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Theorem bezoutlema 11594
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 9034 . . 3  |-  1  e.  ZZ
2 0z 9019 . . 3  |-  0  e.  ZZ
3 bezoutlema.b . . . . . . 7  |-  ( th 
->  B  e.  NN0 )
43nn0cnd 8986 . . . . . 6  |-  ( th 
->  B  e.  CC )
54mul01d 8119 . . . . 5  |-  ( th 
->  ( B  x.  0 )  =  0 )
65oveq2d 5756 . . . 4  |-  ( th 
->  ( ( A  x.  1 )  +  ( B  x.  0 ) )  =  ( ( A  x.  1 )  +  0 ) )
7 bezoutlema.a . . . . . . 7  |-  ( th 
->  A  e.  NN0 )
87nn0cnd 8986 . . . . . 6  |-  ( th 
->  A  e.  CC )
9 1cnd 7746 . . . . . 6  |-  ( th 
->  1  e.  CC )
108, 9mulcld 7750 . . . . 5  |-  ( th 
->  ( A  x.  1 )  e.  CC )
1110addid1d 7875 . . . 4  |-  ( th 
->  ( ( A  x.  1 )  +  0 )  =  ( A  x.  1 ) )
128mulid1d 7747 . . . 4  |-  ( th 
->  ( A  x.  1 )  =  A )
136, 11, 123eqtrrd 2153 . . 3  |-  ( th 
->  A  =  (
( A  x.  1 )  +  ( B  x.  0 ) ) )
14 oveq2 5748 . . . . . 6  |-  ( s  =  1  ->  ( A  x.  s )  =  ( A  x.  1 ) )
1514oveq1d 5755 . . . . 5  |-  ( s  =  1  ->  (
( A  x.  s
)  +  ( B  x.  t ) )  =  ( ( A  x.  1 )  +  ( B  x.  t
) ) )
1615eqeq2d 2127 . . . 4  |-  ( s  =  1  ->  ( A  =  ( ( A  x.  s )  +  ( B  x.  t ) )  <->  A  =  ( ( A  x.  1 )  +  ( B  x.  t ) ) ) )
17 oveq2 5748 . . . . . 6  |-  ( t  =  0  ->  ( B  x.  t )  =  ( B  x.  0 ) )
1817oveq2d 5756 . . . . 5  |-  ( t  =  0  ->  (
( A  x.  1 )  +  ( B  x.  t ) )  =  ( ( A  x.  1 )  +  ( B  x.  0 ) ) )
1918eqeq2d 2127 . . . 4  |-  ( t  =  0  ->  ( A  =  ( ( A  x.  1 )  +  ( B  x.  t ) )  <->  A  =  ( ( A  x.  1 )  +  ( B  x.  0 ) ) ) )
2016, 19rspc2ev 2776 . . 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 1302 . 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 2122 . . . . . 6  |-  ( r  =  A  ->  (
r  =  ( ( A  x.  s )  +  ( B  x.  t ) )  <->  A  =  ( ( A  x.  s )  +  ( B  x.  t ) ) ) )
24232rexbidv 2435 . . . . 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 191 . . . 4  |-  ( r  =  A  ->  ( ph 
<->  E. s  e.  ZZ  E. t  e.  ZZ  A  =  ( ( A  x.  s )  +  ( B  x.  t
) ) ) )
2625sbcieg 2911 . . 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 166 1  |-  ( th 
->  [. A  /  r ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1314    e. wcel 1463   E.wrex 2392   [.wsbc 2880  (class class class)co 5740   0cc0 7584   1c1 7585    + caddc 7587    x. cmul 7589   NN0cn0 8931   ZZcz 9008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8681  df-n0 8932  df-z 9009
This theorem is referenced by:  bezoutlemex  11596
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