ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  bezoutlemb Unicode version

Theorem bezoutlemb 11082
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 8731 . . 3  |-  0  e.  ZZ
2 1z 8746 . . 3  |-  1  e.  ZZ
3 bezoutlema.a . . . . . . 7  |-  ( th 
->  A  e.  NN0 )
43nn0cnd 8698 . . . . . 6  |-  ( th 
->  A  e.  CC )
54mul01d 7850 . . . . 5  |-  ( th 
->  ( A  x.  0 )  =  0 )
65oveq1d 5649 . . . 4  |-  ( th 
->  ( ( A  x.  0 )  +  ( B  x.  1 ) )  =  ( 0  +  ( B  x.  1 ) ) )
7 bezoutlema.b . . . . . . 7  |-  ( th 
->  B  e.  NN0 )
87nn0cnd 8698 . . . . . 6  |-  ( th 
->  B  e.  CC )
9 1cnd 7483 . . . . . 6  |-  ( th 
->  1  e.  CC )
108, 9mulcld 7487 . . . . 5  |-  ( th 
->  ( B  x.  1 )  e.  CC )
1110addid2d 7611 . . . 4  |-  ( th 
->  ( 0  +  ( B  x.  1 ) )  =  ( B  x.  1 ) )
128mulid1d 7484 . . . 4  |-  ( th 
->  ( B  x.  1 )  =  B )
136, 11, 123eqtrrd 2125 . . 3  |-  ( th 
->  B  =  (
( A  x.  0 )  +  ( B  x.  1 ) ) )
14 oveq2 5642 . . . . . 6  |-  ( s  =  0  ->  ( A  x.  s )  =  ( A  x.  0 ) )
1514oveq1d 5649 . . . . 5  |-  ( s  =  0  ->  (
( A  x.  s
)  +  ( B  x.  t ) )  =  ( ( A  x.  0 )  +  ( B  x.  t
) ) )
1615eqeq2d 2099 . . . 4  |-  ( s  =  0  ->  ( B  =  ( ( A  x.  s )  +  ( B  x.  t ) )  <->  B  =  ( ( A  x.  0 )  +  ( B  x.  t ) ) ) )
17 oveq2 5642 . . . . . 6  |-  ( t  =  1  ->  ( B  x.  t )  =  ( B  x.  1 ) )
1817oveq2d 5650 . . . . 5  |-  ( t  =  1  ->  (
( A  x.  0 )  +  ( B  x.  t ) )  =  ( ( A  x.  0 )  +  ( B  x.  1 ) ) )
1918eqeq2d 2099 . . . 4  |-  ( t  =  1  ->  ( B  =  ( ( A  x.  0 )  +  ( B  x.  t ) )  <->  B  =  ( ( A  x.  0 )  +  ( B  x.  1 ) ) ) )
2016, 19rspc2ev 2735 . . 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 1277 . 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 2094 . . . . . 6  |-  ( r  =  B  ->  (
r  =  ( ( A  x.  s )  +  ( B  x.  t ) )  <->  B  =  ( ( A  x.  s )  +  ( B  x.  t ) ) ) )
24232rexbidv 2403 . . . . 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 190 . . . 4  |-  ( r  =  B  ->  ( ph 
<->  E. s  e.  ZZ  E. t  e.  ZZ  B  =  ( ( A  x.  s )  +  ( B  x.  t
) ) ) )
2625sbcieg 2869 . . 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 165 1  |-  ( th 
->  [. B  /  r ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289    e. wcel 1438   E.wrex 2360   [.wsbc 2838  (class class class)co 5634   0cc0 7329   1c1 7330    + caddc 7332    x. cmul 7334   NN0cn0 8643   ZZcz 8720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721
This theorem is referenced by:  bezoutlemex  11083
  Copyright terms: Public domain W3C validator