Theorem nn0ind-raph 7426

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 7310	. 2 |- (A e. NN0 <-> (A e. NN \/ A = 0))
2		dfsbcq 2455	. . . 4 |- (z = 1 -> ([z / x]ph <-> [1 / x]ph))
3		ax-17 1317	. . . . 5 |- (ch -> A.xch)
4		nn0ind-raph.2	. . . . 5 |- (x = y -> (ph <-> ch))
5	3, 4	sbhypf 2452	. . . 4 |- (z = y -> ([z / x]ph <-> ch))
6		ax-17 1317	. . . . 5 |- (th -> A.xth)
7		nn0ind-raph.3	. . . . 5 |- (x = (y + 1) -> (ph <-> th))
8	6, 7	sbhypf 2452	. . . 4 |- (z = (y + 1) -> ([z / x]ph <-> th))
9		ax-17 1317	. . . . 5 |- (ta -> A.xta)
10		nn0ind-raph.4	. . . . 5 |- (x = A -> (ph <-> ta))
11	9, 10	sbhypf 2452	. . . 4 |- (z = A -> ([z / x]ph <-> ta))
12		1re 6598	. . . . . . 7 |- 1 e. RR
13	12	elisseti 2301	. . . . . 6 |- 1 e. _V
14	13	hbsbc1v 2464	. . . . 5 |- ([1 / x]ph -> A.x[1 / x]ph)
15		0nn0 7322	. . . . . . . 8 |- 0 e. NN0
16	15	elisseti 2301	. . . . . . 7 |- 0 e. _V
17		eleq1a 1966	. . . . . . . . . . . 12 |- (0 e. NN0 -> (y = 0 -> y e. NN0))
18	15, 17	ax-mp 7	. . . . . . . . . . 11 |- (y = 0 -> y e. NN0)
19		nn0ind-raph.5	. . . . . . . . . . . . . . 15 |- ps
20		nn0ind-raph.1	. . . . . . . . . . . . . . 15 |- (x = 0 -> (ph <-> ps))
21	19, 20	mpbiri 211	. . . . . . . . . . . . . 14 |- (x = 0 -> ph)
22		eqeq2 1893	. . . . . . . . . . . . . . . 16 |- (y = 0 -> (x = y <-> x = 0))
23	22, 4	syl6bir 232	. . . . . . . . . . . . . . 15 |- (y = 0 -> (x = 0 -> (ph <-> ch)))
24	23	pm5.74d 645	. . . . . . . . . . . . . 14 |- (y = 0 -> ((x = 0 -> ph) <-> (x = 0 -> ch)))
25	21, 24	mpbii 210	. . . . . . . . . . . . 13 |- (y = 0 -> (x = 0 -> ch))
26	25	com12 14	. . . . . . . . . . . 12 |- (x = 0 -> (y = 0 -> ch))
27	16, 26	vtocle 2359	. . . . . . . . . . 11 |- (y = 0 -> ch)
28		nn0ind-raph.6	. . . . . . . . . . 11 |- (y e. NN0 -> (ch -> th))
29	18, 27, 28	sylc 83	. . . . . . . . . 10 |- (y = 0 -> th)
30	29	adantr 425	. . . . . . . . 9 |- ((y = 0 /\ x = 1) -> th)
31		opreq1 4889	. . . . . . . . . . . . 13 |- (y = 0 -> (y + 1) = (0 + 1))
32		ax1cn 6422	. . . . . . . . . . . . . 14 |- 1 e. CC
33	32	addid2i 6484	. . . . . . . . . . . . 13 |- (0 + 1) = 1
34	31, 33	syl6eq 1944	. . . . . . . . . . . 12 |- (y = 0 -> (y + 1) = 1)
35	34	eqeq2d 1895	. . . . . . . . . . 11 |- (y = 0 -> (x = (y + 1) <-> x = 1))
36	35, 7	syl6bir 232	. . . . . . . . . 10 |- (y = 0 -> (x = 1 -> (ph <-> th)))
37	36	imp 377	. . . . . . . . 9 |- ((y = 0 /\ x = 1) -> (ph <-> th))
38	30, 37	mpbird 213	. . . . . . . 8 |- ((y = 0 /\ x = 1) -> ph)
39	38	ex 402	. . . . . . 7 |- (y = 0 -> (x = 1 -> ph))
40	16, 39	vtocle 2359	. . . . . 6 |- (x = 1 -> ph)
41		sbceq1a 2456	. . . . . 6 |- (x = 1 -> (ph <-> [1 / x]ph))
42	40, 41	mpbid 212	. . . . 5 |- (x = 1 -> [1 / x]ph)
43	14, 13, 42	vtoclef 2358	. . . 4 |- [1 / x]ph
44		nnnn0 7315	. . . . 5 |- (y e. NN -> y e. NN0)
45	44, 28	syl 12	. . . 4 |- (y e. NN -> (ch -> th))
46	2, 5, 8, 11, 43, 45	nnind 7120	. . 3 |- (A e. NN -> ta)
47		ax-17 1317	. . . . . 6 |- (0 = A -> A.x0 = A)
48	47, 9	hbim 1354	. . . . 5 |- ((0 = A -> ta) -> A.x(0 = A -> ta))
49		eqeq1 1890	. . . . . 6 |- (x = 0 -> (x = A <-> 0 = A))
50	20	bicomd 580	. . . . . . . . 9 |- (x = 0 -> (ps <-> ph))
51	50, 10	sylan9bb 599	. . . . . . . 8 |- ((x = 0 /\ x = A) -> (ps <-> ta))
52	19, 51	mpbii 210	. . . . . . 7 |- ((x = 0 /\ x = A) -> ta)
53	52	ex 402	. . . . . 6 |- (x = 0 -> (x = A -> ta))
54	49, 53	sylbird 222	. . . . 5 |- (x = 0 -> (0 = A -> ta))
55	48, 16, 54	vtoclef 2358	. . . 4 |- (0 = A -> ta)
56	55	eqcoms 1887	. . 3 |- (A = 0 -> ta)
57	46, 56	jaoi 368	. 2 |- ((A e. NN \/ A = 0) -> ta)
58	1, 57	sylbi 216	1 |- (A e. NN0 -> ta)

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  [wsbc 1534  (class class class)co 4884  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389  NNcn 6449  NN0cn0 6450

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-sub 6511  df-neg 6513  df-n 7108  df-n0 7309

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