Theorem nn0ind-raph 10742

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 10581	. 2 |- ( A e. NN0 <-> ( A e. NN \/ A = 0 ) )
2		dfsbcq2 3189	. . . 4 |- ( z = 1 -> ( [ z / x ] ph <-> [. 1 / x ]. ph ) )
3		nfv 1673	. . . . 5 |- F/ x ch
4		nn0ind-raph.2	. . . . 5 |- ( x = y -> ( ph <-> ch ) )
5	3, 4	sbhypf 3019	. . . 4 |- ( z = y -> ( [ z / x ] ph <-> ch ) )
6		nfv 1673	. . . . 5 |- F/ x th
7		nn0ind-raph.3	. . . . 5 |- ( x = ( y + 1 ) -> ( ph <-> th ) )
8	6, 7	sbhypf 3019	. . . 4 |- ( z = ( y + 1 ) -> ( [ z / x ] ph <-> th ) )
9		nfv 1673	. . . . 5 |- F/ x ta
10		nn0ind-raph.4	. . . . 5 |- ( x = A -> ( ph <-> ta ) )
11	9, 10	sbhypf 3019	. . . 4 |- ( z = A -> ( [ z / x ] ph <-> ta ) )
12		nfsbc1v 3206	. . . . 5 |- F/ x [. 1 / x ]. ph
13		1ex 9381	. . . . 5 |- 1 e. _V
14		c0ex 9380	. . . . . . 7 |- 0 e. _V
15		0nn0 10594	. . . . . . . . . . . 12 |- 0 e. NN0
16		eleq1a 2512	. . . . . . . . . . . 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 2452	. . . . . . . . . . . . . . . 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 3046	. . . . . . . . . . 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 6098	. . . . . . . . . . . . 13 |- ( y = 0 -> ( y + 1 ) = ( 0 + 1 ) )
31		0p1e1 10433	. . . . . . . . . . . . 13 |- ( 0 + 1 ) = 1
32	30, 31	syl6eq 2491	. . . . . . . . . . . 12 |- ( y = 0 -> ( y + 1 ) = 1 )
33	32	eqeq2d 2454	. . . . . . . . . . 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 3046	. . . . . 6 |- ( x = 1 -> ph )
39		sbceq1a 3197	. . . . . 6 |- ( x = 1 -> ( ph <-> [. 1 / x ]. ph ) )
40	38, 39	mpbid 210	. . . . 5 |- ( x = 1 -> [. 1 / x ]. ph )
41	12, 13, 40	vtoclef 3045	. . . 4 |- [. 1 / x ]. ph
42		nnnn0 10586	. . . . 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 10340	. . 3 |- ( A e. NN -> ta )
45		nfv 1673	. . . . 5 |- F/ x ( 0 = A -> ta )
46		eqeq1 2449	. . . . . 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 3045	. . . 4 |- ( 0 = A -> ta )
53	52	eqcoms 2446	. . 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 1369   [wsb 1700   e. wcel 1756   [.wsbc 3186  (class class class)co 6091   0cc0 9282   1c1 9283   + caddc 9285   NNcn 10322   NN0cn0 10579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-om 6477  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-ltxr 9423  df-nn 10323  df-n0 10580

This theorem is referenced by: (None)