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Theorem 2nn0ind 27008
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 10259 . . . 4  |-  ( A  e.  NN0  ->  ( A  +  1 )  e.  NN )
2 oveq1 6088 . . . . . . 7  |-  ( a  =  1  ->  (
a  -  1 )  =  ( 1  -  1 ) )
3 dfsbcq 3163 . . . . . . 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 3163 . . . . . 6  |-  ( a  =  1  ->  ( [. a  /  x ]. ph  <->  [. 1  /  x ]. ph ) )
64, 5anbi12d 692 . . . . 5  |-  ( a  =  1  ->  (
( [. ( a  - 
1 )  /  x ]. ph  /\  [. a  /  x ]. ph )  <->  (
[. ( 1  -  1 )  /  x ]. ph  /\  [. 1  /  x ]. ph )
) )
7 oveq1 6088 . . . . . . 7  |-  ( a  =  y  ->  (
a  -  1 )  =  ( y  - 
1 ) )
8 dfsbcq 3163 . . . . . . 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 3163 . . . . . 6  |-  ( a  =  y  ->  ( [. a  /  x ]. ph  <->  [. y  /  x ]. ph ) )
119, 10anbi12d 692 . . . . 5  |-  ( a  =  y  ->  (
( [. ( a  - 
1 )  /  x ]. ph  /\  [. a  /  x ]. ph )  <->  (
[. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
) )
12 oveq1 6088 . . . . . . 7  |-  ( a  =  ( y  +  1 )  ->  (
a  -  1 )  =  ( ( y  +  1 )  - 
1 ) )
13 dfsbcq 3163 . . . . . . 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 3163 . . . . . 6  |-  ( a  =  ( y  +  1 )  ->  ( [. a  /  x ]. ph  <->  [. ( y  +  1 )  /  x ]. ph ) )
1614, 15anbi12d 692 . . . . 5  |-  ( a  =  ( y  +  1 )  ->  (
( [. ( a  - 
1 )  /  x ]. ph  /\  [. a  /  x ]. ph )  <->  (
[. ( ( y  +  1 )  - 
1 )  /  x ]. ph  /\  [. (
y  +  1 )  /  x ]. ph )
) )
17 oveq1 6088 . . . . . . 7  |-  ( a  =  ( A  + 
1 )  ->  (
a  -  1 )  =  ( ( A  +  1 )  - 
1 ) )
18 dfsbcq 3163 . . . . . . 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 3163 . . . . . 6  |-  ( a  =  ( A  + 
1 )  ->  ( [. a  /  x ]. ph  <->  [. ( A  + 
1 )  /  x ]. ph ) )
2119, 20anbi12d 692 . . . . 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 6106 . . . . . . . 8  |-  ( 1  -  1 )  e. 
_V
24 1m1e0 10068 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
2524eqeq2i 2446 . . . . . . . . 9  |-  ( x  =  ( 1  -  1 )  <->  x  = 
0 )
26 2nn0ind.4 . . . . . . . . 9  |-  ( x  =  0  ->  ( ph 
<->  ps ) )
2725, 26sylbi 188 . . . . . . . 8  |-  ( x  =  ( 1  -  1 )  ->  ( ph 
<->  ps ) )
2823, 27sbcie 3195 . . . . . . 7  |-  ( [. ( 1  -  1 )  /  x ]. ph  <->  ps )
2922, 28mpbir 201 . . . . . 6  |-  [. (
1  -  1 )  /  x ]. ph
30 2nn0ind.2 . . . . . . 7  |-  ch
31 1ex 9086 . . . . . . . 8  |-  1  e.  _V
32 2nn0ind.5 . . . . . . . 8  |-  ( x  =  1  ->  ( ph 
<->  ch ) )
3331, 32sbcie 3195 . . . . . . 7  |-  ( [.
1  /  x ]. ph  <->  ch )
3430, 33mpbir 201 . . . . . 6  |-  [. 1  /  x ]. ph
3529, 34pm3.2i 442 . . . . 5  |-  ( [. ( 1  -  1 )  /  x ]. ph 
/\  [. 1  /  x ]. ph )
36 simprr 734 . . . . . . . 8  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  [. y  /  x ]. ph )
37 nncn 10008 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  y  e.  CC )
38 ax-1cn 9048 . . . . . . . . . . 11  |-  1  e.  CC
39 pncan 9311 . . . . . . . . . . 11  |-  ( ( y  e.  CC  /\  1  e.  CC )  ->  ( ( y  +  1 )  -  1 )  =  y )
4037, 38, 39sylancl 644 . . . . . . . . . 10  |-  ( y  e.  NN  ->  (
( y  +  1 )  -  1 )  =  y )
4140adantr 452 . . . . . . . . 9  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  ( (
y  +  1 )  -  1 )  =  y )
42 dfsbcq 3163 . . . . . . . . 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 224 . . . . . . 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 6106 . . . . . . . . . . 11  |-  ( y  -  1 )  e. 
_V
47 2nn0ind.6 . . . . . . . . . . 11  |-  ( x  =  ( y  - 
1 )  ->  ( ph 
<->  th ) )
4846, 47sbcie 3195 . . . . . . . . . 10  |-  ( [. ( y  -  1 )  /  x ]. ph  <->  th )
49 vex 2959 . . . . . . . . . . 11  |-  y  e. 
_V
50 2nn0ind.7 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( ph 
<->  ta ) )
5149, 50sbcie 3195 . . . . . . . . . 10  |-  ( [. y  /  x ]. ph  <->  ta )
5248, 51anbi12i 679 . . . . . . . . 9  |-  ( (
[. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )  <->  ( th  /\  ta )
)
53 ovex 6106 . . . . . . . . . 10  |-  ( y  +  1 )  e. 
_V
54 2nn0ind.8 . . . . . . . . . 10  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  et ) )
5553, 54sbcie 3195 . . . . . . . . 9  |-  ( [. ( y  +  1 )  /  x ]. ph  <->  et )
5645, 52, 553imtr4g 262 . . . . . . . 8  |-  ( y  e.  NN  ->  (
( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )  ->  [. ( y  +  1 )  /  x ]. ph ) )
5756imp 419 . . . . . . 7  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  [. ( y  +  1 )  /  x ]. ph )
5844, 57jca 519 . . . . . 6  |-  ( ( y  e.  NN  /\  ( [. ( y  - 
1 )  /  x ]. ph  /\  [. y  /  x ]. ph )
)  ->  ( [. ( ( y  +  1 )  -  1 )  /  x ]. ph 
/\  [. ( y  +  1 )  /  x ]. ph ) )
5958ex 424 . . . . 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 10018 . . . 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 10231 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  CC )
63 pncan 9311 . . . . . . 7  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
6462, 38, 63sylancl 644 . . . . . 6  |-  ( A  e.  NN0  ->  ( ( A  +  1 )  -  1 )  =  A )
65 dfsbcq 3163 . . . . . 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 471 . . . 4  |-  ( ( A  e.  NN0  /\  [. ( ( A  + 
1 )  -  1 )  /  x ]. ph )  ->  [. A  /  x ]. ph )
6867adantrr 698 . . 3  |-  ( ( A  e.  NN0  /\  ( [. ( ( A  +  1 )  - 
1 )  /  x ]. ph  /\  [. ( A  +  1 )  /  x ]. ph )
)  ->  [. A  /  x ]. ph )
6961, 68mpdan 650 . 2  |-  ( A  e.  NN0  ->  [. A  /  x ]. ph )
70 2nn0ind.9 . . 3  |-  ( x  =  A  ->  ( ph 
<->  rh ) )
7170sbcieg 3193 . 2  |-  ( A  e.  NN0  ->  ( [. A  /  x ]. ph  <->  rh )
)
7269, 71mpbid 202 1  |-  ( A  e.  NN0  ->  rh )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   [.wsbc 3161  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    - cmin 9291   NNcn 10000   NN0cn0 10221
This theorem is referenced by:  jm2.18  27059  jm2.15nn0  27074  jm2.16nn0  27075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-nn 10001  df-n0 10222
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