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Theorem mulcnsrec 9019
Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 6969, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 9017.

Note: This is the last lemma (from which the axioms will be derived) in the construction of real and complex numbers. The construction starts at cnpi 8719. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)

Assertion
Ref Expression
mulcnsrec  |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. )
)  ->  ( [ <. A ,  B >. ] `'  _E  x.  [ <. C ,  D >. ] `'  _E  )  =  [ <. ( ( A  .R  C )  +R  ( -1R  .R  ( B  .R  D ) ) ) ,  ( ( B  .R  C )  +R  ( A  .R  D
) ) >. ] `'  _E  )

Proof of Theorem mulcnsrec
StepHypRef Expression
1 mulcnsr 9011 . 2  |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. )
)  ->  ( <. A ,  B >.  x.  <. C ,  D >. )  =  <. ( ( A  .R  C )  +R  ( -1R  .R  ( B  .R  D ) ) ) ,  ( ( B  .R  C )  +R  ( A  .R  D ) ) >.
)
2 opex 4427 . . . 4  |-  <. A ,  B >.  e.  _V
32ecid 6969 . . 3  |-  [ <. A ,  B >. ] `'  _E  =  <. A ,  B >.
4 opex 4427 . . . 4  |-  <. C ,  D >.  e.  _V
54ecid 6969 . . 3  |-  [ <. C ,  D >. ] `'  _E  =  <. C ,  D >.
63, 5oveq12i 6093 . 2  |-  ( [
<. A ,  B >. ] `'  _E  x.  [ <. C ,  D >. ] `'  _E  )  =  ( <. A ,  B >.  x. 
<. C ,  D >. )
7 opex 4427 . . 3  |-  <. (
( A  .R  C
)  +R  ( -1R 
.R  ( B  .R  D ) ) ) ,  ( ( B  .R  C )  +R  ( A  .R  D
) ) >.  e.  _V
87ecid 6969 . 2  |-  [ <. ( ( A  .R  C
)  +R  ( -1R 
.R  ( B  .R  D ) ) ) ,  ( ( B  .R  C )  +R  ( A  .R  D
) ) >. ] `'  _E  =  <. ( ( A  .R  C )  +R  ( -1R  .R  ( B  .R  D ) ) ) ,  ( ( B  .R  C
)  +R  ( A  .R  D ) )
>.
91, 6, 83eqtr4g 2493 1  |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. )
)  ->  ( [ <. A ,  B >. ] `'  _E  x.  [ <. C ,  D >. ] `'  _E  )  =  [ <. ( ( A  .R  C )  +R  ( -1R  .R  ( B  .R  D ) ) ) ,  ( ( B  .R  C )  +R  ( A  .R  D
) ) >. ] `'  _E  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3817    _E cep 4492   `'ccnv 4877  (class class class)co 6081   [cec 6903   R.cnr 8742   -1Rcm1r 8745    +R cplr 8746    .R cmr 8747    x. cmul 8995
This theorem is referenced by:  axmulcom  9030  axmulass  9032  axdistr  9033
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-eprel 4494  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-ec 6907  df-c 8996  df-mul 9002
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