Theorem nn1suc 11041

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 10035	. . . . . 6 |- 1 e. _V
3		nn1suc.1	. . . . . 6 |- ( x = 1 -> ( ph <-> ps ) )
4	2, 3	sbcie 3470	. . . . 5 |- ( [. 1 / x ]. ph <-> ps )
5	1, 4	mpbir 221	. . . 4 |- [. 1 / x ]. ph
6		1nn 11031	. . . . . . 7 |- 1 e. NN
7		eleq1 2689	. . . . . . 7 |- ( A = 1 -> ( A e. NN <-> 1 e. NN ) )
8	6, 7	mpbiri 248	. . . . . 6 |- ( A = 1 -> A e. NN )
9		nn1suc.4	. . . . . . 7 |- ( x = A -> ( ph <-> th ) )
10	9	sbcieg 3468	. . . . . 6 |- ( A e. NN -> ( [. A / x ]. ph <-> th ) )
11	8, 10	syl 17	. . . . 5 |- ( A = 1 -> ( [. A / x ]. ph <-> th ) )
12		dfsbcq 3437	. . . . 5 |- ( A = 1 -> ( [. A / x ]. ph <-> [. 1 / x ]. ph ) )
13	11, 12	bitr3d 270	. . . 4 |- ( A = 1 -> ( th <-> [. 1 / x ]. ph ) )
14	5, 13	mpbiri 248	. . 3 |- ( A = 1 -> th )
15	14	a1i 11	. 2 |- ( A e. NN -> ( A = 1 -> th ) )
16		ovex 6678	. . . . . 6 |- ( y + 1 ) e. _V
17		nn1suc.3	. . . . . 6 |- ( x = ( y + 1 ) -> ( ph <-> ch ) )
18	16, 17	sbcie 3470	. . . . 5 |- ( [. ( y + 1 ) / x ]. ph <-> ch )
19		oveq1 6657	. . . . . 6 |- ( y = ( A - 1 ) -> ( y + 1 ) = ( ( A - 1 ) + 1 ) )
20	19	sbceq1d 3440	. . . . 5 |- ( y = ( A - 1 ) -> ( [. ( y + 1 ) / x ]. ph <-> [. ( ( A - 1 ) + 1 ) / x ]. ph ) )
21	18, 20	syl5bbr 274	. . . 4 |- ( y = ( A - 1 ) -> ( ch <-> [. ( ( A - 1 ) + 1 ) / x ]. ph ) )
22		nn1suc.6	. . . 4 |- ( y e. NN -> ch )
23	21, 22	vtoclga 3272	. . 3 |- ( ( A - 1 ) e. NN -> [. ( ( A - 1 ) + 1 ) / x ]. ph )
24		nncn 11028	. . . . . 6 |- ( A e. NN -> A e. CC )
25		ax-1cn 9994	. . . . . 6 |- 1 e. CC
26		npcan 10290	. . . . . 6 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A - 1 ) + 1 ) = A )
27	24, 25, 26	sylancl 694	. . . . 5 |- ( A e. NN -> ( ( A - 1 ) + 1 ) = A )
28	27	sbceq1d 3440	. . . 4 |- ( A e. NN -> ( [. ( ( A - 1 ) + 1 ) / x ]. ph <-> [. A / x ]. ph ) )
29	28, 10	bitrd 268	. . 3 |- ( A e. NN -> ( [. ( ( A - 1 ) + 1 ) / x ]. ph <-> th ) )
30	23, 29	syl5ib 234	. 2 |- ( A e. NN -> ( ( A - 1 ) e. NN -> th ) )
31		nn1m1nn 11040	. 2 |- ( A e. NN -> ( A = 1 \/ ( A - 1 ) e. NN ) )
32	15, 30, 31	mpjaod 396	1 |- ( A e. NN -> th )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   [.wsbc 3435  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   - cmin 10266   NNcn 11020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-nn 11021

This theorem is referenced by:  opsqrlem6  29004