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

Theorem bezoutlema 11898
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 9198 . . 3  |-  1  e.  ZZ
2 0z 9183 . . 3  |-  0  e.  ZZ
3 bezoutlema.b . . . . . . 7  |-  ( th 
->  B  e.  NN0 )
43nn0cnd 9150 . . . . . 6  |-  ( th 
->  B  e.  CC )
54mul01d 8272 . . . . 5  |-  ( th 
->  ( B  x.  0 )  =  0 )
65oveq2d 5842 . . . 4  |-  ( th 
->  ( ( A  x.  1 )  +  ( B  x.  0 ) )  =  ( ( A  x.  1 )  +  0 ) )
7 bezoutlema.a . . . . . . 7  |-  ( th 
->  A  e.  NN0 )
87nn0cnd 9150 . . . . . 6  |-  ( th 
->  A  e.  CC )
9 1cnd 7896 . . . . . 6  |-  ( th 
->  1  e.  CC )
108, 9mulcld 7900 . . . . 5  |-  ( th 
->  ( A  x.  1 )  e.  CC )
1110addid1d 8028 . . . 4  |-  ( th 
->  ( ( A  x.  1 )  +  0 )  =  ( A  x.  1 ) )
128mulid1d 7897 . . . 4  |-  ( th 
->  ( A  x.  1 )  =  A )
136, 11, 123eqtrrd 2195 . . 3  |-  ( th 
->  A  =  (
( A  x.  1 )  +  ( B  x.  0 ) ) )
14 oveq2 5834 . . . . . 6  |-  ( s  =  1  ->  ( A  x.  s )  =  ( A  x.  1 ) )
1514oveq1d 5841 . . . . 5  |-  ( s  =  1  ->  (
( A  x.  s
)  +  ( B  x.  t ) )  =  ( ( A  x.  1 )  +  ( B  x.  t
) ) )
1615eqeq2d 2169 . . . 4  |-  ( s  =  1  ->  ( A  =  ( ( A  x.  s )  +  ( B  x.  t ) )  <->  A  =  ( ( A  x.  1 )  +  ( B  x.  t ) ) ) )
17 oveq2 5834 . . . . . 6  |-  ( t  =  0  ->  ( B  x.  t )  =  ( B  x.  0 ) )
1817oveq2d 5842 . . . . 5  |-  ( t  =  0  ->  (
( A  x.  1 )  +  ( B  x.  t ) )  =  ( ( A  x.  1 )  +  ( B  x.  0 ) ) )
1918eqeq2d 2169 . . . 4  |-  ( t  =  0  ->  ( A  =  ( ( A  x.  1 )  +  ( B  x.  t ) )  <->  A  =  ( ( A  x.  1 )  +  ( B  x.  0 ) ) ) )
2016, 19rspc2ev 2831 . . 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 1323 . 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 2164 . . . . . 6  |-  ( r  =  A  ->  (
r  =  ( ( A  x.  s )  +  ( B  x.  t ) )  <->  A  =  ( ( A  x.  s )  +  ( B  x.  t ) ) ) )
24232rexbidv 2482 . . . . 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, 24syl5bb 191 . . . 4  |-  ( r  =  A  ->  ( ph 
<->  E. s  e.  ZZ  E. t  e.  ZZ  A  =  ( ( A  x.  s )  +  ( B  x.  t
) ) ) )
2625sbcieg 2969 . . 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 166 1  |-  ( th 
->  [. A  /  r ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1335    e. wcel 2128   E.wrex 2436   [.wsbc 2937  (class class class)co 5826   0cc0 7734   1c1 7735    + caddc 7737    x. cmul 7739   NN0cn0 9095   ZZcz 9172
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-addcom 7834  ax-mulcom 7835  ax-addass 7836  ax-mulass 7837  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-1rid 7841  ax-0id 7842  ax-rnegex 7843  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-ltadd 7850
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-br 3968  df-opab 4028  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-iota 5137  df-fun 5174  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-inn 8839  df-n0 9096  df-z 9173
This theorem is referenced by:  bezoutlemex  11900
  Copyright terms: Public domain W3C validator