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Theorem bezoutlemb 11615
Description: Lemma for Bézout's identity. The is-bezout condition is satisfied by  B. (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
bezoutlemb  |-  ( th 
->  [. B  /  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 bezoutlemb
StepHypRef Expression
1 0z 9033 . . 3  |-  0  e.  ZZ
2 1z 9048 . . 3  |-  1  e.  ZZ
3 bezoutlema.a . . . . . . 7  |-  ( th 
->  A  e.  NN0 )
43nn0cnd 9000 . . . . . 6  |-  ( th 
->  A  e.  CC )
54mul01d 8123 . . . . 5  |-  ( th 
->  ( A  x.  0 )  =  0 )
65oveq1d 5757 . . . 4  |-  ( th 
->  ( ( A  x.  0 )  +  ( B  x.  1 ) )  =  ( 0  +  ( B  x.  1 ) ) )
7 bezoutlema.b . . . . . . 7  |-  ( th 
->  B  e.  NN0 )
87nn0cnd 9000 . . . . . 6  |-  ( th 
->  B  e.  CC )
9 1cnd 7750 . . . . . 6  |-  ( th 
->  1  e.  CC )
108, 9mulcld 7754 . . . . 5  |-  ( th 
->  ( B  x.  1 )  e.  CC )
1110addid2d 7880 . . . 4  |-  ( th 
->  ( 0  +  ( B  x.  1 ) )  =  ( B  x.  1 ) )
128mulid1d 7751 . . . 4  |-  ( th 
->  ( B  x.  1 )  =  B )
136, 11, 123eqtrrd 2155 . . 3  |-  ( th 
->  B  =  (
( A  x.  0 )  +  ( B  x.  1 ) ) )
14 oveq2 5750 . . . . . 6  |-  ( s  =  0  ->  ( A  x.  s )  =  ( A  x.  0 ) )
1514oveq1d 5757 . . . . 5  |-  ( s  =  0  ->  (
( A  x.  s
)  +  ( B  x.  t ) )  =  ( ( A  x.  0 )  +  ( B  x.  t
) ) )
1615eqeq2d 2129 . . . 4  |-  ( s  =  0  ->  ( B  =  ( ( A  x.  s )  +  ( B  x.  t ) )  <->  B  =  ( ( A  x.  0 )  +  ( B  x.  t ) ) ) )
17 oveq2 5750 . . . . . 6  |-  ( t  =  1  ->  ( B  x.  t )  =  ( B  x.  1 ) )
1817oveq2d 5758 . . . . 5  |-  ( t  =  1  ->  (
( A  x.  0 )  +  ( B  x.  t ) )  =  ( ( A  x.  0 )  +  ( B  x.  1 ) ) )
1918eqeq2d 2129 . . . 4  |-  ( t  =  1  ->  ( B  =  ( ( A  x.  0 )  +  ( B  x.  t ) )  <->  B  =  ( ( A  x.  0 )  +  ( B  x.  1 ) ) ) )
2016, 19rspc2ev 2778 . . 3  |-  ( ( 0  e.  ZZ  /\  1  e.  ZZ  /\  B  =  ( ( A  x.  0 )  +  ( B  x.  1 ) ) )  ->  E. s  e.  ZZ  E. t  e.  ZZ  B  =  ( ( A  x.  s )  +  ( B  x.  t
) ) )
211, 2, 13, 20mp3an12i 1304 . 2  |-  ( th 
->  E. s  e.  ZZ  E. t  e.  ZZ  B  =  ( ( 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 2124 . . . . . 6  |-  ( r  =  B  ->  (
r  =  ( ( A  x.  s )  +  ( B  x.  t ) )  <->  B  =  ( ( A  x.  s )  +  ( B  x.  t ) ) ) )
24232rexbidv 2437 . . . . 5  |-  ( r  =  B  ->  ( E. s  e.  ZZ  E. t  e.  ZZ  r  =  ( ( A  x.  s )  +  ( B  x.  t
) )  <->  E. s  e.  ZZ  E. t  e.  ZZ  B  =  ( ( A  x.  s
)  +  ( B  x.  t ) ) ) )
2522, 24syl5bb 191 . . . 4  |-  ( r  =  B  ->  ( ph 
<->  E. s  e.  ZZ  E. t  e.  ZZ  B  =  ( ( A  x.  s )  +  ( B  x.  t
) ) ) )
2625sbcieg 2913 . . 3  |-  ( B  e.  NN0  ->  ( [. B  /  r ]. ph  <->  E. s  e.  ZZ  E. t  e.  ZZ  B  =  ( ( A  x.  s
)  +  ( B  x.  t ) ) ) )
277, 26syl 14 . 2  |-  ( th 
->  ( [. B  / 
r ]. ph  <->  E. s  e.  ZZ  E. t  e.  ZZ  B  =  ( ( A  x.  s
)  +  ( B  x.  t ) ) ) )
2821, 27mpbird 166 1  |-  ( th 
->  [. B  /  r ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316    e. wcel 1465   E.wrex 2394   [.wsbc 2882  (class class class)co 5742   0cc0 7588   1c1 7589    + caddc 7591    x. cmul 7593   NN0cn0 8945   ZZcz 9022
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8689  df-n0 8946  df-z 9023
This theorem is referenced by:  bezoutlemex  11616
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