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Theorem 2nn0ind 35864
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 10933 . . . 4  |-  ( A  e.  NN0  ->  ( A  +  1 )  e.  NN )
2 oveq1 6315 . . . . . . 7  |-  ( a  =  1  ->  (
a  -  1 )  =  ( 1  -  1 ) )
32sbceq1d 3260 . . . . . 6  |-  ( a  =  1  ->  ( [. ( a  -  1 )  /  x ]. ph  <->  [. ( 1  -  1 )  /  x ]. ph ) )
4 dfsbcq 3257 . . . . . 6  |-  ( a  =  1  ->  ( [. a  /  x ]. ph  <->  [. 1  /  x ]. ph ) )
53, 4anbi12d 725 . . . . 5  |-  ( a  =  1  ->  (
( [. ( a  - 
1 )  /  x ]. ph  /\  [. a  /  x ]. ph )  <->  (
[. ( 1  -  1 )  /  x ]. ph  /\  [. 1  /  x ]. ph )
) )
6 oveq1 6315 . . . . . . 7  |-  ( a  =  y  ->  (
a  -  1 )  =  ( y  - 
1 ) )
76sbceq1d 3260 . . . . . 6  |-  ( a  =  y  ->  ( [. ( a  -  1 )  /  x ]. ph  <->  [. ( y  -  1 )  /  x ]. ph ) )
8 dfsbcq 3257 . . . . . 6  |-  ( a  =  y  ->  ( [. a  /  x ]. ph  <->  [. y  /  x ]. ph ) )
97, 8anbi12d 725 . . . . 5  |-  ( a  =  y  ->  (
( [. ( a  - 
1 )  /  x ]. ph  /\  [. a  /  x ]. ph )  <->  (
[. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
) )
10 oveq1 6315 . . . . . . 7  |-  ( a  =  ( y  +  1 )  ->  (
a  -  1 )  =  ( ( y  +  1 )  - 
1 ) )
1110sbceq1d 3260 . . . . . 6  |-  ( a  =  ( y  +  1 )  ->  ( [. ( a  -  1 )  /  x ]. ph  <->  [. ( ( y  +  1 )  -  1 )  /  x ]. ph ) )
12 dfsbcq 3257 . . . . . 6  |-  ( a  =  ( y  +  1 )  ->  ( [. a  /  x ]. ph  <->  [. ( y  +  1 )  /  x ]. ph ) )
1311, 12anbi12d 725 . . . . 5  |-  ( a  =  ( y  +  1 )  ->  (
( [. ( a  - 
1 )  /  x ]. ph  /\  [. a  /  x ]. ph )  <->  (
[. ( ( y  +  1 )  - 
1 )  /  x ]. ph  /\  [. (
y  +  1 )  /  x ]. ph )
) )
14 oveq1 6315 . . . . . . 7  |-  ( a  =  ( A  + 
1 )  ->  (
a  -  1 )  =  ( ( A  +  1 )  - 
1 ) )
1514sbceq1d 3260 . . . . . 6  |-  ( a  =  ( A  + 
1 )  ->  ( [. ( a  -  1 )  /  x ]. ph  <->  [. ( ( A  + 
1 )  -  1 )  /  x ]. ph ) )
16 dfsbcq 3257 . . . . . 6  |-  ( a  =  ( A  + 
1 )  ->  ( [. a  /  x ]. ph  <->  [. ( A  + 
1 )  /  x ]. ph ) )
1715, 16anbi12d 725 . . . . 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 6336 . . . . . . . 8  |-  ( 1  -  1 )  e. 
_V
20 1m1e0 10700 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
2120eqeq2i 2483 . . . . . . . . 9  |-  ( x  =  ( 1  -  1 )  <->  x  = 
0 )
22 2nn0ind.4 . . . . . . . . 9  |-  ( x  =  0  ->  ( ph 
<->  ps ) )
2321, 22sylbi 200 . . . . . . . 8  |-  ( x  =  ( 1  -  1 )  ->  ( ph 
<->  ps ) )
2419, 23sbcie 3290 . . . . . . 7  |-  ( [. ( 1  -  1 )  /  x ]. ph  <->  ps )
2518, 24mpbir 214 . . . . . 6  |-  [. (
1  -  1 )  /  x ]. ph
26 2nn0ind.2 . . . . . . 7  |-  ch
27 1ex 9656 . . . . . . . 8  |-  1  e.  _V
28 2nn0ind.5 . . . . . . . 8  |-  ( x  =  1  ->  ( ph 
<->  ch ) )
2927, 28sbcie 3290 . . . . . . 7  |-  ( [.
1  /  x ]. ph  <->  ch )
3026, 29mpbir 214 . . . . . 6  |-  [. 1  /  x ]. ph
3125, 30pm3.2i 462 . . . . 5  |-  ( [. ( 1  -  1 )  /  x ]. ph 
/\  [. 1  /  x ]. ph )
32 simprr 774 . . . . . . . 8  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  [. y  /  x ]. ph )
33 nncn 10639 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  y  e.  CC )
34 ax-1cn 9615 . . . . . . . . . . 11  |-  1  e.  CC
35 pncan 9901 . . . . . . . . . . 11  |-  ( ( y  e.  CC  /\  1  e.  CC )  ->  ( ( y  +  1 )  -  1 )  =  y )
3633, 34, 35sylancl 675 . . . . . . . . . 10  |-  ( y  e.  NN  ->  (
( y  +  1 )  -  1 )  =  y )
3736adantr 472 . . . . . . . . 9  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  ( (
y  +  1 )  -  1 )  =  y )
3837sbceq1d 3260 . . . . . . . 8  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  ( [. ( ( y  +  1 )  -  1 )  /  x ]. ph  <->  [. y  /  x ]. ph ) )
3932, 38mpbird 240 . . . . . . 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 6336 . . . . . . . . . . 11  |-  ( y  -  1 )  e. 
_V
42 2nn0ind.6 . . . . . . . . . . 11  |-  ( x  =  ( y  - 
1 )  ->  ( ph 
<->  th ) )
4341, 42sbcie 3290 . . . . . . . . . 10  |-  ( [. ( y  -  1 )  /  x ]. ph  <->  th )
44 vex 3034 . . . . . . . . . . 11  |-  y  e. 
_V
45 2nn0ind.7 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( ph 
<->  ta ) )
4644, 45sbcie 3290 . . . . . . . . . 10  |-  ( [. y  /  x ]. ph  <->  ta )
4743, 46anbi12i 711 . . . . . . . . 9  |-  ( (
[. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )  <->  ( th  /\  ta )
)
48 ovex 6336 . . . . . . . . . 10  |-  ( y  +  1 )  e. 
_V
49 2nn0ind.8 . . . . . . . . . 10  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  et ) )
5048, 49sbcie 3290 . . . . . . . . 9  |-  ( [. ( y  +  1 )  /  x ]. ph  <->  et )
5140, 47, 503imtr4g 278 . . . . . . . 8  |-  ( y  e.  NN  ->  (
( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )  ->  [. ( y  +  1 )  /  x ]. ph ) )
5251imp 436 . . . . . . 7  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  [. ( y  +  1 )  /  x ]. ph )
5339, 52jca 541 . . . . . 6  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  ( [. ( ( y  +  1 )  -  1 )  /  x ]. ph 
/\  [. ( y  +  1 )  /  x ]. ph ) )
5453ex 441 . . . . 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 10649 . . . 4  |-  ( ( A  +  1 )  e.  NN  ->  ( [. ( ( A  + 
1 )  -  1 )  /  x ]. ph 
/\  [. ( A  + 
1 )  /  x ]. ph ) )
561, 55syl 17 . . 3  |-  ( A  e.  NN0  ->  ( [. ( ( A  + 
1 )  -  1 )  /  x ]. ph 
/\  [. ( A  + 
1 )  /  x ]. ph ) )
57 nn0cn 10903 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  CC )
58 pncan 9901 . . . . . . 7  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
5957, 34, 58sylancl 675 . . . . . 6  |-  ( A  e.  NN0  ->  ( ( A  +  1 )  -  1 )  =  A )
6059sbceq1d 3260 . . . . 5  |-  ( A  e.  NN0  ->  ( [. ( ( A  + 
1 )  -  1 )  /  x ]. ph  <->  [. A  /  x ]. ph ) )
6160biimpa 492 . . . 4  |-  ( ( A  e.  NN0  /\  [. ( ( A  + 
1 )  -  1 )  /  x ]. ph )  ->  [. A  /  x ]. ph )
6261adantrr 731 . . 3  |-  ( ( A  e.  NN0  /\  ( [. ( ( A  +  1 )  - 
1 )  /  x ]. ph  /\  [. ( A  +  1 )  /  x ]. ph )
)  ->  [. A  /  x ]. ph )
6356, 62mpdan 681 . 2  |-  ( A  e.  NN0  ->  [. A  /  x ]. ph )
64 2nn0ind.9 . . 3  |-  ( x  =  A  ->  ( ph 
<->  rh ) )
6564sbcieg 3288 . 2  |-  ( A  e.  NN0  ->  ( [. A  /  x ]. ph  <->  rh )
)
6663, 65mpbid 215 1  |-  ( A  e.  NN0  ->  rh )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   [.wsbc 3255  (class class class)co 6308   CCcc 9555   0cc0 9557   1c1 9558    + caddc 9560    - cmin 9880   NNcn 10631   NN0cn0 10893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-ltxr 9698  df-sub 9882  df-nn 10632  df-n0 10894
This theorem is referenced by:  jm2.18  35914  jm2.15nn0  35929  jm2.16nn0  35930
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