Theorem nn1suc 10343

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 9381	. . . . . 6 |- 1 e. _V
3		nn1suc.1	. . . . . 6 |- ( x = 1 -> ( ph <-> ps ) )
4	2, 3	sbcie 3221	. . . . 5 |- ( [. 1 / x ]. ph <-> ps )
5	1, 4	mpbir 209	. . . 4 |- [. 1 / x ]. ph
6		1nn 10333	. . . . . . 7 |- 1 e. NN
7		eleq1 2503	. . . . . . 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 3219	. . . . . 6 |- ( A e. NN -> ( [. A / x ]. ph <-> th ) )
11	8, 10	syl 16	. . . . 5 |- ( A = 1 -> ( [. A / x ]. ph <-> th ) )
12		dfsbcq 3188	. . . . 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 6116	. . . . . 6 |- ( y + 1 ) e. _V
17		nn1suc.3	. . . . . 6 |- ( x = ( y + 1 ) -> ( ph <-> ch ) )
18	16, 17	sbcie 3221	. . . . 5 |- ( [. ( y + 1 ) / x ]. ph <-> ch )
19		oveq1 6098	. . . . . 6 |- ( y = ( A - 1 ) -> ( y + 1 ) = ( ( A - 1 ) + 1 ) )
20		dfsbcq 3188	. . . . . 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 3036	. . 3 |- ( ( A - 1 ) e. NN -> [. ( ( A - 1 ) + 1 ) / x ]. ph )
25		nncn 10330	. . . . . 6 |- ( A e. NN -> A e. CC )
26		ax-1cn 9340	. . . . . 6 |- 1 e. CC
27		npcan 9619	. . . . . 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 3188	. . . . 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 10342	. 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 1369   e. wcel 1756   [.wsbc 3186  (class class class)co 6091   CCcc 9280   1c1 9283   + caddc 9285   - cmin 9595   NNcn 10322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-ltxr 9423  df-sub 9597  df-nn 10323

This theorem is referenced by:  opsqrlem6  25549