Theorem indpi 6326

Description: Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.)

Hypotheses

Ref	Expression
indpi.1	|- (x = 1o -> (ph <-> ps))
indpi.2	|- (x = y -> (ph <-> ch))
indpi.3	|- (x = (y +N 1o) -> (ph <-> th))
indpi.4	|- (x = A -> (ph <-> ta))
indpi.5	|- ps
indpi.6	|- (y e. N. -> (ch -> th))

Assertion

Ref	Expression
indpi	|- (A e. N. -> 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 indpi

Step	Hyp	Ref	Expression
1		indpi.4	. 2 |- (x = A -> (ph <-> ta))
2		elni 6292	. . 3 |- (x e. N. <-> (x e. om /\ x =/= (/)))
3		eqid 2037	. . . . . . . . . 10 |- (/) = (/)
4	3	orci 649	. . . . . . . . 9 |- ( (/) = (/) \/ [. (/) / x ].ph)
5		nfv 1418	. . . . . . . . . . 11 |- F/x (/) = (/)
6		nfsbc1v 2776	. . . . . . . . . . 11 |- F/x [. (/) / x ].ph
7	5, 6	nfor 1463	. . . . . . . . . 10 |- F/x( (/) = (/) \/ [. (/) / x ].ph)
8		0ex 3875	. . . . . . . . . 10 |- (/) e. _V
9		eqeq1 2043	. . . . . . . . . . 11 |- (x = (/) -> (x = (/) <-> (/) = (/)))
10		sbceq1a 2767	. . . . . . . . . . 11 |- (x = (/) -> (ph <-> [. (/) / x ].ph))
11	9, 10	orbi12d 706	. . . . . . . . . 10 |- (x = (/) -> ((x = (/) \/ ph) <-> ( (/) = (/) \/ [. (/) / x ].ph)))
12	7, 8, 11	elabf 2680	. . . . . . . . 9 |- ( (/) e. {x | (x = (/) \/ ph) } <-> ( (/) = (/) \/ [. (/) / x ].ph))
13	4, 12	mpbir 134	. . . . . . . 8 |- (/) e. {x | (x = (/) \/ ph) }
14		suceq 4105	. . . . . . . . . . . . . 14 |- (y = (/) -> suc y = suc (/))
15		df-1o 5940	. . . . . . . . . . . . . 14 |- 1o = suc (/)
16	14, 15	syl6eqr 2087	. . . . . . . . . . . . 13 |- (y = (/) -> suc y = 1o)
17		indpi.5	. . . . . . . . . . . . . . 15 |- ps
18	17	olci 650	. . . . . . . . . . . . . 14 |- ( 1o = (/) \/ ps)
19		1onn 6029	. . . . . . . . . . . . . . . 16 |- 1o e. om
20	19	elexi 2561	. . . . . . . . . . . . . . 15 |- 1o e. _V
21		eqeq1 2043	. . . . . . . . . . . . . . . 16 |- (x = 1o -> (x = (/) <-> 1o = (/)))
22		indpi.1	. . . . . . . . . . . . . . . 16 |- (x = 1o -> (ph <-> ps))
23	21, 22	orbi12d 706	. . . . . . . . . . . . . . 15 |- (x = 1o -> ((x = (/) \/ ph) <-> ( 1o = (/) \/ ps)))
24	20, 23	elab 2681	. . . . . . . . . . . . . 14 |- ( 1o e. {x | (x = (/) \/ ph) } <-> ( 1o = (/) \/ ps))
25	18, 24	mpbir 134	. . . . . . . . . . . . 13 |- 1o e. {x | (x = (/) \/ ph) }
26	16, 25	syl6eqel 2125	. . . . . . . . . . . 12 |- (y = (/) -> suc y e. {x | (x = (/) \/ ph) })
27	26	a1d 22	. . . . . . . . . . 11 |- (y = (/) -> (y e. {x | (x = (/) \/ ph) } -> suc y e. {x | (x = (/) \/ ph) }))
28	27	a1i 9	. . . . . . . . . 10 |- (y e. om -> (y = (/) -> (y e. {x | (x = (/) \/ ph) } -> suc y e. {x | (x = (/) \/ ph) })))
29		indpi.6	. . . . . . . . . . . 12 |- (y e. N. -> (ch -> th))
30		elni 6292	. . . . . . . . . . . . . . . 16 |- (y e. N. <-> (y e. om /\ y =/= (/)))
31	30	simprbi 260	. . . . . . . . . . . . . . 15 |- (y e. N. -> y =/= (/))
32	31	neneqd 2221	. . . . . . . . . . . . . 14 |- (y e. N. -> -. y = (/))
33		biorf 662	. . . . . . . . . . . . . 14 |- (-. y = (/) -> (ch <-> (y = (/) \/ ch)))
34	32, 33	syl 14	. . . . . . . . . . . . 13 |- (y e. N. -> (ch <-> (y = (/) \/ ch)))
35		vex 2554	. . . . . . . . . . . . . 14 |- y e. _V
36		eqeq1 2043	. . . . . . . . . . . . . . 15 |- (x = y -> (x = (/) <-> y = (/)))
37		indpi.2	. . . . . . . . . . . . . . 15 |- (x = y -> (ph <-> ch))
38	36, 37	orbi12d 706	. . . . . . . . . . . . . 14 |- (x = y -> ((x = (/) \/ ph) <-> (y = (/) \/ ch)))
39	35, 38	elab 2681	. . . . . . . . . . . . 13 |- (y e. {x | (x = (/) \/ ph) } <-> (y = (/) \/ ch))
40	34, 39	syl6bbr 187	. . . . . . . . . . . 12 |- (y e. N. -> (ch <-> y e. {x | (x = (/) \/ ph) }))
41		1pi 6299	. . . . . . . . . . . . . . . . . 18 |- 1o e. N.
42		addclpi 6311	. . . . . . . . . . . . . . . . . 18 |- ((y e. N. /\ 1o e. N.) -> (y +N 1o) e. N.)
43	41, 42	mpan2 401	. . . . . . . . . . . . . . . . 17 |- (y e. N. -> (y +N 1o) e. N.)
44		elni 6292	. . . . . . . . . . . . . . . . 17 |- ((y +N 1o) e. N. <-> ((y +N 1o) e. om /\ (y +N 1o) =/= (/)))
45	43, 44	sylib 127	. . . . . . . . . . . . . . . 16 |- (y e. N. -> ((y +N 1o) e. om /\ (y +N 1o) =/= (/)))
46	45	simprd 107	. . . . . . . . . . . . . . 15 |- (y e. N. -> (y +N 1o) =/= (/))
47	46	neneqd 2221	. . . . . . . . . . . . . 14 |- (y e. N. -> -. (y +N 1o) = (/))
48		biorf 662	. . . . . . . . . . . . . 14 |- (-. (y +N 1o) = (/) -> (th <-> ((y +N 1o) = (/) \/ th)))
49	47, 48	syl 14	. . . . . . . . . . . . 13 |- (y e. N. -> (th <-> ((y +N 1o) = (/) \/ th)))
50		eqeq1 2043	. . . . . . . . . . . . . . . 16 |- (x = (y +N 1o) -> (x = (/) <-> (y +N 1o) = (/)))
51		indpi.3	. . . . . . . . . . . . . . . 16 |- (x = (y +N 1o) -> (ph <-> th))
52	50, 51	orbi12d 706	. . . . . . . . . . . . . . 15 |- (x = (y +N 1o) -> ((x = (/) \/ ph) <-> ((y +N 1o) = (/) \/ th)))
53	52	elabg 2682	. . . . . . . . . . . . . 14 |- ((y +N 1o) e. N. -> ((y +N 1o) e. {x | (x = (/) \/ ph) } <-> ((y +N 1o) = (/) \/ th)))
54	43, 53	syl 14	. . . . . . . . . . . . 13 |- (y e. N. -> ((y +N 1o) e. {x | (x = (/) \/ ph) } <-> ((y +N 1o) = (/) \/ th)))
55		addpiord 6300	. . . . . . . . . . . . . . . 16 |- ((y e. N. /\ 1o e. N.) -> (y +N 1o) = (y +o 1o))
56	41, 55	mpan2 401	. . . . . . . . . . . . . . 15 |- (y e. N. -> (y +N 1o) = (y +o 1o))
57		pion 6294	. . . . . . . . . . . . . . . 16 |- (y e. N. -> y e. On)
58		oa1suc 5986	. . . . . . . . . . . . . . . 16 |- (y e. On -> (y +o 1o) = suc y)
59	57, 58	syl 14	. . . . . . . . . . . . . . 15 |- (y e. N. -> (y +o 1o) = suc y)
60	56, 59	eqtrd 2069	. . . . . . . . . . . . . 14 |- (y e. N. -> (y +N 1o) = suc y)
61	60	eleq1d 2103	. . . . . . . . . . . . 13 |- (y e. N. -> ((y +N 1o) e. {x | (x = (/) \/ ph) } <-> suc y e. {x | (x = (/) \/ ph) }))
62	49, 54, 61	3bitr2d 205	. . . . . . . . . . . 12 |- (y e. N. -> (th <-> suc y e. {x | (x = (/) \/ ph) }))
63	29, 40, 62	3imtr3d 191	. . . . . . . . . . 11 |- (y e. N. -> (y e. {x | (x = (/) \/ ph) } -> suc y e. {x | (x = (/) \/ ph) }))
64	63	a1i 9	. . . . . . . . . 10 |- (y e. om -> (y e. N. -> (y e. {x | (x = (/) \/ ph) } -> suc y e. {x | (x = (/) \/ ph) })))
65		nndceq0 4282	. . . . . . . . . . . 12 |- (y e. om -> DECID y = (/))
66		df-dc 742	. . . . . . . . . . . 12 |- (DECID y = (/) <-> (y = (/) \/ -. y = (/)))
67	65, 66	sylib 127	. . . . . . . . . . 11 |- (y e. om -> (y = (/) \/ -. y = (/)))
68		idd 21	. . . . . . . . . . . . . . 15 |- (y e. om -> (y = (/) -> y = (/)))
69	68	necon3bd 2242	. . . . . . . . . . . . . 14 |- (y e. om -> (-. y = (/) -> y =/= (/)))
70	69	anc2li 312	. . . . . . . . . . . . 13 |- (y e. om -> (-. y = (/) -> (y e. om /\ y =/= (/))))
71	70, 30	syl6ibr 151	. . . . . . . . . . . 12 |- (y e. om -> (-. y = (/) -> y e. N.))
72	71	orim2d 701	. . . . . . . . . . 11 |- (y e. om -> ((y = (/) \/ -. y = (/)) -> (y = (/) \/ y e. N.)))
73	67, 72	mpd 13	. . . . . . . . . 10 |- (y e. om -> (y = (/) \/ y e. N.))
74	28, 64, 73	mpjaod 637	. . . . . . . . 9 |- (y e. om -> (y e. {x | (x = (/) \/ ph) } -> suc y e. {x | (x = (/) \/ ph) }))
75	74	rgen 2368	. . . . . . . 8 |- A.y e. om (y e. {x | (x = (/) \/ ph) } -> suc y e. {x | (x = (/) \/ ph) })
76		peano5 4264	. . . . . . . 8 |- (( (/) e. {x | (x = (/) \/ ph) } /\ A.y e. om (y e. {x | (x = (/) \/ ph) } -> suc y e. {x | (x = (/) \/ ph) })) -> om C_ {x | (x = (/) \/ ph) })
77	13, 75, 76	mp2an 402	. . . . . . 7 |- om C_ {x | (x = (/) \/ ph) }
78	77	sseli 2935	. . . . . 6 |- (x e. om -> x e. {x | (x = (/) \/ ph) })
79		abid 2025	. . . . . 6 |- (x e. {x | (x = (/) \/ ph) } <-> (x = (/) \/ ph))
80	78, 79	sylib 127	. . . . 5 |- (x e. om -> (x = (/) \/ ph))
81	80	adantr 261	. . . 4 |- ((x e. om /\ x =/= (/)) -> (x = (/) \/ ph))
82		df-ne 2203	. . . . . 6 |- (x =/= (/) <-> -. x = (/))
83		biorf 662	. . . . . 6 |- (-. x = (/) -> (ph <-> (x = (/) \/ ph)))
84	82, 83	sylbi 114	. . . . 5 |- (x =/= (/) -> (ph <-> (x = (/) \/ ph)))
85	84	adantl 262	. . . 4 |- ((x e. om /\ x =/= (/)) -> (ph <-> (x = (/) \/ ph)))
86	81, 85	mpbird 156	. . 3 |- ((x e. om /\ x =/= (/)) -> ph)
87	2, 86	sylbi 114	. 2 |- (x e. N. -> ph)
88	1, 87	vtoclga 2613	1 |- (A e. N. -> ta)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 628  DECID wdc 741   = wceq 1242   e. wcel 1390   {cab 2023   =/= wne 2201  A.wral 2300   [.wsbc 2758   C_ wss 2911   (/)c0 3218   Oncon0 4066   suc csuc 4068   omcom 4256  (class class class)co 5455   1oc1o 5933   +o coa 5937   N.cnpi 6256   +N cpli 6257

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-ni 6288  df-pli 6289

This theorem is referenced by:  pitonn  6744