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Theorem bezoutlema 10768
Description: Lemma for Bézout's identity. The is-bezout condition is satisfied by  A. (Contributed by Jim Kingdon, 30-Dec-2021.)
Hypotheses
Ref Expression
bezoutlema.is-bezout  |-  ( ph  <->  E. s  e.  ZZ  E. t  e.  ZZ  r  =  ( ( A  x.  s )  +  ( B  x.  t
) ) )
bezoutlema.a  |-  ( th 
->  A  e.  NN0 )
bezoutlema.b  |-  ( th 
->  B  e.  NN0 )
Assertion
Ref Expression
bezoutlema  |-  ( th 
->  [. A  /  r ]. ph )
Distinct variable groups:    A, r, s, t    B, r, s, t
Allowed substitution hints:    ph( t, s, r)    th( t, s, r)

Proof of Theorem bezoutlema
StepHypRef Expression
1 1z 8672 . . 3  |-  1  e.  ZZ
2 0z 8657 . . 3  |-  0  e.  ZZ
3 bezoutlema.b . . . . . . 7  |-  ( th 
->  B  e.  NN0 )
43nn0cnd 8620 . . . . . 6  |-  ( th 
->  B  e.  CC )
54mul01d 7774 . . . . 5  |-  ( th 
->  ( B  x.  0 )  =  0 )
65oveq2d 5607 . . . 4  |-  ( th 
->  ( ( A  x.  1 )  +  ( B  x.  0 ) )  =  ( ( A  x.  1 )  +  0 ) )
7 bezoutlema.a . . . . . . 7  |-  ( th 
->  A  e.  NN0 )
87nn0cnd 8620 . . . . . 6  |-  ( th 
->  A  e.  CC )
9 1cnd 7407 . . . . . 6  |-  ( th 
->  1  e.  CC )
108, 9mulcld 7411 . . . . 5  |-  ( th 
->  ( A  x.  1 )  e.  CC )
1110addid1d 7534 . . . 4  |-  ( th 
->  ( ( A  x.  1 )  +  0 )  =  ( A  x.  1 ) )
128mulid1d 7408 . . . 4  |-  ( th 
->  ( A  x.  1 )  =  A )
136, 11, 123eqtrrd 2120 . . 3  |-  ( th 
->  A  =  (
( A  x.  1 )  +  ( B  x.  0 ) ) )
14 oveq2 5599 . . . . . 6  |-  ( s  =  1  ->  ( A  x.  s )  =  ( A  x.  1 ) )
1514oveq1d 5606 . . . . 5  |-  ( s  =  1  ->  (
( A  x.  s
)  +  ( B  x.  t ) )  =  ( ( A  x.  1 )  +  ( B  x.  t
) ) )
1615eqeq2d 2094 . . . 4  |-  ( s  =  1  ->  ( A  =  ( ( A  x.  s )  +  ( B  x.  t ) )  <->  A  =  ( ( A  x.  1 )  +  ( B  x.  t ) ) ) )
17 oveq2 5599 . . . . . 6  |-  ( t  =  0  ->  ( B  x.  t )  =  ( B  x.  0 ) )
1817oveq2d 5607 . . . . 5  |-  ( t  =  0  ->  (
( A  x.  1 )  +  ( B  x.  t ) )  =  ( ( A  x.  1 )  +  ( B  x.  0 ) ) )
1918eqeq2d 2094 . . . 4  |-  ( t  =  0  ->  ( A  =  ( ( A  x.  1 )  +  ( B  x.  t ) )  <->  A  =  ( ( A  x.  1 )  +  ( B  x.  0 ) ) ) )
2016, 19rspc2ev 2725 . . 3  |-  ( ( 1  e.  ZZ  /\  0  e.  ZZ  /\  A  =  ( ( A  x.  1 )  +  ( B  x.  0 ) ) )  ->  E. s  e.  ZZ  E. t  e.  ZZ  A  =  ( ( A  x.  s )  +  ( B  x.  t
) ) )
211, 2, 13, 20mp3an12i 1273 . 2  |-  ( th 
->  E. s  e.  ZZ  E. t  e.  ZZ  A  =  ( ( A  x.  s )  +  ( B  x.  t
) ) )
22 bezoutlema.is-bezout . . . . 5  |-  ( ph  <->  E. s  e.  ZZ  E. t  e.  ZZ  r  =  ( ( A  x.  s )  +  ( B  x.  t
) ) )
23 eqeq1 2089 . . . . . 6  |-  ( r  =  A  ->  (
r  =  ( ( A  x.  s )  +  ( B  x.  t ) )  <->  A  =  ( ( A  x.  s )  +  ( B  x.  t ) ) ) )
24232rexbidv 2397 . . . . 5  |-  ( r  =  A  ->  ( E. s  e.  ZZ  E. t  e.  ZZ  r  =  ( ( A  x.  s )  +  ( B  x.  t
) )  <->  E. s  e.  ZZ  E. t  e.  ZZ  A  =  ( ( A  x.  s
)  +  ( B  x.  t ) ) ) )
2522, 24syl5bb 190 . . . 4  |-  ( r  =  A  ->  ( ph 
<->  E. s  e.  ZZ  E. t  e.  ZZ  A  =  ( ( A  x.  s )  +  ( B  x.  t
) ) ) )
2625sbcieg 2857 . . 3  |-  ( A  e.  NN0  ->  ( [. A  /  r ]. ph  <->  E. s  e.  ZZ  E. t  e.  ZZ  A  =  ( ( A  x.  s
)  +  ( B  x.  t ) ) ) )
277, 26syl 14 . 2  |-  ( th 
->  ( [. A  / 
r ]. ph  <->  E. s  e.  ZZ  E. t  e.  ZZ  A  =  ( ( A  x.  s
)  +  ( B  x.  t ) ) ) )
2821, 27mpbird 165 1  |-  ( th 
->  [. A  /  r ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434   E.wrex 2354   [.wsbc 2826  (class class class)co 5591   0cc0 7253   1c1 7254    + caddc 7256    x. cmul 7258   NN0cn0 8565   ZZcz 8646
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 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-cnex 7339  ax-resscn 7340  ax-1cn 7341  ax-1re 7342  ax-icn 7343  ax-addcl 7344  ax-addrcl 7345  ax-mulcl 7346  ax-addcom 7348  ax-mulcom 7349  ax-addass 7350  ax-mulass 7351  ax-distr 7352  ax-i2m1 7353  ax-0lt1 7354  ax-1rid 7355  ax-0id 7356  ax-rnegex 7357  ax-cnre 7359  ax-pre-ltirr 7360  ax-pre-ltwlin 7361  ax-pre-lttrn 7362  ax-pre-ltadd 7364
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 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-br 3812  df-opab 3866  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-iota 4934  df-fun 4971  df-fv 4977  df-riota 5547  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-pnf 7427  df-mnf 7428  df-xr 7429  df-ltxr 7430  df-le 7431  df-sub 7558  df-neg 7559  df-inn 8317  df-n0 8566  df-z 8647
This theorem is referenced by:  bezoutlemex  10770
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