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Theorem 2nn0ind 27046
Description: Induction on natural numbers 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.
StepHypRef Expression
1 nn0p1nn 10290 . . . 4  |-  ( A  e.  NN0  ->  ( A  +  1 )  e.  NN )
2 oveq1 6117 . . . . . . 7  |-  ( a  =  1  ->  (
a  -  1 )  =  ( 1  -  1 ) )
3 dfsbcq 3169 . . . . . . 7  |-  ( ( a  -  1 )  =  ( 1  -  1 )  ->  ( [. ( a  -  1 )  /  x ]. ph  <->  [. ( 1  -  1 )  /  x ]. ph ) )
42, 3syl 16 . . . . . 6  |-  ( a  =  1  ->  ( [. ( a  -  1 )  /  x ]. ph  <->  [. ( 1  -  1 )  /  x ]. ph ) )
5 dfsbcq 3169 . . . . . 6  |-  ( a  =  1  ->  ( [. a  /  x ]. ph  <->  [. 1  /  x ]. ph ) )
64, 5anbi12d 693 . . . . 5  |-  ( a  =  1  ->  (
( [. ( a  - 
1 )  /  x ]. ph  /\  [. a  /  x ]. ph )  <->  (
[. ( 1  -  1 )  /  x ]. ph  /\  [. 1  /  x ]. ph )
) )
7 oveq1 6117 . . . . . . 7  |-  ( a  =  y  ->  (
a  -  1 )  =  ( y  - 
1 ) )
8 dfsbcq 3169 . . . . . . 7  |-  ( ( a  -  1 )  =  ( y  - 
1 )  ->  ( [. ( a  -  1 )  /  x ]. ph  <->  [. ( y  -  1 )  /  x ]. ph ) )
97, 8syl 16 . . . . . 6  |-  ( a  =  y  ->  ( [. ( a  -  1 )  /  x ]. ph  <->  [. ( y  -  1 )  /  x ]. ph ) )
10 dfsbcq 3169 . . . . . 6  |-  ( a  =  y  ->  ( [. a  /  x ]. ph  <->  [. y  /  x ]. ph ) )
119, 10anbi12d 693 . . . . 5  |-  ( a  =  y  ->  (
( [. ( a  - 
1 )  /  x ]. ph  /\  [. a  /  x ]. ph )  <->  (
[. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
) )
12 oveq1 6117 . . . . . . 7  |-  ( a  =  ( y  +  1 )  ->  (
a  -  1 )  =  ( ( y  +  1 )  - 
1 ) )
13 dfsbcq 3169 . . . . . . 7  |-  ( ( a  -  1 )  =  ( ( y  +  1 )  - 
1 )  ->  ( [. ( a  -  1 )  /  x ]. ph  <->  [. ( ( y  +  1 )  -  1 )  /  x ]. ph ) )
1412, 13syl 16 . . . . . 6  |-  ( a  =  ( y  +  1 )  ->  ( [. ( a  -  1 )  /  x ]. ph  <->  [. ( ( y  +  1 )  -  1 )  /  x ]. ph ) )
15 dfsbcq 3169 . . . . . 6  |-  ( a  =  ( y  +  1 )  ->  ( [. a  /  x ]. ph  <->  [. ( y  +  1 )  /  x ]. ph ) )
1614, 15anbi12d 693 . . . . 5  |-  ( a  =  ( y  +  1 )  ->  (
( [. ( a  - 
1 )  /  x ]. ph  /\  [. a  /  x ]. ph )  <->  (
[. ( ( y  +  1 )  - 
1 )  /  x ]. ph  /\  [. (
y  +  1 )  /  x ]. ph )
) )
17 oveq1 6117 . . . . . . 7  |-  ( a  =  ( A  + 
1 )  ->  (
a  -  1 )  =  ( ( A  +  1 )  - 
1 ) )
18 dfsbcq 3169 . . . . . . 7  |-  ( ( a  -  1 )  =  ( ( A  +  1 )  - 
1 )  ->  ( [. ( a  -  1 )  /  x ]. ph  <->  [. ( ( A  + 
1 )  -  1 )  /  x ]. ph ) )
1917, 18syl 16 . . . . . 6  |-  ( a  =  ( A  + 
1 )  ->  ( [. ( a  -  1 )  /  x ]. ph  <->  [. ( ( A  + 
1 )  -  1 )  /  x ]. ph ) )
20 dfsbcq 3169 . . . . . 6  |-  ( a  =  ( A  + 
1 )  ->  ( [. a  /  x ]. ph  <->  [. ( A  + 
1 )  /  x ]. ph ) )
2119, 20anbi12d 693 . . . . 5  |-  ( a  =  ( A  + 
1 )  ->  (
( [. ( a  - 
1 )  /  x ]. ph  /\  [. a  /  x ]. ph )  <->  (
[. ( ( A  +  1 )  - 
1 )  /  x ]. ph  /\  [. ( A  +  1 )  /  x ]. ph )
) )
22 2nn0ind.1 . . . . . . 7  |-  ps
23 ovex 6135 . . . . . . . 8  |-  ( 1  -  1 )  e. 
_V
24 1m1e0 10099 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
2524eqeq2i 2452 . . . . . . . . 9  |-  ( x  =  ( 1  -  1 )  <->  x  = 
0 )
26 2nn0ind.4 . . . . . . . . 9  |-  ( x  =  0  ->  ( ph 
<->  ps ) )
2725, 26sylbi 189 . . . . . . . 8  |-  ( x  =  ( 1  -  1 )  ->  ( ph 
<->  ps ) )
2823, 27sbcie 3201 . . . . . . 7  |-  ( [. ( 1  -  1 )  /  x ]. ph  <->  ps )
2922, 28mpbir 202 . . . . . 6  |-  [. (
1  -  1 )  /  x ]. ph
30 2nn0ind.2 . . . . . . 7  |-  ch
31 1ex 9117 . . . . . . . 8  |-  1  e.  _V
32 2nn0ind.5 . . . . . . . 8  |-  ( x  =  1  ->  ( ph 
<->  ch ) )
3331, 32sbcie 3201 . . . . . . 7  |-  ( [.
1  /  x ]. ph  <->  ch )
3430, 33mpbir 202 . . . . . 6  |-  [. 1  /  x ]. ph
3529, 34pm3.2i 443 . . . . 5  |-  ( [. ( 1  -  1 )  /  x ]. ph 
/\  [. 1  /  x ]. ph )
36 simprr 735 . . . . . . . 8  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  [. y  /  x ]. ph )
37 nncn 10039 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  y  e.  CC )
38 ax-1cn 9079 . . . . . . . . . . 11  |-  1  e.  CC
39 pncan 9342 . . . . . . . . . . 11  |-  ( ( y  e.  CC  /\  1  e.  CC )  ->  ( ( y  +  1 )  -  1 )  =  y )
4037, 38, 39sylancl 645 . . . . . . . . . 10  |-  ( y  e.  NN  ->  (
( y  +  1 )  -  1 )  =  y )
4140adantr 453 . . . . . . . . 9  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  ( (
y  +  1 )  -  1 )  =  y )
42 dfsbcq 3169 . . . . . . . . 9  |-  ( ( ( y  +  1 )  -  1 )  =  y  ->  ( [. ( ( y  +  1 )  -  1 )  /  x ]. ph  <->  [. y  /  x ]. ph ) )
4341, 42syl 16 . . . . . . . 8  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  ( [. ( ( y  +  1 )  -  1 )  /  x ]. ph  <->  [. y  /  x ]. ph ) )
4436, 43mpbird 225 . . . . . . 7  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  [. ( ( y  +  1 )  -  1 )  /  x ]. ph )
45 2nn0ind.3 . . . . . . . . 9  |-  ( y  e.  NN  ->  (
( th  /\  ta )  ->  et ) )
46 ovex 6135 . . . . . . . . . . 11  |-  ( y  -  1 )  e. 
_V
47 2nn0ind.6 . . . . . . . . . . 11  |-  ( x  =  ( y  - 
1 )  ->  ( ph 
<->  th ) )
4846, 47sbcie 3201 . . . . . . . . . 10  |-  ( [. ( y  -  1 )  /  x ]. ph  <->  th )
49 vex 2965 . . . . . . . . . . 11  |-  y  e. 
_V
50 2nn0ind.7 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( ph 
<->  ta ) )
5149, 50sbcie 3201 . . . . . . . . . 10  |-  ( [. y  /  x ]. ph  <->  ta )
5248, 51anbi12i 680 . . . . . . . . 9  |-  ( (
[. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )  <->  ( th  /\  ta )
)
53 ovex 6135 . . . . . . . . . 10  |-  ( y  +  1 )  e. 
_V
54 2nn0ind.8 . . . . . . . . . 10  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  et ) )
5553, 54sbcie 3201 . . . . . . . . 9  |-  ( [. ( y  +  1 )  /  x ]. ph  <->  et )
5645, 52, 553imtr4g 263 . . . . . . . 8  |-  ( y  e.  NN  ->  (
( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )  ->  [. ( y  +  1 )  /  x ]. ph ) )
5756imp 420 . . . . . . 7  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  [. ( y  +  1 )  /  x ]. ph )
5844, 57jca 520 . . . . . 6  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  ( [. ( ( y  +  1 )  -  1 )  /  x ]. ph 
/\  [. ( y  +  1 )  /  x ]. ph ) )
5958ex 425 . . . . 5  |-  ( y  e.  NN  ->  (
( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )  ->  ( [. ( ( y  +  1 )  -  1 )  /  x ]. ph  /\  [. (
y  +  1 )  /  x ]. ph )
) )
606, 11, 16, 21, 35, 59nnind 10049 . . . 4  |-  ( ( A  +  1 )  e.  NN  ->  ( [. ( ( A  + 
1 )  -  1 )  /  x ]. ph 
/\  [. ( A  + 
1 )  /  x ]. ph ) )
611, 60syl 16 . . 3  |-  ( A  e.  NN0  ->  ( [. ( ( A  + 
1 )  -  1 )  /  x ]. ph 
/\  [. ( A  + 
1 )  /  x ]. ph ) )
62 nn0cn 10262 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  CC )
63 pncan 9342 . . . . . . 7  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
6462, 38, 63sylancl 645 . . . . . 6  |-  ( A  e.  NN0  ->  ( ( A  +  1 )  -  1 )  =  A )
65 dfsbcq 3169 . . . . . 6  |-  ( ( ( A  +  1 )  -  1 )  =  A  ->  ( [. ( ( A  + 
1 )  -  1 )  /  x ]. ph  <->  [. A  /  x ]. ph ) )
6664, 65syl 16 . . . . 5  |-  ( A  e.  NN0  ->  ( [. ( ( A  + 
1 )  -  1 )  /  x ]. ph  <->  [. A  /  x ]. ph ) )
6766biimpa 472 . . . 4  |-  ( ( A  e.  NN0  /\  [. ( ( A  + 
1 )  -  1 )  /  x ]. ph )  ->  [. A  /  x ]. ph )
6867adantrr 699 . . 3  |-  ( ( A  e.  NN0  /\  ( [. ( ( A  +  1 )  - 
1 )  /  x ]. ph  /\  [. ( A  +  1 )  /  x ]. ph )
)  ->  [. A  /  x ]. ph )
6961, 68mpdan 651 . 2  |-  ( A  e.  NN0  ->  [. A  /  x ]. ph )
70 2nn0ind.9 . . 3  |-  ( x  =  A  ->  ( ph 
<->  rh ) )
7170sbcieg 3199 . 2  |-  ( A  e.  NN0  ->  ( [. A  /  x ]. ph  <->  rh )
)
7269, 71mpbid 203 1  |-  ( A  e.  NN0  ->  rh )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727   [.wsbc 3167  (class class class)co 6110   CCcc 9019   0cc0 9021   1c1 9022    + caddc 9024    - cmin 9322   NNcn 10031   NN0cn0 10252
This theorem is referenced by:  jm2.18  27097  jm2.15nn0  27112  jm2.16nn0  27113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-ltxr 9156  df-sub 9324  df-nn 10032  df-n0 10253
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