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Theorem bezoutlema 12002
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 9281 . . 3  |-  1  e.  ZZ
2 0z 9266 . . 3  |-  0  e.  ZZ
3 bezoutlema.b . . . . . . 7  |-  ( th 
->  B  e.  NN0 )
43nn0cnd 9233 . . . . . 6  |-  ( th 
->  B  e.  CC )
54mul01d 8352 . . . . 5  |-  ( th 
->  ( B  x.  0 )  =  0 )
65oveq2d 5893 . . . 4  |-  ( th 
->  ( ( A  x.  1 )  +  ( B  x.  0 ) )  =  ( ( A  x.  1 )  +  0 ) )
7 bezoutlema.a . . . . . . 7  |-  ( th 
->  A  e.  NN0 )
87nn0cnd 9233 . . . . . 6  |-  ( th 
->  A  e.  CC )
9 1cnd 7975 . . . . . 6  |-  ( th 
->  1  e.  CC )
108, 9mulcld 7980 . . . . 5  |-  ( th 
->  ( A  x.  1 )  e.  CC )
1110addid1d 8108 . . . 4  |-  ( th 
->  ( ( A  x.  1 )  +  0 )  =  ( A  x.  1 ) )
128mulridd 7976 . . . 4  |-  ( th 
->  ( A  x.  1 )  =  A )
136, 11, 123eqtrrd 2215 . . 3  |-  ( th 
->  A  =  (
( A  x.  1 )  +  ( B  x.  0 ) ) )
14 oveq2 5885 . . . . . 6  |-  ( s  =  1  ->  ( A  x.  s )  =  ( A  x.  1 ) )
1514oveq1d 5892 . . . . 5  |-  ( s  =  1  ->  (
( A  x.  s
)  +  ( B  x.  t ) )  =  ( ( A  x.  1 )  +  ( B  x.  t
) ) )
1615eqeq2d 2189 . . . 4  |-  ( s  =  1  ->  ( A  =  ( ( A  x.  s )  +  ( B  x.  t ) )  <->  A  =  ( ( A  x.  1 )  +  ( B  x.  t ) ) ) )
17 oveq2 5885 . . . . . 6  |-  ( t  =  0  ->  ( B  x.  t )  =  ( B  x.  0 ) )
1817oveq2d 5893 . . . . 5  |-  ( t  =  0  ->  (
( A  x.  1 )  +  ( B  x.  t ) )  =  ( ( A  x.  1 )  +  ( B  x.  0 ) ) )
1918eqeq2d 2189 . . . 4  |-  ( t  =  0  ->  ( A  =  ( ( A  x.  1 )  +  ( B  x.  t ) )  <->  A  =  ( ( A  x.  1 )  +  ( B  x.  0 ) ) ) )
2016, 19rspc2ev 2858 . . 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 1341 . 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 2184 . . . . . 6  |-  ( r  =  A  ->  (
r  =  ( ( A  x.  s )  +  ( B  x.  t ) )  <->  A  =  ( ( A  x.  s )  +  ( B  x.  t ) ) ) )
24232rexbidv 2502 . . . . 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, 24bitrid 192 . . . 4  |-  ( r  =  A  ->  ( ph 
<->  E. s  e.  ZZ  E. t  e.  ZZ  A  =  ( ( A  x.  s )  +  ( B  x.  t
) ) ) )
2625sbcieg 2997 . . 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 167 1  |-  ( th 
->  [. A  /  r ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    e. wcel 2148   E.wrex 2456   [.wsbc 2964  (class class class)co 5877   0cc0 7813   1c1 7814    + caddc 7816    x. cmul 7818   NN0cn0 9178   ZZcz 9255
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256
This theorem is referenced by:  bezoutlemex  12004
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