Theorem nn0ind-raph 6385

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 hypothesis. (Contributed by Raph Levien, 10-Apr-2004. Raph says: "This seems a bit painful. I wonder if an explicit substitution version would be easier.")

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

Proof of Theorem nn0ind-raph

Step	Hyp	Ref	Expression
1		elnn0 6269	. 2 |- (A e. NN0 <-> (A e. NN \/ A = 0))
2		dfsbcq 1988	. . . 4 |- (z = 1 -> ([z / x]ph <-> [1 / x]ph))
3		ax-17 1007	. . . . 5 |- (ch -> A.xch)
4		nn0ind-raph.2	. . . . 5 |- (x = y -> (ph <-> ch))
5	3, 4	sbhypf 1985	. . . 4 |- (z = y -> ([z / x]ph <-> ch))
6		ax-17 1007	. . . . 5 |- (th -> A.xth)
7		nn0ind-raph.3	. . . . 5 |- (x = (y + 1) -> (ph <-> th))
8	6, 7	sbhypf 1985	. . . 4 |- (z = (y + 1) -> ([z / x]ph <-> th))
9		ax-17 1007	. . . . 5 |- (ta -> A.xta)
10		nn0ind-raph.4	. . . . 5 |- (x = A -> (ph <-> ta))
11	9, 10	sbhypf 1985	. . . 4 |- (z = A -> ([z / x]ph <-> ta))
12		1re 5589	. . . . . . 7 |- 1 e. RR
13	12	elisseti 1864	. . . . . 6 |- 1 e. V
14	13	hbsbc1v 1995	. . . . 5 |- ([1 / x]ph -> A.x[1 / x]ph)
15		0nn0 6281	. . . . . . . 8 |- 0 e. NN0
16	15	elisseti 1864	. . . . . . 7 |- 0 e. V
17		nn0ind-raph.6	. . . . . . . . . . 11 |- (y e. NN0 -> (ch -> th))
18		eleq1a 1586	. . . . . . . . . . . 12 |- (0 e. NN0 -> (y = 0 -> y e. NN0))
19	15, 18	ax-mp 7	. . . . . . . . . . 11 |- (y = 0 -> y e. NN0)
20		nn0ind-raph.5	. . . . . . . . . . . . . . 15 |- ps
21		nn0ind-raph.1	. . . . . . . . . . . . . . 15 |- (x = 0 -> (ph <-> ps))
22	20, 21	mpbiri 192	. . . . . . . . . . . . . 14 |- (x = 0 -> ph)
23		eqeq2 1527	. . . . . . . . . . . . . . . 16 |- (y = 0 -> (x = y <-> x = 0))
24	23, 4	syl6bir 213	. . . . . . . . . . . . . . 15 |- (y = 0 -> (x = 0 -> (ph <-> ch)))
25	24	pm5.74d 588	. . . . . . . . . . . . . 14 |- (y = 0 -> ((x = 0 -> ph) <-> (x = 0 -> ch)))
26	22, 25	mpbii 191	. . . . . . . . . . . . 13 |- (y = 0 -> (x = 0 -> ch))
27	26	com12 11	. . . . . . . . . . . 12 |- (x = 0 -> (y = 0 -> ch))
28	16, 27	vtocle 1904	. . . . . . . . . . 11 |- (y = 0 -> ch)
29	17, 19, 28	sylc 68	. . . . . . . . . 10 |- (y = 0 -> th)
30	29	adantr 389	. . . . . . . . 9 |- ((y = 0 /\ x = 1) -> th)
31		opreq1 4026	. . . . . . . . . . . . 13 |- (y = 0 -> (y + 1) = (0 + 1))
32		ax1cn 5423	. . . . . . . . . . . . . 14 |- 1 e. CC
33	32	addid2i 5485	. . . . . . . . . . . . 13 |- (0 + 1) = 1
34	31, 33	syl6eq 1566	. . . . . . . . . . . 12 |- (y = 0 -> (y + 1) = 1)
35	34	eqeq2d 1529	. . . . . . . . . . 11 |- (y = 0 -> (x = (y + 1) <-> x = 1))
36	35, 7	syl6bir 213	. . . . . . . . . 10 |- (y = 0 -> (x = 1 -> (ph <-> th)))
37	36	imp 348	. . . . . . . . 9 |- ((y = 0 /\ x = 1) -> (ph <-> th))
38	30, 37	mpbird 194	. . . . . . . 8 |- ((y = 0 /\ x = 1) -> ph)
39	38	ex 371	. . . . . . 7 |- (y = 0 -> (x = 1 -> ph))
40	16, 39	vtocle 1904	. . . . . 6 |- (x = 1 -> ph)
41		sbceq1a 1989	. . . . . 6 |- (x = 1 -> (ph <-> [1 / x]ph))
42	40, 41	mpbid 193	. . . . 5 |- (x = 1 -> [1 / x]ph)
43	14, 13, 42	vtoclef 1903	. . . 4 |- [1 / x]ph
44		nnnn0 6274	. . . . 5 |- (y e. NN -> y e. NN0)
45	44, 17	syl 10	. . . 4 |- (y e. NN -> (ch -> th))
46	2, 5, 8, 11, 43, 45	nnind 6082	. . 3 |- (A e. NN -> ta)
47		ax-17 1007	. . . . . 6 |- (0 = A -> A.x0 = A)
48	47, 9	hbim 1043	. . . . 5 |- ((0 = A -> ta) -> A.x(0 = A -> ta))
49		eqeq1 1524	. . . . . 6 |- (x = 0 -> (x = A <-> 0 = A))
50	21	bicomd 524	. . . . . . . . 9 |- (x = 0 -> (ps <-> ph))
51	50, 10	sylan9bb 543	. . . . . . . 8 |- ((x = 0 /\ x = A) -> (ps <-> ta))
52	20, 51	mpbii 191	. . . . . . 7 |- ((x = 0 /\ x = A) -> ta)
53	52	ex 371	. . . . . 6 |- (x = 0 -> (x = A -> ta))
54	49, 53	sylbird 203	. . . . 5 |- (x = 0 -> (0 = A -> ta))
55	48, 16, 54	vtoclef 1903	. . . 4 |- (0 = A -> ta)
56	55	eqcoms 1521	. . 3 |- (A = 0 -> ta)
57	46, 56	jaoi 339	. 2 |- ((A e. NN \/ A = 0) -> ta)
58	1, 57	sylbi 197	1 |- (A e. NN0 -> ta)

Syntax hints:   -> wi 3   <-> wb 144   \/ wo 220   /\ wa 221   = wceq 992   e. wcel 994  [wsbc 1207  (class class class)co 4021  RRcr 5387  0cc0 5388  1c1 5389   + caddc 5391  NNcn 5450  NN0cn0 5451

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-ltr 5324  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400  df-sub 5510  df-neg 5512  df-n 6070  df-n0 6268

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