Theorem 2nn0ind 37510

Description: Induction on nonnegative integers with two base cases, for use with Lucas-type sequences. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Hypotheses

Ref	Expression
2nn0ind.1	|- ps
2nn0ind.2	|- ch
2nn0ind.3	|- ( y e. NN -> ( ( th /\ ta ) -> et ) )
2nn0ind.4	|- ( x = 0 -> ( ph <-> ps ) )
2nn0ind.5	|- ( x = 1 -> ( ph <-> ch ) )
2nn0ind.6	|- ( x = ( y - 1 ) -> ( ph <-> th ) )
2nn0ind.7	|- ( x = y -> ( ph <-> ta ) )
2nn0ind.8	|- ( x = ( y + 1 ) -> ( ph <-> et ) )
2nn0ind.9	|- ( x = A -> ( ph <-> rh ) )

Assertion

Ref	Expression
2nn0ind	|- ( A e. NN0 -> rh )

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

Allowed substitution hints:   ph( x)   ps( y)   ch( y)   th( y)   ta( y)   et( y)   rh( y)   A( y)

Proof of Theorem 2nn0ind

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0p1nn 11332	. . . 4 |- ( A e. NN0 -> ( A + 1 ) e. NN )
2		oveq1 6657	. . . . . . 7 |- ( a = 1 -> ( a - 1 ) = ( 1 - 1 ) )
3	2	sbceq1d 3440	. . . . . 6 |- ( a = 1 -> ( [. ( a - 1 ) / x ]. ph <-> [. ( 1 - 1 ) / x ]. ph ) )
4		dfsbcq 3437	. . . . . 6 |- ( a = 1 -> ( [. a / x ]. ph <-> [. 1 / x ]. ph ) )
5	3, 4	anbi12d 747	. . . . 5 |- ( a = 1 -> ( ( [. ( a - 1 ) / x ]. ph /\ [. a / x ]. ph ) <-> ( [. ( 1 - 1 ) / x ]. ph /\ [. 1 / x ]. ph ) ) )
6		oveq1 6657	. . . . . . 7 |- ( a = y -> ( a - 1 ) = ( y - 1 ) )
7	6	sbceq1d 3440	. . . . . 6 |- ( a = y -> ( [. ( a - 1 ) / x ]. ph <-> [. ( y - 1 ) / x ]. ph ) )
8		dfsbcq 3437	. . . . . 6 |- ( a = y -> ( [. a / x ]. ph <-> [. y / x ]. ph ) )
9	7, 8	anbi12d 747	. . . . 5 |- ( a = y -> ( ( [. ( a - 1 ) / x ]. ph /\ [. a / x ]. ph ) <-> ( [. ( y - 1 ) / x ]. ph /\ [. y / x ]. ph ) ) )
10		oveq1 6657	. . . . . . 7 |- ( a = ( y + 1 ) -> ( a - 1 ) = ( ( y + 1 ) - 1 ) )
11	10	sbceq1d 3440	. . . . . 6 |- ( a = ( y + 1 ) -> ( [. ( a - 1 ) / x ]. ph <-> [. ( ( y + 1 ) - 1 ) / x ]. ph ) )
12		dfsbcq 3437	. . . . . 6 |- ( a = ( y + 1 ) -> ( [. a / x ]. ph <-> [. ( y + 1 ) / x ]. ph ) )
13	11, 12	anbi12d 747	. . . . 5 |- ( a = ( y + 1 ) -> ( ( [. ( a - 1 ) / x ]. ph /\ [. a / x ]. ph ) <-> ( [. ( ( y + 1 ) - 1 ) / x ]. ph /\ [. ( y + 1 ) / x ]. ph ) ) )
14		oveq1 6657	. . . . . . 7 |- ( a = ( A + 1 ) -> ( a - 1 ) = ( ( A + 1 ) - 1 ) )
15	14	sbceq1d 3440	. . . . . 6 |- ( a = ( A + 1 ) -> ( [. ( a - 1 ) / x ]. ph <-> [. ( ( A + 1 ) - 1 ) / x ]. ph ) )
16		dfsbcq 3437	. . . . . 6 |- ( a = ( A + 1 ) -> ( [. a / x ]. ph <-> [. ( A + 1 ) / x ]. ph ) )
17	15, 16	anbi12d 747	. . . . 5 |- ( a = ( A + 1 ) -> ( ( [. ( a - 1 ) / x ]. ph /\ [. a / x ]. ph ) <-> ( [. ( ( A + 1 ) - 1 ) / x ]. ph /\ [. ( A + 1 ) / x ]. ph ) ) )
18		2nn0ind.1	. . . . . . 7 |- ps
19		ovex 6678	. . . . . . . 8 |- ( 1 - 1 ) e. _V
20		1m1e0 11089	. . . . . . . . . 10 |- ( 1 - 1 ) = 0
21	20	eqeq2i 2634	. . . . . . . . 9 |- ( x = ( 1 - 1 ) <-> x = 0 )
22		2nn0ind.4	. . . . . . . . 9 |- ( x = 0 -> ( ph <-> ps ) )
23	21, 22	sylbi 207	. . . . . . . 8 |- ( x = ( 1 - 1 ) -> ( ph <-> ps ) )
24	19, 23	sbcie 3470	. . . . . . 7 |- ( [. ( 1 - 1 ) / x ]. ph <-> ps )
25	18, 24	mpbir 221	. . . . . 6 |- [. ( 1 - 1 ) / x ]. ph
26		2nn0ind.2	. . . . . . 7 |- ch
27		1ex 10035	. . . . . . . 8 |- 1 e. _V
28		2nn0ind.5	. . . . . . . 8 |- ( x = 1 -> ( ph <-> ch ) )
29	27, 28	sbcie 3470	. . . . . . 7 |- ( [. 1 / x ]. ph <-> ch )
30	26, 29	mpbir 221	. . . . . 6 |- [. 1 / x ]. ph
31	25, 30	pm3.2i 471	. . . . 5 |- ( [. ( 1 - 1 ) / x ]. ph /\ [. 1 / x ]. ph )
32		simprr 796	. . . . . . . 8 |- ( ( y e. NN /\ ( [. ( y - 1 ) / x ]. ph /\ [. y / x ]. ph ) ) -> [. y / x ]. ph )
33		nncn 11028	. . . . . . . . . . 11 |- ( y e. NN -> y e. CC )
34		ax-1cn 9994	. . . . . . . . . . 11 |- 1 e. CC
35		pncan 10287	. . . . . . . . . . 11 |- ( ( y e. CC /\ 1 e. CC ) -> ( ( y + 1 ) - 1 ) = y )
36	33, 34, 35	sylancl 694	. . . . . . . . . 10 |- ( y e. NN -> ( ( y + 1 ) - 1 ) = y )
37	36	adantr 481	. . . . . . . . 9 |- ( ( y e. NN /\ ( [. ( y - 1 ) / x ]. ph /\ [. y / x ]. ph ) ) -> ( ( y + 1 ) - 1 ) = y )
38	37	sbceq1d 3440	. . . . . . . 8 |- ( ( y e. NN /\ ( [. ( y - 1 ) / x ]. ph /\ [. y / x ]. ph ) ) -> ( [. ( ( y + 1 ) - 1 ) / x ]. ph <-> [. y / x ]. ph ) )
39	32, 38	mpbird 247	. . . . . . 7 |- ( ( y e. NN /\ ( [. ( y - 1 ) / x ]. ph /\ [. y / x ]. ph ) ) -> [. ( ( y + 1 ) - 1 ) / x ]. ph )
40		2nn0ind.3	. . . . . . . . 9 |- ( y e. NN -> ( ( th /\ ta ) -> et ) )
41		ovex 6678	. . . . . . . . . . 11 |- ( y - 1 ) e. _V
42		2nn0ind.6	. . . . . . . . . . 11 |- ( x = ( y - 1 ) -> ( ph <-> th ) )
43	41, 42	sbcie 3470	. . . . . . . . . 10 |- ( [. ( y - 1 ) / x ]. ph <-> th )
44		vex 3203	. . . . . . . . . . 11 |- y e. _V
45		2nn0ind.7	. . . . . . . . . . 11 |- ( x = y -> ( ph <-> ta ) )
46	44, 45	sbcie 3470	. . . . . . . . . 10 |- ( [. y / x ]. ph <-> ta )
47	43, 46	anbi12i 733	. . . . . . . . 9 |- ( ( [. ( y - 1 ) / x ]. ph /\ [. y / x ]. ph ) <-> ( th /\ ta ) )
48		ovex 6678	. . . . . . . . . 10 |- ( y + 1 ) e. _V
49		2nn0ind.8	. . . . . . . . . 10 |- ( x = ( y + 1 ) -> ( ph <-> et ) )
50	48, 49	sbcie 3470	. . . . . . . . 9 |- ( [. ( y + 1 ) / x ]. ph <-> et )
51	40, 47, 50	3imtr4g 285	. . . . . . . 8 |- ( y e. NN -> ( ( [. ( y - 1 ) / x ]. ph /\ [. y / x ]. ph ) -> [. ( y + 1 ) / x ]. ph ) )
52	51	imp 445	. . . . . . 7 |- ( ( y e. NN /\ ( [. ( y - 1 ) / x ]. ph /\ [. y / x ]. ph ) ) -> [. ( y + 1 ) / x ]. ph )
53	39, 52	jca 554	. . . . . 6 |- ( ( y e. NN /\ ( [. ( y - 1 ) / x ]. ph /\ [. y / x ]. ph ) ) -> ( [. ( ( y + 1 ) - 1 ) / x ]. ph /\ [. ( y + 1 ) / x ]. ph ) )
54	53	ex 450	. . . . 5 |- ( y e. NN -> ( ( [. ( y - 1 ) / x ]. ph /\ [. y / x ]. ph ) -> ( [. ( ( y + 1 ) - 1 ) / x ]. ph /\ [. ( y + 1 ) / x ]. ph ) ) )
55	5, 9, 13, 17, 31, 54	nnind 11038	. . . 4 |- ( ( A + 1 ) e. NN -> ( [. ( ( A + 1 ) - 1 ) / x ]. ph /\ [. ( A + 1 ) / x ]. ph ) )
56	1, 55	syl 17	. . 3 |- ( A e. NN0 -> ( [. ( ( A + 1 ) - 1 ) / x ]. ph /\ [. ( A + 1 ) / x ]. ph ) )
57		nn0cn 11302	. . . . . . 7 |- ( A e. NN0 -> A e. CC )
58		pncan 10287	. . . . . . 7 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A + 1 ) - 1 ) = A )
59	57, 34, 58	sylancl 694	. . . . . 6 |- ( A e. NN0 -> ( ( A + 1 ) - 1 ) = A )
60	59	sbceq1d 3440	. . . . 5 |- ( A e. NN0 -> ( [. ( ( A + 1 ) - 1 ) / x ]. ph <-> [. A / x ]. ph ) )
61	60	biimpa 501	. . . 4 |- ( ( A e. NN0 /\ [. ( ( A + 1 ) - 1 ) / x ]. ph ) -> [. A / x ]. ph )
62	61	adantrr 753	. . 3 |- ( ( A e. NN0 /\ ( [. ( ( A + 1 ) - 1 ) / x ]. ph /\ [. ( A + 1 ) / x ]. ph ) ) -> [. A / x ]. ph )
63	56, 62	mpdan 702	. 2 |- ( A e. NN0 -> [. A / x ]. ph )
64		2nn0ind.9	. . 3 |- ( x = A -> ( ph <-> rh ) )
65	64	sbcieg 3468	. 2 |- ( A e. NN0 -> ( [. A / x ]. ph <-> rh ) )
66	63, 65	mpbid 222	1 |- ( A e. NN0 -> rh )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   [.wsbc 3435  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NNcn 11020   NN0cn0 11292

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  df-n0 11293

This theorem is referenced by:  jm2.18  37555  jm2.15nn0  37570  jm2.16nn0  37571