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Theorem zindd 9067
Description: Principle of Mathematical Induction on all integers, deduction version. The first five hypotheses give the substitutions; the last three are the basis, the induction, and the extension to negative numbers. (Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
zindd.1  |-  ( x  =  0  ->  ( ph 
<->  ps ) )
zindd.2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
zindd.3  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  ta ) )
zindd.4  |-  ( x  =  -u y  ->  ( ph 
<->  th ) )
zindd.5  |-  ( x  =  A  ->  ( ph 
<->  et ) )
zindd.6  |-  ( ze 
->  ps )
zindd.7  |-  ( ze 
->  ( y  e.  NN0  ->  ( ch  ->  ta ) ) )
zindd.8  |-  ( ze 
->  ( y  e.  NN  ->  ( ch  ->  th )
) )
Assertion
Ref Expression
zindd  |-  ( ze 
->  ( A  e.  ZZ  ->  et ) )
Distinct variable groups:    x, A    ch, x    et, x    ph, y    ps, x    ta, x    th, x    x, y, ze
Allowed substitution hints:    ph( x)    ps( y)    ch( y)    th( y)    ta( y)    et( y)    A( y)

Proof of Theorem zindd
StepHypRef Expression
1 znegcl 8983 . . . . . . 7  |-  ( y  e.  ZZ  ->  -u y  e.  ZZ )
2 elznn0nn 8966 . . . . . . 7  |-  ( -u y  e.  ZZ  <->  ( -u y  e.  NN0  \/  ( -u y  e.  RR  /\  -u -u y  e.  NN ) ) )
31, 2sylib 121 . . . . . 6  |-  ( y  e.  ZZ  ->  ( -u y  e.  NN0  \/  ( -u y  e.  RR  /\  -u -u y  e.  NN ) ) )
4 simpr 109 . . . . . . 7  |-  ( (
-u y  e.  RR  /\  -u -u y  e.  NN )  ->  -u -u y  e.  NN )
54orim2i 733 . . . . . 6  |-  ( (
-u y  e.  NN0  \/  ( -u y  e.  RR  /\  -u -u y  e.  NN ) )  -> 
( -u y  e.  NN0  \/  -u -u y  e.  NN ) )
63, 5syl 14 . . . . 5  |-  ( y  e.  ZZ  ->  ( -u y  e.  NN0  \/  -u -u y  e.  NN ) )
7 zcn 8957 . . . . . . . 8  |-  ( y  e.  ZZ  ->  y  e.  CC )
87negnegd 7981 . . . . . . 7  |-  ( y  e.  ZZ  ->  -u -u y  =  y )
98eleq1d 2181 . . . . . 6  |-  ( y  e.  ZZ  ->  ( -u -u y  e.  NN  <->  y  e.  NN ) )
109orbi2d 762 . . . . 5  |-  ( y  e.  ZZ  ->  (
( -u y  e.  NN0  \/  -u -u y  e.  NN ) 
<->  ( -u y  e. 
NN0  \/  y  e.  NN ) ) )
116, 10mpbid 146 . . . 4  |-  ( y  e.  ZZ  ->  ( -u y  e.  NN0  \/  y  e.  NN )
)
12 zindd.1 . . . . . . . 8  |-  ( x  =  0  ->  ( ph 
<->  ps ) )
1312imbi2d 229 . . . . . . 7  |-  ( x  =  0  ->  (
( ze  ->  ph )  <->  ( ze  ->  ps )
) )
14 zindd.2 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
1514imbi2d 229 . . . . . . 7  |-  ( x  =  y  ->  (
( ze  ->  ph )  <->  ( ze  ->  ch )
) )
16 zindd.3 . . . . . . . 8  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  ta ) )
1716imbi2d 229 . . . . . . 7  |-  ( x  =  ( y  +  1 )  ->  (
( ze  ->  ph )  <->  ( ze  ->  ta )
) )
18 zindd.4 . . . . . . . 8  |-  ( x  =  -u y  ->  ( ph 
<->  th ) )
1918imbi2d 229 . . . . . . 7  |-  ( x  =  -u y  ->  (
( ze  ->  ph )  <->  ( ze  ->  th )
) )
20 zindd.6 . . . . . . 7  |-  ( ze 
->  ps )
21 zindd.7 . . . . . . . . 9  |-  ( ze 
->  ( y  e.  NN0  ->  ( ch  ->  ta ) ) )
2221com12 30 . . . . . . . 8  |-  ( y  e.  NN0  ->  ( ze 
->  ( ch  ->  ta ) ) )
2322a2d 26 . . . . . . 7  |-  ( y  e.  NN0  ->  ( ( ze  ->  ch )  ->  ( ze  ->  ta ) ) )
2413, 15, 17, 19, 20, 23nn0ind 9063 . . . . . 6  |-  ( -u y  e.  NN0  ->  ( ze  ->  th ) )
2524com12 30 . . . . 5  |-  ( ze 
->  ( -u y  e. 
NN0  ->  th ) )
26 nnnn0 8882 . . . . . . . 8  |-  ( y  e.  NN  ->  y  e.  NN0 )
2713, 15, 17, 15, 20, 23nn0ind 9063 . . . . . . . 8  |-  ( y  e.  NN0  ->  ( ze 
->  ch ) )
2826, 27syl 14 . . . . . . 7  |-  ( y  e.  NN  ->  ( ze  ->  ch ) )
2928com12 30 . . . . . 6  |-  ( ze 
->  ( y  e.  NN  ->  ch ) )
30 zindd.8 . . . . . 6  |-  ( ze 
->  ( y  e.  NN  ->  ( ch  ->  th )
) )
3129, 30mpdd 41 . . . . 5  |-  ( ze 
->  ( y  e.  NN  ->  th ) )
3225, 31jaod 689 . . . 4  |-  ( ze 
->  ( ( -u y  e.  NN0  \/  y  e.  NN )  ->  th )
)
3311, 32syl5 32 . . 3  |-  ( ze 
->  ( y  e.  ZZ  ->  th ) )
3433ralrimiv 2476 . 2  |-  ( ze 
->  A. y  e.  ZZ  th )
35 znegcl 8983 . . . . 5  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
36 negeq 7872 . . . . . . . . 9  |-  ( y  =  -u x  ->  -u y  =  -u -u x )
37 zcn 8957 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  x  e.  CC )
3837negnegd 7981 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  -u -u x  =  x )
3936, 38sylan9eqr 2167 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  -u y  =  x )
4039eqcomd 2118 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  x  =  -u y )
4140, 18syl 14 . . . . . 6  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  ( ph  <->  th )
)
4241bicomd 140 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  ( th  <->  ph ) )
4335, 42rspcdv 2761 . . . 4  |-  ( x  e.  ZZ  ->  ( A. y  e.  ZZ  th 
->  ph ) )
4443com12 30 . . 3  |-  ( A. y  e.  ZZ  th  ->  ( x  e.  ZZ  ->  ph ) )
4544ralrimiv 2476 . 2  |-  ( A. y  e.  ZZ  th  ->  A. x  e.  ZZ  ph )
46 zindd.5 . . 3  |-  ( x  =  A  ->  ( ph 
<->  et ) )
4746rspccv 2755 . 2  |-  ( A. x  e.  ZZ  ph  ->  ( A  e.  ZZ  ->  et ) )
4834, 45, 473syl 17 1  |-  ( ze 
->  ( A  e.  ZZ  ->  et ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    = wceq 1312    e. wcel 1461   A.wral 2388  (class class class)co 5726   RRcr 7540   0cc0 7541   1c1 7542    + caddc 7544   -ucneg 7851   NNcn 8624   NN0cn0 8875   ZZcz 8952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-addcom 7639  ax-addass 7641  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-0id 7647  ax-rnegex 7648  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-ltadd 7655
This theorem depends on definitions:  df-bi 116  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-br 3894  df-opab 3948  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-inn 8625  df-n0 8876  df-z 8953
This theorem is referenced by:  efexp  11233
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