Theorem nn1suc 10339

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 9377	. . . . . 6 |- 1 e. _V
3		nn1suc.1	. . . . . 6 |- ( x = 1 -> ( ph <-> ps ) )
4	2, 3	sbcie 3218	. . . . 5 |- ( [. 1 / x ]. ph <-> ps )
5	1, 4	mpbir 209	. . . 4 |- [. 1 / x ]. ph
6		1nn 10329	. . . . . . 7 |- 1 e. NN
7		eleq1 2501	. . . . . . 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 3216	. . . . . 6 |- ( A e. NN -> ( [. A / x ]. ph <-> th ) )
11	8, 10	syl 16	. . . . 5 |- ( A = 1 -> ( [. A / x ]. ph <-> th ) )
12		dfsbcq 3185	. . . . 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 6115	. . . . . 6 |- ( y + 1 ) e. _V
17		nn1suc.3	. . . . . 6 |- ( x = ( y + 1 ) -> ( ph <-> ch ) )
18	16, 17	sbcie 3218	. . . . 5 |- ( [. ( y + 1 ) / x ]. ph <-> ch )
19		oveq1 6097	. . . . . 6 |- ( y = ( A - 1 ) -> ( y + 1 ) = ( ( A - 1 ) + 1 ) )
20		dfsbcq 3185	. . . . . 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 3033	. . 3 |- ( ( A - 1 ) e. NN -> [. ( ( A - 1 ) + 1 ) / x ]. ph )
25		nncn 10326	. . . . . 6 |- ( A e. NN -> A e. CC )
26		ax-1cn 9336	. . . . . 6 |- 1 e. CC
27		npcan 9615	. . . . . 6 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A - 1 ) + 1 ) = A )
28	25, 26, 27	sylancl 657	. . . . 5 |- ( A e. NN -> ( ( A - 1 ) + 1 ) = A )
29		dfsbcq 3185	. . . . 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 10338	. 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 1364   e. wcel 1761   [.wsbc 3183  (class class class)co 6090   CCcc 9276   1c1 9279   + caddc 9281   - cmin 9591   NNcn 10318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-ltxr 9419  df-sub 9593  df-nn 10319

This theorem is referenced by:  opsqrlem6  25484