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Theorem 2nn0ind 30336
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.
StepHypRef Expression
1 nn0p1nn 10824 . . . 4  |-  ( A  e.  NN0  ->  ( A  +  1 )  e.  NN )
2 oveq1 6282 . . . . . . 7  |-  ( a  =  1  ->  (
a  -  1 )  =  ( 1  -  1 ) )
3 dfsbcq 3326 . . . . . . 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 3326 . . . . . 6  |-  ( a  =  1  ->  ( [. a  /  x ]. ph  <->  [. 1  /  x ]. ph ) )
64, 5anbi12d 710 . . . . 5  |-  ( a  =  1  ->  (
( [. ( a  - 
1 )  /  x ]. ph  /\  [. a  /  x ]. ph )  <->  (
[. ( 1  -  1 )  /  x ]. ph  /\  [. 1  /  x ]. ph )
) )
7 oveq1 6282 . . . . . . 7  |-  ( a  =  y  ->  (
a  -  1 )  =  ( y  - 
1 ) )
8 dfsbcq 3326 . . . . . . 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 3326 . . . . . 6  |-  ( a  =  y  ->  ( [. a  /  x ]. ph  <->  [. y  /  x ]. ph ) )
119, 10anbi12d 710 . . . . 5  |-  ( a  =  y  ->  (
( [. ( a  - 
1 )  /  x ]. ph  /\  [. a  /  x ]. ph )  <->  (
[. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
) )
12 oveq1 6282 . . . . . . 7  |-  ( a  =  ( y  +  1 )  ->  (
a  -  1 )  =  ( ( y  +  1 )  - 
1 ) )
13 dfsbcq 3326 . . . . . . 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 3326 . . . . . 6  |-  ( a  =  ( y  +  1 )  ->  ( [. a  /  x ]. ph  <->  [. ( y  +  1 )  /  x ]. ph ) )
1614, 15anbi12d 710 . . . . 5  |-  ( a  =  ( y  +  1 )  ->  (
( [. ( a  - 
1 )  /  x ]. ph  /\  [. a  /  x ]. ph )  <->  (
[. ( ( y  +  1 )  - 
1 )  /  x ]. ph  /\  [. (
y  +  1 )  /  x ]. ph )
) )
17 oveq1 6282 . . . . . . 7  |-  ( a  =  ( A  + 
1 )  ->  (
a  -  1 )  =  ( ( A  +  1 )  - 
1 ) )
18 dfsbcq 3326 . . . . . . 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 3326 . . . . . 6  |-  ( a  =  ( A  + 
1 )  ->  ( [. a  /  x ]. ph  <->  [. ( A  + 
1 )  /  x ]. ph ) )
2119, 20anbi12d 710 . . . . 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 6300 . . . . . . . 8  |-  ( 1  -  1 )  e. 
_V
24 1m1e0 10593 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
2524eqeq2i 2478 . . . . . . . . 9  |-  ( x  =  ( 1  -  1 )  <->  x  = 
0 )
26 2nn0ind.4 . . . . . . . . 9  |-  ( x  =  0  ->  ( ph 
<->  ps ) )
2725, 26sylbi 195 . . . . . . . 8  |-  ( x  =  ( 1  -  1 )  ->  ( ph 
<->  ps ) )
2823, 27sbcie 3359 . . . . . . 7  |-  ( [. ( 1  -  1 )  /  x ]. ph  <->  ps )
2922, 28mpbir 209 . . . . . 6  |-  [. (
1  -  1 )  /  x ]. ph
30 2nn0ind.2 . . . . . . 7  |-  ch
31 1ex 9580 . . . . . . . 8  |-  1  e.  _V
32 2nn0ind.5 . . . . . . . 8  |-  ( x  =  1  ->  ( ph 
<->  ch ) )
3331, 32sbcie 3359 . . . . . . 7  |-  ( [.
1  /  x ]. ph  <->  ch )
3430, 33mpbir 209 . . . . . 6  |-  [. 1  /  x ]. ph
3529, 34pm3.2i 455 . . . . 5  |-  ( [. ( 1  -  1 )  /  x ]. ph 
/\  [. 1  /  x ]. ph )
36 simprr 756 . . . . . . . 8  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  [. y  /  x ]. ph )
37 nncn 10533 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  y  e.  CC )
38 ax-1cn 9539 . . . . . . . . . . 11  |-  1  e.  CC
39 pncan 9815 . . . . . . . . . . 11  |-  ( ( y  e.  CC  /\  1  e.  CC )  ->  ( ( y  +  1 )  -  1 )  =  y )
4037, 38, 39sylancl 662 . . . . . . . . . 10  |-  ( y  e.  NN  ->  (
( y  +  1 )  -  1 )  =  y )
4140adantr 465 . . . . . . . . 9  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  ( (
y  +  1 )  -  1 )  =  y )
42 dfsbcq 3326 . . . . . . . . 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 232 . . . . . . 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 6300 . . . . . . . . . . 11  |-  ( y  -  1 )  e. 
_V
47 2nn0ind.6 . . . . . . . . . . 11  |-  ( x  =  ( y  - 
1 )  ->  ( ph 
<->  th ) )
4846, 47sbcie 3359 . . . . . . . . . 10  |-  ( [. ( y  -  1 )  /  x ]. ph  <->  th )
49 vex 3109 . . . . . . . . . . 11  |-  y  e. 
_V
50 2nn0ind.7 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( ph 
<->  ta ) )
5149, 50sbcie 3359 . . . . . . . . . 10  |-  ( [. y  /  x ]. ph  <->  ta )
5248, 51anbi12i 697 . . . . . . . . 9  |-  ( (
[. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )  <->  ( th  /\  ta )
)
53 ovex 6300 . . . . . . . . . 10  |-  ( y  +  1 )  e. 
_V
54 2nn0ind.8 . . . . . . . . . 10  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  et ) )
5553, 54sbcie 3359 . . . . . . . . 9  |-  ( [. ( y  +  1 )  /  x ]. ph  <->  et )
5645, 52, 553imtr4g 270 . . . . . . . 8  |-  ( y  e.  NN  ->  (
( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )  ->  [. ( y  +  1 )  /  x ]. ph ) )
5756imp 429 . . . . . . 7  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  [. ( y  +  1 )  /  x ]. ph )
5844, 57jca 532 . . . . . 6  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  ( [. ( ( y  +  1 )  -  1 )  /  x ]. ph 
/\  [. ( y  +  1 )  /  x ]. ph ) )
5958ex 434 . . . . 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 10543 . . . 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 10794 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  CC )
63 pncan 9815 . . . . . . 7  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
6462, 38, 63sylancl 662 . . . . . 6  |-  ( A  e.  NN0  ->  ( ( A  +  1 )  -  1 )  =  A )
65 dfsbcq 3326 . . . . . 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 484 . . . 4  |-  ( ( A  e.  NN0  /\  [. ( ( A  + 
1 )  -  1 )  /  x ]. ph )  ->  [. A  /  x ]. ph )
6867adantrr 716 . . 3  |-  ( ( A  e.  NN0  /\  ( [. ( ( A  +  1 )  - 
1 )  /  x ]. ph  /\  [. ( A  +  1 )  /  x ]. ph )
)  ->  [. A  /  x ]. ph )
6961, 68mpdan 668 . 2  |-  ( A  e.  NN0  ->  [. A  /  x ]. ph )
70 2nn0ind.9 . . 3  |-  ( x  =  A  ->  ( ph 
<->  rh ) )
7170sbcieg 3357 . 2  |-  ( A  e.  NN0  ->  ( [. A  /  x ]. ph  <->  rh )
)
7269, 71mpbid 210 1  |-  ( A  e.  NN0  ->  rh )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   [.wsbc 3324  (class class class)co 6275   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    - cmin 9794   NNcn 10525   NN0cn0 10784
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9796  df-nn 10526  df-n0 10785
This theorem is referenced by:  jm2.18  30387  jm2.15nn0  30402  jm2.16nn0  30403
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