / x ]. ph ) )
3		nfv 1683	. . . . 5 |- F/ x ch
4		nn0ind-raph.2	. . . . 5 |- ( x = y -> ( ph <-> ch ) )
5	3, 4	sbhypf 3160	. . . 4 |- ( z = y -> ( [ z / x ] ph <-> ch ) )
6		nfv 1683	. . . . 5 |- F/ x th
7		nn0ind-raph.3	. . . . 5 |- ( x = ( y + 1 ) -> ( ph <-> th ) )
8	6, 7	sbhypf 3160	. . . 4 |- ( z = ( y + 1 ) -> ( [ z / x ] ph <-> th ) )
9		nfv 1683	. . . . 5 |- F/ x ta
10		nn0ind-raph.4	. . . . 5 |- ( x = A -> ( ph <-> ta ) )
11	9, 10	sbhypf 3160	. . . 4 |- ( z = A -> ( [ z / x ] ph <-> ta ) )
12		nfsbc1v 3351	. . . . 5 |- F/ x [. 1 / x ]. ph
13		1ex 9587	. . . . 5 |- 1 e. _V
14		c0ex 9586	. . . . . . 7 |- 0 e. _V
15		0nn0 10806	. . . . . . . . . . . 12 |- 0 e. NN0
16		eleq1a 2550	. . . . . . . . . . . 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 233	. . . . . . . . . . . . . 14 |- ( x = 0 -> ph )
21		eqeq2 2482	. . . . . . . . . . . . . . . 16 |- ( y = 0 -> ( x = y <-> x = 0 ) )
22	21, 4	syl6bir 229	. . . . . . . . . . . . . . 15 |- ( y = 0 -> ( x = 0 -> ( ph <-> ch ) ) )
23	22	pm5.74d 247	. . . . . . . . . . . . . 14 |- ( y = 0 -> ( ( x = 0 -> ph ) <-> ( x = 0 -> ch ) ) )
24	20, 23	mpbii 211	. . . . . . . . . . . . 13 |- ( y = 0 -> ( x = 0 -> ch ) )
25	24	com12 31	. . . . . . . . . . . 12 |- ( x = 0 -> ( y = 0 -> ch ) )
26	14, 25	vtocle 3187	. . . . . . . . . . 11 |- ( y = 0 -> ch )
27		nn0ind-raph.6	. . . . . . . . . . 11 |- ( y e. NN0 -> ( ch -> th ) )
28	17, 26, 27	sylc 60	. . . . . . . . . 10 |- ( y = 0 -> th )
29	28	adantr 465	. . . . . . . . 9 |- ( ( y = 0 /\ x = 1 ) -> th )
30		oveq1 6289	. . . . . . . . . . . . 13 |- ( y = 0 -> ( y + 1 ) = ( 0 + 1 ) )
31		0p1e1 10643	. . . . . . . . . . . . 13 |- ( 0 + 1 ) = 1
32	30, 31	syl6eq 2524	. . . . . . . . . . . 12 |- ( y = 0 -> ( y + 1 ) = 1 )
33	32	eqeq2d 2481	. . . . . . . . . . 11 |- ( y = 0 -> ( x = ( y + 1 ) <-> x = 1 ) )
34	33, 7	syl6bir 229	. . . . . . . . . 10 |- ( y = 0 -> ( x = 1 -> ( ph <-> th ) ) )
35	34	imp 429	. . . . . . . . 9 |- ( ( y = 0 /\ x = 1 ) -> ( ph <-> th ) )
36	29, 35	mpbird 232	. . . . . . . 8 |- ( ( y = 0 /\ x = 1 ) -> ph )
37	36	ex 434	. . . . . . 7 |- ( y = 0 -> ( x = 1 -> ph ) )
38	14, 37	vtocle 3187	. . . . . 6 |- ( x = 1 -> ph )
39		sbceq1a 3342	. . . . . 6 |- ( x = 1 -> ( ph <-> [. 1 / x ]. ph ) )
40	38, 39	mpbid 210	. . . . 5 |- ( x = 1 -> [. 1 / x ]. ph )
41	12, 13, 40	vtoclef 3186	. . . 4 |- [. 1 / x ]. ph
42		nnnn0 10798	. . . . 5 |- ( y e. NN -> y e. NN0 )
43	42, 27	syl 16	. . . 4 |- ( y e. NN -> ( ch -> th ) )
44	2, 5, 8, 11, 41, 43	nnind 10550	. . 3 |- ( A e. NN -> ta )
45		nfv 1683	. . . . 5 |- F/ x ( 0 = A -> ta )
46		eqeq1 2471	. . . . . 6 |- ( x = 0 -> ( x = A <-> 0 = A ) )
47	19	bicomd 201	. . . . . . . . 9 |- ( x = 0 -> ( ps <-> ph ) )
48	47, 10	sylan9bb 699	. . . . . . . 8 |- ( ( x = 0 /\ x = A ) -> ( ps <-> ta ) )
49	18, 48	mpbii 211	. . . . . . 7 |- ( ( x = 0 /\ x = A ) -> ta )
50	49	ex 434	. . . . . 6 |- ( x = 0 -> ( x = A -> ta ) )
51	46, 50	sylbird 235	. . . . 5 |- ( x = 0 -> ( 0 = A -> ta ) )
52	45, 14, 51	vtoclef 3186	. . . 4 |- ( 0 = A -> ta )
53	52	eqcoms 2479	. . 3 |- ( A = 0 -> ta )
54	44, 53	jaoi 379	. 2 |- ( ( A e. NN \/ A = 0 ) -> ta )
55	1, 54	sylbi 195	1 |- ( A e. NN0 -> ta )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1379   [wsb 1711   e. wcel 1767   [.wsbc 3331  (class class class)co 6282   0cc0 9488   1c1 9489   + caddc 9491   NNcn 10532   NN0cn0 10791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-ltxr 9629  df-nn 10533  df-n0 10792

This theorem is referenced by: (None)