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Theorem bezoutlemb 11695
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 9072 . . 3  |-  0  e.  ZZ
2 1z 9087 . . 3  |-  1  e.  ZZ
3 bezoutlema.a . . . . . . 7  |-  ( th 
->  A  e.  NN0 )
43nn0cnd 9039 . . . . . 6  |-  ( th 
->  A  e.  CC )
54mul01d 8162 . . . . 5  |-  ( th 
->  ( A  x.  0 )  =  0 )
65oveq1d 5789 . . . 4  |-  ( th 
->  ( ( A  x.  0 )  +  ( B  x.  1 ) )  =  ( 0  +  ( B  x.  1 ) ) )
7 bezoutlema.b . . . . . . 7  |-  ( th 
->  B  e.  NN0 )
87nn0cnd 9039 . . . . . 6  |-  ( th 
->  B  e.  CC )
9 1cnd 7789 . . . . . 6  |-  ( th 
->  1  e.  CC )
108, 9mulcld 7793 . . . . 5  |-  ( th 
->  ( B  x.  1 )  e.  CC )
1110addid2d 7919 . . . 4  |-  ( th 
->  ( 0  +  ( B  x.  1 ) )  =  ( B  x.  1 ) )
128mulid1d 7790 . . . 4  |-  ( th 
->  ( B  x.  1 )  =  B )
136, 11, 123eqtrrd 2177 . . 3  |-  ( th 
->  B  =  (
( A  x.  0 )  +  ( B  x.  1 ) ) )
14 oveq2 5782 . . . . . 6  |-  ( s  =  0  ->  ( A  x.  s )  =  ( A  x.  0 ) )
1514oveq1d 5789 . . . . 5  |-  ( s  =  0  ->  (
( A  x.  s
)  +  ( B  x.  t ) )  =  ( ( A  x.  0 )  +  ( B  x.  t
) ) )
1615eqeq2d 2151 . . . 4  |-  ( s  =  0  ->  ( B  =  ( ( A  x.  s )  +  ( B  x.  t ) )  <->  B  =  ( ( A  x.  0 )  +  ( B  x.  t ) ) ) )
17 oveq2 5782 . . . . . 6  |-  ( t  =  1  ->  ( B  x.  t )  =  ( B  x.  1 ) )
1817oveq2d 5790 . . . . 5  |-  ( t  =  1  ->  (
( A  x.  0 )  +  ( B  x.  t ) )  =  ( ( A  x.  0 )  +  ( B  x.  1 ) ) )
1918eqeq2d 2151 . . . 4  |-  ( t  =  1  ->  ( B  =  ( ( A  x.  0 )  +  ( B  x.  t ) )  <->  B  =  ( ( A  x.  0 )  +  ( B  x.  1 ) ) ) )
2016, 19rspc2ev 2804 . . 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 1319 . 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 2146 . . . . . 6  |-  ( r  =  B  ->  (
r  =  ( ( A  x.  s )  +  ( B  x.  t ) )  <->  B  =  ( ( A  x.  s )  +  ( B  x.  t ) ) ) )
24232rexbidv 2460 . . . . 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 2941 . . 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 1331    e. wcel 1480   E.wrex 2417   [.wsbc 2909  (class class class)co 5774   0cc0 7627   1c1 7628    + caddc 7630    x. cmul 7632   NN0cn0 8984   ZZcz 9061
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-ltadd 7743
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-inn 8728  df-n0 8985  df-z 9062
This theorem is referenced by:  bezoutlemex  11696
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