Theorem nn0ind-raph 6214

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 6101	. 2 |- (A e. NN0 <-> (A e. NN \/ A = 0))
2		dfsbcq 1943	. . . 4 |- (z = 1 -> ([z / x]ph <-> [1 / x]ph))
3		nn0ind-raph.2	. . . . 5 |- (x = y -> (ph <-> ch))
4	3	sbhyp 1940	. . . 4 |- (z = y -> ([z / x]ph <-> ch))
5		nn0ind-raph.3	. . . . 5 |- (x = (y + 1) -> (ph <-> th))
6	5	sbhyp 1940	. . . 4 |- (z = (y + 1) -> ([z / x]ph <-> th))
7		nn0ind-raph.4	. . . . 5 |- (x = A -> (ph <-> ta))
8	7	sbhyp 1940	. . . 4 |- (z = A -> ([z / x]ph <-> ta))
9		1re 5435	. . . . . . 7 |- 1 e. RR
10	9	elisseti 1818	. . . . . 6 |- 1 e. V
11	10	hbsbc1v 1950	. . . . 5 |- ([1 / x]ph -> A.x[1 / x]ph)
12		0nn0 6113	. . . . . . . 8 |- 0 e. NN0
13	12	elisseti 1818	. . . . . . 7 |- 0 e. V
14		nn0ind-raph.6	. . . . . . . . . . 11 |- (y e. NN0 -> (ch -> th))
15		eleq1a 1543	. . . . . . . . . . . 12 |- (0 e. NN0 -> (y = 0 -> y e. NN0))
16	12, 15	ax-mp 7	. . . . . . . . . . 11 |- (y = 0 -> y e. NN0)
17		nn0ind-raph.5	. . . . . . . . . . . . . . 15 |- ps
18		nn0ind-raph.1	. . . . . . . . . . . . . . 15 |- (x = 0 -> (ph <-> ps))
19	17, 18	mpbiri 194	. . . . . . . . . . . . . 14 |- (x = 0 -> ph)
20		eqeq2 1484	. . . . . . . . . . . . . . . 16 |- (y = 0 -> (x = y <-> x = 0))
21	20, 3	syl6bir 215	. . . . . . . . . . . . . . 15 |- (y = 0 -> (x = 0 -> (ph <-> ch)))
22	21	pm5.74d 585	. . . . . . . . . . . . . 14 |- (y = 0 -> ((x = 0 -> ph) <-> (x = 0 -> ch)))
23	19, 22	mpbii 193	. . . . . . . . . . . . 13 |- (y = 0 -> (x = 0 -> ch))
24	23	com12 11	. . . . . . . . . . . 12 |- (x = 0 -> (y = 0 -> ch))
25	13, 24	vtocle 1858	. . . . . . . . . . 11 |- (y = 0 -> ch)
26	14, 16, 25	sylc 68	. . . . . . . . . 10 |- (y = 0 -> th)
27	26	adantr 389	. . . . . . . . 9 |- ((y = 0 /\ x = 1) -> th)
28		opreq1 3968	. . . . . . . . . . . . 13 |- (y = 0 -> (y + 1) = (0 + 1))
29		ax1cn 5269	. . . . . . . . . . . . . 14 |- 1 e. CC
30	29	addid2 5331	. . . . . . . . . . . . 13 |- (0 + 1) = 1
31	28, 30	syl6eq 1523	. . . . . . . . . . . 12 |- (y = 0 -> (y + 1) = 1)
32	31	eqeq2d 1486	. . . . . . . . . . 11 |- (y = 0 -> (x = (y + 1) <-> x = 1))
33	32, 5	syl6bir 215	. . . . . . . . . 10 |- (y = 0 -> (x = 1 -> (ph <-> th)))
34	33	imp 350	. . . . . . . . 9 |- ((y = 0 /\ x = 1) -> (ph <-> th))
35	27, 34	mpbird 196	. . . . . . . 8 |- ((y = 0 /\ x = 1) -> ph)
36	35	ex 373	. . . . . . 7 |- (y = 0 -> (x = 1 -> ph))
37	13, 36	vtocle 1858	. . . . . 6 |- (x = 1 -> ph)
38		sbceq1a 1944	. . . . . 6 |- (x = 1 -> (ph <-> [1 / x]ph))
39	37, 38	mpbid 195	. . . . 5 |- (x = 1 -> [1 / x]ph)
40	11, 10, 39	vtoclef 1857	. . . 4 |- [1 / x]ph
41		nnnn0t 6106	. . . . 5 |- (y e. NN -> y e. NN0)
42	41, 14	syl 10	. . . 4 |- (y e. NN -> (ch -> th))
43	2, 4, 6, 8, 40, 42	nnind 5937	. . 3 |- (A e. NN -> ta)
44		ax-17 971	. . . . . 6 |- (0 = A -> A.x0 = A)
45		ax-17 971	. . . . . 6 |- (ta -> A.xta)
46	44, 45	hbim 1007	. . . . 5 |- ((0 = A -> ta) -> A.x(0 = A -> ta))
47		eqeq1 1481	. . . . . 6 |- (x = 0 -> (x = A <-> 0 = A))
48	18	bicomd 521	. . . . . . . . 9 |- (x = 0 -> (ps <-> ph))
49	48, 7	sylan9bb 540	. . . . . . . 8 |- ((x = 0 /\ x = A) -> (ps <-> ta))
50	17, 49	mpbii 193	. . . . . . 7 |- ((x = 0 /\ x = A) -> ta)
51	50	ex 373	. . . . . 6 |- (x = 0 -> (x = A -> ta))
52	47, 51	sylbird 205	. . . . 5 |- (x = 0 -> (0 = A -> ta))
53	46, 13, 52	vtoclef 1857	. . . 4 |- (0 = A -> ta)
54	53	eqcoms 1478	. . 3 |- (A = 0 -> ta)
55	43, 54	jaoi 341	. 2 |- ((A e. NN \/ A = 0) -> ta)
56	1, 55	sylbi 199	1 |- (A e. NN0 -> ta)

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 956   e. wcel 958  [wsbc 1170  (class class class)co 3963  RRcr 5233  0cc0 5234  1c1 5235   + caddc 5237  NNcn 5296  NN0cn0 5297

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-sub 5356  df-neg 5358  df-n 5925  df-n0 6100

Copyright terms: Public domain