Theorem nn1suc 10557

Description: If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)

Hypotheses

Ref	Expression
nn1suc.1	|- ( x = 1 -> ( ph <-> ps ) )
nn1suc.3	|- ( x = ( y + 1 ) -> ( ph <-> ch ) )
nn1suc.4	|- ( x = A -> ( ph <-> th ) )
nn1suc.5	|- ps
nn1suc.6	|- ( y e. NN -> ch )

Assertion

Ref	Expression
nn1suc	|- ( A e. NN -> th )

Distinct variable groups:   x, y, A   ps, x   ch, x   th, x   ph, y

Allowed substitution hints:   ph( x)   ps( y)   ch( y)   th( y)

Proof of Theorem nn1suc

Step	Hyp	Ref	Expression
1		nn1suc.5	. . . . 5 |- ps
2		1ex 9591	. . . . . 6 |- 1 e. _V
3		nn1suc.1	. . . . . 6 |- ( x = 1 -> ( ph <-> ps ) )
4	2, 3	sbcie 3366	. . . . 5 |- ( [. 1 / x ]. ph <-> ps )
5	1, 4	mpbir 209	. . . 4 |- [. 1 / x ]. ph
6		1nn 10547	. . . . . . 7 |- 1 e. NN
7		eleq1 2539	. . . . . . 7 |- ( A = 1 -> ( A e. NN <-> 1 e. NN ) )
8	6, 7	mpbiri 233	. . . . . 6 |- ( A = 1 -> A e. NN )
9		nn1suc.4	. . . . . . 7 |- ( x = A -> ( ph <-> th ) )
10	9	sbcieg 3364	. . . . . 6 |- ( A e. NN -> ( [. A / x ]. ph <-> th ) )
11	8, 10	syl 16	. . . . 5 |- ( A = 1 -> ( [. A / x ]. ph <-> th ) )
12		dfsbcq 3333	. . . . 5 |- ( A = 1 -> ( [. A / x ]. ph <-> [. 1 / x ]. ph ) )
13	11, 12	bitr3d 255	. . . 4 |- ( A = 1 -> ( th <-> [. 1 / x ]. ph ) )
14	5, 13	mpbiri 233	. . 3 |- ( A = 1 -> th )
15	14	a1i 11	. 2 |- ( A e. NN -> ( A = 1 -> th ) )
16		ovex 6309	. . . . . 6 |- ( y + 1 ) e. _V
17		nn1suc.3	. . . . . 6 |- ( x = ( y + 1 ) -> ( ph <-> ch ) )
18	16, 17	sbcie 3366	. . . . 5 |- ( [. ( y + 1 ) / x ]. ph <-> ch )
19		oveq1 6291	. . . . . 6 |- ( y = ( A - 1 ) -> ( y + 1 ) = ( ( A - 1 ) + 1 ) )
20		dfsbcq 3333	. . . . . 6 |- ( ( y + 1 ) = ( ( A - 1 ) + 1 ) -> ( [. ( y + 1 ) / x ]. ph <-> [. ( ( A - 1 ) + 1 ) / x ]. ph ) )
21	19, 20	syl 16	. . . . 5 |- ( y = ( A - 1 ) -> ( [. ( y + 1 ) / x ]. ph <-> [. ( ( A - 1 ) + 1 ) / x ]. ph ) )
22	18, 21	syl5bbr 259	. . . 4 |- ( y = ( A - 1 ) -> ( ch <-> [. ( ( A - 1 ) + 1 ) / x ]. ph ) )
23		nn1suc.6	. . . 4 |- ( y e. NN -> ch )
24	22, 23	vtoclga 3177	. . 3 |- ( ( A - 1 ) e. NN -> [. ( ( A - 1 ) + 1 ) / x ]. ph )
25		nncn 10544	. . . . . 6 |- ( A e. NN -> A e. CC )
26		ax-1cn 9550	. . . . . 6 |- 1 e. CC
27		npcan 9829	. . . . . 6 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A - 1 ) + 1 ) = A )
28	25, 26, 27	sylancl 662	. . . . 5 |- ( A e. NN -> ( ( A - 1 ) + 1 ) = A )
29		dfsbcq 3333	. . . . 5 |- ( ( ( A - 1 ) + 1 ) = A -> ( [. ( ( A - 1 ) + 1 ) / x ]. ph <-> [. A / x ]. ph ) )
30	28, 29	syl 16	. . . 4 |- ( A e. NN -> ( [. ( ( A - 1 ) + 1 ) / x ]. ph <-> [. A / x ]. ph ) )
31	30, 10	bitrd 253	. . 3 |- ( A e. NN -> ( [. ( ( A - 1 ) + 1 ) / x ]. ph <-> th ) )
32	24, 31	syl5ib 219	. 2 |- ( A e. NN -> ( ( A - 1 ) e. NN -> th ) )
33		nn1m1nn 10556	. 2 |- ( A e. NN -> ( A = 1 \/ ( A - 1 ) e. NN ) )
34	15, 32, 33	mpjaod 381	1 |- ( A e. NN -> th )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1379   e. wcel 1767   [.wsbc 3331  (class class class)co 6284   CCcc 9490   1c1 9493   + caddc 9495   - cmin 9805   NNcn 10536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-ltxr 9633  df-sub 9807  df-nn 10537

This theorem is referenced by:  opsqrlem6  26768