Theorem nn0ind-raph 6170

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 6056	. 2 |- (A e. NN0 <-> (A e. NN \/ A = 0))
2		dfsbcq 1939	. . . 4 |- (z = 1 -> ([z / x]ph <-> [1 / x]ph))
3		nn0ind-raph.2	. . . . 5 |- (x = y -> (ph <-> ch))
4	3	sbhyp 1936	. . . 4 |- (z = y -> ([z / x]ph <-> ch))
5		nn0ind-raph.3	. . . . 5 |- (x = (y + 1) -> (ph <-> th))
6	5	sbhyp 1936	. . . 4 |- (z = (y + 1) -> ([z / x]ph <-> th))
7		nn0ind-raph.4	. . . . 5 |- (x = A -> (ph <-> ta))
8	7	sbhyp 1936	. . . 4 |- (z = A -> ([z / x]ph <-> ta))
9		1re 5415	. . . . . . 7 |- 1 e. RR
10	9	elisseti 1814	. . . . . 6 |- 1 e. V
11	10	hbsbc1v 1946	. . . . 5 |- ([1 / x]ph -> A.x[1 / x]ph)
12		0nn0 6068	. . . . . . . 8 |- 0 e. NN0
13	12	elisseti 1814	. . . . . . 7 |- 0 e. V
14		nn0ind-raph.6	. . . . . . . . . . 11 |- (y e. NN0 -> (ch -> th))
15		eleq1a 1540	. . . . . . . . . . . 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 1481	. . . . . . . . . . . . . . . 16 |- (y = 0 -> (x = y <-> x = 0))
21	20, 3	syl6bir 215	. . . . . . . . . . . . . . 15 |- (y = 0 -> (x = 0 -> (ph <-> ch)))
22	21	pm5.74d 584	. . . . . . . . . . . . . 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 1854	. . . . . . . . . . 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 3959	. . . . . . . . . . . . 13 |- (y = 0 -> (y + 1) = (0 + 1))
29		ax1cn 5249	. . . . . . . . . . . . . 14 |- 1 e. CC
30	29	addid2 5311	. . . . . . . . . . . . 13 |- (0 + 1) = 1
31	28, 30	syl6eq 1520	. . . . . . . . . . . 12 |- (y = 0 -> (y + 1) = 1)
32	31	eqeq2d 1483	. . . . . . . . . . 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 1854	. . . . . 6 |- (x = 1 -> ph)
38		sbceq1a 1940	. . . . . 6 |- (x = 1 -> (ph <-> [1 / x]ph))
39	37, 38	mpbid 195	. . . . 5 |- (x = 1 -> [1 / x]ph)
40	11, 10, 39	vtoclef 1853	. . . 4 |- [1 / x]ph
41		nnnn0t 6061	. . . . 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 5893	. . 3 |- (A e. NN -> ta)
44		ax-17 969	. . . . . 6 |- (0 = A -> A.x0 = A)
45		ax-17 969	. . . . . 6 |- (ta -> A.xta)
46	44, 45	hbim 1005	. . . . 5 |- ((0 = A -> ta) -> A.x(0 = A -> ta))
47		eqeq1 1478	. . . . . 6 |- (x = 0 -> (x = A <-> 0 = A))
48	18	bicomd 520	. . . . . . . . 9 |- (x = 0 -> (ps <-> ph))
49	48, 7	sylan9bb 539	. . . . . . . 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 1853	. . . 4 |- (0 = A -> ta)
54	53	eqcoms 1475	. . 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 954   e. wcel 956  [wsbc 1168  (class class class)co 3954  RRcr 5213  0cc0 5214  1c1 5215   + caddc 5217  NNcn 5276  NN0cn0 5277

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-sub 5336  df-neg 5338  df-n 5881  df-n0 6055

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