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Theorem zindd 8834
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 8751 . . . . . . 7  |-  ( y  e.  ZZ  ->  -u y  e.  ZZ )
2 elznn0nn 8734 . . . . . . 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 713 . . . . . 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 8725 . . . . . . . 8  |-  ( y  e.  ZZ  ->  y  e.  CC )
87negnegd 7763 . . . . . . 7  |-  ( y  e.  ZZ  ->  -u -u y  =  y )
98eleq1d 2156 . . . . . 6  |-  ( y  e.  ZZ  ->  ( -u -u y  e.  NN  <->  y  e.  NN ) )
109orbi2d 739 . . . . 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 8830 . . . . . 6  |-  ( -u y  e.  NN0  ->  ( ze  ->  th ) )
2524com12 30 . . . . 5  |-  ( ze 
->  ( -u y  e. 
NN0  ->  th ) )
26 nnnn0 8650 . . . . . . . 8  |-  ( y  e.  NN  ->  y  e.  NN0 )
2713, 15, 17, 15, 20, 23nn0ind 8830 . . . . . . . 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 672 . . . 4  |-  ( ze 
->  ( ( -u y  e.  NN0  \/  y  e.  NN )  ->  th )
)
3311, 32syl5 32 . . 3  |-  ( ze 
->  ( y  e.  ZZ  ->  th ) )
3433ralrimiv 2445 . 2  |-  ( ze 
->  A. y  e.  ZZ  th )
35 znegcl 8751 . . . . 5  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
36 negeq 7654 . . . . . . . . 9  |-  ( y  =  -u x  ->  -u y  =  -u -u x )
37 zcn 8725 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  x  e.  CC )
3837negnegd 7763 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  -u -u x  =  x )
3936, 38sylan9eqr 2142 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  -u y  =  x )
4039eqcomd 2093 . . . . . . 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 2725 . . . 4  |-  ( x  e.  ZZ  ->  ( A. y  e.  ZZ  th 
->  ph ) )
4443com12 30 . . 3  |-  ( A. y  e.  ZZ  th  ->  ( x  e.  ZZ  ->  ph ) )
4544ralrimiv 2445 . 2  |-  ( A. y  e.  ZZ  th  ->  A. x  e.  ZZ  ph )
46 zindd.5 . . 3  |-  ( x  =  A  ->  ( ph 
<->  et ) )
4746rspccv 2719 . 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 664    = wceq 1289    e. wcel 1438   A.wral 2359  (class class class)co 5634   RRcr 7328   0cc0 7329   1c1 7330    + caddc 7332   -ucneg 7633   NNcn 8394   NN0cn0 8643   ZZcz 8720
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721
This theorem is referenced by: (None)
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