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Theorem 2nn0ind 30856
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 10841 . . . 4  |-  ( A  e.  NN0  ->  ( A  +  1 )  e.  NN )
2 oveq1 6288 . . . . . . 7  |-  ( a  =  1  ->  (
a  -  1 )  =  ( 1  -  1 ) )
32sbceq1d 3318 . . . . . 6  |-  ( a  =  1  ->  ( [. ( a  -  1 )  /  x ]. ph  <->  [. ( 1  -  1 )  /  x ]. ph ) )
4 dfsbcq 3315 . . . . . 6  |-  ( a  =  1  ->  ( [. a  /  x ]. ph  <->  [. 1  /  x ]. ph ) )
53, 4anbi12d 710 . . . . 5  |-  ( a  =  1  ->  (
( [. ( a  - 
1 )  /  x ]. ph  /\  [. a  /  x ]. ph )  <->  (
[. ( 1  -  1 )  /  x ]. ph  /\  [. 1  /  x ]. ph )
) )
6 oveq1 6288 . . . . . . 7  |-  ( a  =  y  ->  (
a  -  1 )  =  ( y  - 
1 ) )
76sbceq1d 3318 . . . . . 6  |-  ( a  =  y  ->  ( [. ( a  -  1 )  /  x ]. ph  <->  [. ( y  -  1 )  /  x ]. ph ) )
8 dfsbcq 3315 . . . . . 6  |-  ( a  =  y  ->  ( [. a  /  x ]. ph  <->  [. y  /  x ]. ph ) )
97, 8anbi12d 710 . . . . 5  |-  ( a  =  y  ->  (
( [. ( a  - 
1 )  /  x ]. ph  /\  [. a  /  x ]. ph )  <->  (
[. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
) )
10 oveq1 6288 . . . . . . 7  |-  ( a  =  ( y  +  1 )  ->  (
a  -  1 )  =  ( ( y  +  1 )  - 
1 ) )
1110sbceq1d 3318 . . . . . 6  |-  ( a  =  ( y  +  1 )  ->  ( [. ( a  -  1 )  /  x ]. ph  <->  [. ( ( y  +  1 )  -  1 )  /  x ]. ph ) )
12 dfsbcq 3315 . . . . . 6  |-  ( a  =  ( y  +  1 )  ->  ( [. a  /  x ]. ph  <->  [. ( y  +  1 )  /  x ]. ph ) )
1311, 12anbi12d 710 . . . . 5  |-  ( a  =  ( y  +  1 )  ->  (
( [. ( a  - 
1 )  /  x ]. ph  /\  [. a  /  x ]. ph )  <->  (
[. ( ( y  +  1 )  - 
1 )  /  x ]. ph  /\  [. (
y  +  1 )  /  x ]. ph )
) )
14 oveq1 6288 . . . . . . 7  |-  ( a  =  ( A  + 
1 )  ->  (
a  -  1 )  =  ( ( A  +  1 )  - 
1 ) )
1514sbceq1d 3318 . . . . . 6  |-  ( a  =  ( A  + 
1 )  ->  ( [. ( a  -  1 )  /  x ]. ph  <->  [. ( ( A  + 
1 )  -  1 )  /  x ]. ph ) )
16 dfsbcq 3315 . . . . . 6  |-  ( a  =  ( A  + 
1 )  ->  ( [. a  /  x ]. ph  <->  [. ( A  + 
1 )  /  x ]. ph ) )
1715, 16anbi12d 710 . . . . 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 6309 . . . . . . . 8  |-  ( 1  -  1 )  e. 
_V
20 1m1e0 10610 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
2120eqeq2i 2461 . . . . . . . . 9  |-  ( x  =  ( 1  -  1 )  <->  x  = 
0 )
22 2nn0ind.4 . . . . . . . . 9  |-  ( x  =  0  ->  ( ph 
<->  ps ) )
2321, 22sylbi 195 . . . . . . . 8  |-  ( x  =  ( 1  -  1 )  ->  ( ph 
<->  ps ) )
2419, 23sbcie 3348 . . . . . . 7  |-  ( [. ( 1  -  1 )  /  x ]. ph  <->  ps )
2518, 24mpbir 209 . . . . . 6  |-  [. (
1  -  1 )  /  x ]. ph
26 2nn0ind.2 . . . . . . 7  |-  ch
27 1ex 9594 . . . . . . . 8  |-  1  e.  _V
28 2nn0ind.5 . . . . . . . 8  |-  ( x  =  1  ->  ( ph 
<->  ch ) )
2927, 28sbcie 3348 . . . . . . 7  |-  ( [.
1  /  x ]. ph  <->  ch )
3026, 29mpbir 209 . . . . . 6  |-  [. 1  /  x ]. ph
3125, 30pm3.2i 455 . . . . 5  |-  ( [. ( 1  -  1 )  /  x ]. ph 
/\  [. 1  /  x ]. ph )
32 simprr 757 . . . . . . . 8  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  [. y  /  x ]. ph )
33 nncn 10550 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  y  e.  CC )
34 ax-1cn 9553 . . . . . . . . . . 11  |-  1  e.  CC
35 pncan 9831 . . . . . . . . . . 11  |-  ( ( y  e.  CC  /\  1  e.  CC )  ->  ( ( y  +  1 )  -  1 )  =  y )
3633, 34, 35sylancl 662 . . . . . . . . . 10  |-  ( y  e.  NN  ->  (
( y  +  1 )  -  1 )  =  y )
3736adantr 465 . . . . . . . . 9  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  ( (
y  +  1 )  -  1 )  =  y )
3837sbceq1d 3318 . . . . . . . 8  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  ( [. ( ( y  +  1 )  -  1 )  /  x ]. ph  <->  [. y  /  x ]. ph ) )
3932, 38mpbird 232 . . . . . . 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 6309 . . . . . . . . . . 11  |-  ( y  -  1 )  e. 
_V
42 2nn0ind.6 . . . . . . . . . . 11  |-  ( x  =  ( y  - 
1 )  ->  ( ph 
<->  th ) )
4341, 42sbcie 3348 . . . . . . . . . 10  |-  ( [. ( y  -  1 )  /  x ]. ph  <->  th )
44 vex 3098 . . . . . . . . . . 11  |-  y  e. 
_V
45 2nn0ind.7 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( ph 
<->  ta ) )
4644, 45sbcie 3348 . . . . . . . . . 10  |-  ( [. y  /  x ]. ph  <->  ta )
4743, 46anbi12i 697 . . . . . . . . 9  |-  ( (
[. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )  <->  ( th  /\  ta )
)
48 ovex 6309 . . . . . . . . . 10  |-  ( y  +  1 )  e. 
_V
49 2nn0ind.8 . . . . . . . . . 10  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  et ) )
5048, 49sbcie 3348 . . . . . . . . 9  |-  ( [. ( y  +  1 )  /  x ]. ph  <->  et )
5140, 47, 503imtr4g 270 . . . . . . . 8  |-  ( y  e.  NN  ->  (
( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )  ->  [. ( y  +  1 )  /  x ]. ph ) )
5251imp 429 . . . . . . 7  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  [. ( y  +  1 )  /  x ]. ph )
5339, 52jca 532 . . . . . 6  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  ( [. ( ( y  +  1 )  -  1 )  /  x ]. ph 
/\  [. ( y  +  1 )  /  x ]. ph ) )
5453ex 434 . . . . 5  |-  ( y  e.  NN  ->  (
( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )  ->  ( [. ( ( y  +  1 )  -  1 )  /  x ]. ph  /\  [. (
y  +  1 )  /  x ]. ph )
) )
555, 9, 13, 17, 31, 54nnind 10560 . . . 4  |-  ( ( A  +  1 )  e.  NN  ->  ( [. ( ( A  + 
1 )  -  1 )  /  x ]. ph 
/\  [. ( A  + 
1 )  /  x ]. ph ) )
561, 55syl 16 . . 3  |-  ( A  e.  NN0  ->  ( [. ( ( A  + 
1 )  -  1 )  /  x ]. ph 
/\  [. ( A  + 
1 )  /  x ]. ph ) )
57 nn0cn 10811 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  CC )
58 pncan 9831 . . . . . . 7  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
5957, 34, 58sylancl 662 . . . . . 6  |-  ( A  e.  NN0  ->  ( ( A  +  1 )  -  1 )  =  A )
6059sbceq1d 3318 . . . . 5  |-  ( A  e.  NN0  ->  ( [. ( ( A  + 
1 )  -  1 )  /  x ]. ph  <->  [. A  /  x ]. ph ) )
6160biimpa 484 . . . 4  |-  ( ( A  e.  NN0  /\  [. ( ( A  + 
1 )  -  1 )  /  x ]. ph )  ->  [. A  /  x ]. ph )
6261adantrr 716 . . 3  |-  ( ( A  e.  NN0  /\  ( [. ( ( A  +  1 )  - 
1 )  /  x ]. ph  /\  [. ( A  +  1 )  /  x ]. ph )
)  ->  [. A  /  x ]. ph )
6356, 62mpdan 668 . 2  |-  ( A  e.  NN0  ->  [. A  /  x ]. ph )
64 2nn0ind.9 . . 3  |-  ( x  =  A  ->  ( ph 
<->  rh ) )
6564sbcieg 3346 . 2  |-  ( A  e.  NN0  ->  ( [. A  /  x ]. ph  <->  rh )
)
6663, 65mpbid 210 1  |-  ( A  e.  NN0  ->  rh )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   [.wsbc 3313  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496    + caddc 9498    - cmin 9810   NNcn 10542   NN0cn0 10801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-ltxr 9636  df-sub 9812  df-nn 10543  df-n0 10802
This theorem is referenced by:  jm2.18  30905  jm2.15nn0  30920  jm2.16nn0  30921
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