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Theorem zindd 8546
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 8463 . . . . . . 7  |-  ( y  e.  ZZ  ->  -u y  e.  ZZ )
2 elznn0nn 8446 . . . . . . 7  |-  ( -u y  e.  ZZ  <->  ( -u y  e.  NN0  \/  ( -u y  e.  RR  /\  -u -u y  e.  NN ) ) )
31, 2sylib 120 . . . . . 6  |-  ( y  e.  ZZ  ->  ( -u y  e.  NN0  \/  ( -u y  e.  RR  /\  -u -u y  e.  NN ) ) )
4 simpr 108 . . . . . . 7  |-  ( (
-u y  e.  RR  /\  -u -u y  e.  NN )  ->  -u -u y  e.  NN )
54orim2i 711 . . . . . 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 8437 . . . . . . . 8  |-  ( y  e.  ZZ  ->  y  e.  CC )
87negnegd 7477 . . . . . . 7  |-  ( y  e.  ZZ  ->  -u -u y  =  y )
98eleq1d 2148 . . . . . 6  |-  ( y  e.  ZZ  ->  ( -u -u y  e.  NN  <->  y  e.  NN ) )
109orbi2d 737 . . . . 5  |-  ( y  e.  ZZ  ->  (
( -u y  e.  NN0  \/  -u -u y  e.  NN ) 
<->  ( -u y  e. 
NN0  \/  y  e.  NN ) ) )
116, 10mpbid 145 . . . 4  |-  ( y  e.  ZZ  ->  ( -u y  e.  NN0  \/  y  e.  NN )
)
12 zindd.1 . . . . . . . 8  |-  ( x  =  0  ->  ( ph 
<->  ps ) )
1312imbi2d 228 . . . . . . 7  |-  ( x  =  0  ->  (
( ze  ->  ph )  <->  ( ze  ->  ps )
) )
14 zindd.2 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
1514imbi2d 228 . . . . . . 7  |-  ( x  =  y  ->  (
( ze  ->  ph )  <->  ( ze  ->  ch )
) )
16 zindd.3 . . . . . . . 8  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  ta ) )
1716imbi2d 228 . . . . . . 7  |-  ( x  =  ( y  +  1 )  ->  (
( ze  ->  ph )  <->  ( ze  ->  ta )
) )
18 zindd.4 . . . . . . . 8  |-  ( x  =  -u y  ->  ( ph 
<->  th ) )
1918imbi2d 228 . . . . . . 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 8542 . . . . . 6  |-  ( -u y  e.  NN0  ->  ( ze  ->  th ) )
2524com12 30 . . . . 5  |-  ( ze 
->  ( -u y  e. 
NN0  ->  th ) )
26 nnnn0 8362 . . . . . . . 8  |-  ( y  e.  NN  ->  y  e.  NN0 )
2713, 15, 17, 15, 20, 23nn0ind 8542 . . . . . . . 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 40 . . . . 5  |-  ( ze 
->  ( y  e.  NN  ->  th ) )
3225, 31jaod 670 . . . 4  |-  ( ze 
->  ( ( -u y  e.  NN0  \/  y  e.  NN )  ->  th )
)
3311, 32syl5 32 . . 3  |-  ( ze 
->  ( y  e.  ZZ  ->  th ) )
3433ralrimiv 2434 . 2  |-  ( ze 
->  A. y  e.  ZZ  th )
35 znegcl 8463 . . . . 5  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
36 negeq 7368 . . . . . . . . 9  |-  ( y  =  -u x  ->  -u y  =  -u -u x )
37 zcn 8437 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  x  e.  CC )
3837negnegd 7477 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  -u -u x  =  x )
3936, 38sylan9eqr 2136 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  -u y  =  x )
4039eqcomd 2087 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  x  =  -u y )
4140, 18syl 14 . . . . . 6  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  ( ph  <->  th )
)
4241bicomd 139 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  ( th  <->  ph ) )
4335, 42rspcdv 2705 . . . 4  |-  ( x  e.  ZZ  ->  ( A. y  e.  ZZ  th 
->  ph ) )
4443com12 30 . . 3  |-  ( A. y  e.  ZZ  th  ->  ( x  e.  ZZ  ->  ph ) )
4544ralrimiv 2434 . 2  |-  ( A. y  e.  ZZ  th  ->  A. x  e.  ZZ  ph )
46 zindd.5 . . 3  |-  ( x  =  A  ->  ( ph 
<->  et ) )
4746rspccv 2699 . 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 102    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434   A.wral 2349  (class class class)co 5543   RRcr 7042   0cc0 7043   1c1 7044    + caddc 7046   -ucneg 7347   NNcn 8106   NN0cn0 8355   ZZcz 8432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433
This theorem is referenced by: (None)
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