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Theorem bezoutlemb 12028
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 9289 . . 3  |-  0  e.  ZZ
2 1z 9304 . . 3  |-  1  e.  ZZ
3 bezoutlema.a . . . . . . 7  |-  ( th 
->  A  e.  NN0 )
43nn0cnd 9256 . . . . . 6  |-  ( th 
->  A  e.  CC )
54mul01d 8375 . . . . 5  |-  ( th 
->  ( A  x.  0 )  =  0 )
65oveq1d 5907 . . . 4  |-  ( th 
->  ( ( A  x.  0 )  +  ( B  x.  1 ) )  =  ( 0  +  ( B  x.  1 ) ) )
7 bezoutlema.b . . . . . . 7  |-  ( th 
->  B  e.  NN0 )
87nn0cnd 9256 . . . . . 6  |-  ( th 
->  B  e.  CC )
9 1cnd 7998 . . . . . 6  |-  ( th 
->  1  e.  CC )
108, 9mulcld 8003 . . . . 5  |-  ( th 
->  ( B  x.  1 )  e.  CC )
1110addlidd 8132 . . . 4  |-  ( th 
->  ( 0  +  ( B  x.  1 ) )  =  ( B  x.  1 ) )
128mulridd 7999 . . . 4  |-  ( th 
->  ( B  x.  1 )  =  B )
136, 11, 123eqtrrd 2227 . . 3  |-  ( th 
->  B  =  (
( A  x.  0 )  +  ( B  x.  1 ) ) )
14 oveq2 5900 . . . . . 6  |-  ( s  =  0  ->  ( A  x.  s )  =  ( A  x.  0 ) )
1514oveq1d 5907 . . . . 5  |-  ( s  =  0  ->  (
( A  x.  s
)  +  ( B  x.  t ) )  =  ( ( A  x.  0 )  +  ( B  x.  t
) ) )
1615eqeq2d 2201 . . . 4  |-  ( s  =  0  ->  ( B  =  ( ( A  x.  s )  +  ( B  x.  t ) )  <->  B  =  ( ( A  x.  0 )  +  ( B  x.  t ) ) ) )
17 oveq2 5900 . . . . . 6  |-  ( t  =  1  ->  ( B  x.  t )  =  ( B  x.  1 ) )
1817oveq2d 5908 . . . . 5  |-  ( t  =  1  ->  (
( A  x.  0 )  +  ( B  x.  t ) )  =  ( ( A  x.  0 )  +  ( B  x.  1 ) ) )
1918eqeq2d 2201 . . . 4  |-  ( t  =  1  ->  ( B  =  ( ( A  x.  0 )  +  ( B  x.  t ) )  <->  B  =  ( ( A  x.  0 )  +  ( B  x.  1 ) ) ) )
2016, 19rspc2ev 2871 . . 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 1352 . 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 2196 . . . . . 6  |-  ( r  =  B  ->  (
r  =  ( ( A  x.  s )  +  ( B  x.  t ) )  <->  B  =  ( ( A  x.  s )  +  ( B  x.  t ) ) ) )
24232rexbidv 2515 . . . . 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, 24bitrid 192 . . . 4  |-  ( r  =  B  ->  ( ph 
<->  E. s  e.  ZZ  E. t  e.  ZZ  B  =  ( ( A  x.  s )  +  ( B  x.  t
) ) ) )
2625sbcieg 3010 . . 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 167 1  |-  ( th 
->  [. B  /  r ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1364    e. wcel 2160   E.wrex 2469   [.wsbc 2977  (class class class)co 5892   0cc0 7836   1c1 7837    + caddc 7839    x. cmul 7841   NN0cn0 9201   ZZcz 9278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-addcom 7936  ax-mulcom 7937  ax-addass 7938  ax-mulass 7939  ax-distr 7940  ax-i2m1 7941  ax-0lt1 7942  ax-1rid 7943  ax-0id 7944  ax-rnegex 7945  ax-cnre 7947  ax-pre-ltirr 7948  ax-pre-ltwlin 7949  ax-pre-lttrn 7950  ax-pre-ltadd 7952
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5234  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-pnf 8019  df-mnf 8020  df-xr 8021  df-ltxr 8022  df-le 8023  df-sub 8155  df-neg 8156  df-inn 8945  df-n0 9202  df-z 9279
This theorem is referenced by:  bezoutlemex  12029
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