Theorem nn1suc 9001

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 8014	. . . . . 6 |- 1 e. _V
3		nn1suc.1	. . . . . 6 |- ( x = 1 -> ( ph <-> ps ) )
4	2, 3	sbcie 3020	. . . . 5 |- ( [. 1 / x ]. ph <-> ps )
5	1, 4	mpbir 146	. . . 4 |- [. 1 / x ]. ph
6		1nn 8993	. . . . . . 7 |- 1 e. NN
7		eleq1 2256	. . . . . . 7 |- ( A = 1 -> ( A e. NN <-> 1 e. NN ) )
8	6, 7	mpbiri 168	. . . . . 6 |- ( A = 1 -> A e. NN )
9		nn1suc.4	. . . . . . 7 |- ( x = A -> ( ph <-> th ) )
10	9	sbcieg 3018	. . . . . 6 |- ( A e. NN -> ( [. A / x ]. ph <-> th ) )
11	8, 10	syl 14	. . . . 5 |- ( A = 1 -> ( [. A / x ]. ph <-> th ) )
12		dfsbcq 2987	. . . . 5 |- ( A = 1 -> ( [. A / x ]. ph <-> [. 1 / x ]. ph ) )
13	11, 12	bitr3d 190	. . . 4 |- ( A = 1 -> ( th <-> [. 1 / x ]. ph ) )
14	5, 13	mpbiri 168	. . 3 |- ( A = 1 -> th )
15	14	a1i 9	. 2 |- ( A e. NN -> ( A = 1 -> th ) )
16		elisset 2774	. . . 4 |- ( ( A - 1 ) e. NN -> E. y y = ( A - 1 ) )
17		eleq1 2256	. . . . . 6 |- ( y = ( A - 1 ) -> ( y e. NN <-> ( A - 1 ) e. NN ) )
18	17	pm5.32ri 455	. . . . 5 |- ( ( y e. NN /\ y = ( A - 1 ) ) <-> ( ( A - 1 ) e. NN /\ y = ( A - 1 ) ) )
19		nn1suc.6	. . . . . . 7 |- ( y e. NN -> ch )
20	19	adantr 276	. . . . . 6 |- ( ( y e. NN /\ y = ( A - 1 ) ) -> ch )
21		nnre 8989	. . . . . . . . 9 |- ( y e. NN -> y e. RR )
22		peano2re 8155	. . . . . . . . 9 |- ( y e. RR -> ( y + 1 ) e. RR )
23		nn1suc.3	. . . . . . . . . 10 |- ( x = ( y + 1 ) -> ( ph <-> ch ) )
24	23	sbcieg 3018	. . . . . . . . 9 |- ( ( y + 1 ) e. RR -> ( [. ( y + 1 ) / x ]. ph <-> ch ) )
25	21, 22, 24	3syl 17	. . . . . . . 8 |- ( y e. NN -> ( [. ( y + 1 ) / x ]. ph <-> ch ) )
26	25	adantr 276	. . . . . . 7 |- ( ( y e. NN /\ y = ( A - 1 ) ) -> ( [. ( y + 1 ) / x ]. ph <-> ch ) )
27		oveq1 5925	. . . . . . . . 9 |- ( y = ( A - 1 ) -> ( y + 1 ) = ( ( A - 1 ) + 1 ) )
28	27	sbceq1d 2990	. . . . . . . 8 |- ( y = ( A - 1 ) -> ( [. ( y + 1 ) / x ]. ph <-> [. ( ( A - 1 ) + 1 ) / x ]. ph ) )
29	28	adantl 277	. . . . . . 7 |- ( ( y e. NN /\ y = ( A - 1 ) ) -> ( [. ( y + 1 ) / x ]. ph <-> [. ( ( A - 1 ) + 1 ) / x ]. ph ) )
30	26, 29	bitr3d 190	. . . . . 6 |- ( ( y e. NN /\ y = ( A - 1 ) ) -> ( ch <-> [. ( ( A - 1 ) + 1 ) / x ]. ph ) )
31	20, 30	mpbid 147	. . . . 5 |- ( ( y e. NN /\ y = ( A - 1 ) ) -> [. ( ( A - 1 ) + 1 ) / x ]. ph )
32	18, 31	sylbir 135	. . . 4 |- ( ( ( A - 1 ) e. NN /\ y = ( A - 1 ) ) -> [. ( ( A - 1 ) + 1 ) / x ]. ph )
33	16, 32	exlimddv 1910	. . 3 |- ( ( A - 1 ) e. NN -> [. ( ( A - 1 ) + 1 ) / x ]. ph )
34		nncn 8990	. . . . . 6 |- ( A e. NN -> A e. CC )
35		ax-1cn 7965	. . . . . 6 |- 1 e. CC
36		npcan 8228	. . . . . 6 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A - 1 ) + 1 ) = A )
37	34, 35, 36	sylancl 413	. . . . 5 |- ( A e. NN -> ( ( A - 1 ) + 1 ) = A )
38	37	sbceq1d 2990	. . . 4 |- ( A e. NN -> ( [. ( ( A - 1 ) + 1 ) / x ]. ph <-> [. A / x ]. ph ) )
39	38, 10	bitrd 188	. . 3 |- ( A e. NN -> ( [. ( ( A - 1 ) + 1 ) / x ]. ph <-> th ) )
40	33, 39	imbitrid 154	. 2 |- ( A e. NN -> ( ( A - 1 ) e. NN -> th ) )
41		nn1m1nn 9000	. 2 |- ( A e. NN -> ( A = 1 \/ ( A - 1 ) e. NN ) )
42	15, 40, 41	mpjaod 719	1 |- ( A e. NN -> th )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   [.wsbc 2985  (class class class)co 5918   CCcc 7870   RRcr 7871   1c1 7873   + caddc 7875   - cmin 8190   NNcn 8982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-sub 8192  df-inn 8983

This theorem is referenced by: (None)