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Theorem mulcnsrec 9575
Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 7439, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 9573.

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 9276. (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 9567 . 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 4685 . . . 4  |-  <. A ,  B >.  e.  _V
32ecid 7439 . . 3  |-  [ <. A ,  B >. ] `'  _E  =  <. A ,  B >.
4 opex 4685 . . . 4  |-  <. C ,  D >.  e.  _V
54ecid 7439 . . 3  |-  [ <. C ,  D >. ] `'  _E  =  <. C ,  D >.
63, 5oveq12i 6317 . 2  |-  ( [
<. A ,  B >. ] `'  _E  x.  [ <. C ,  D >. ] `'  _E  )  =  ( <. A ,  B >.  x. 
<. C ,  D >. )
7 opex 4685 . . 3  |-  <. (
( A  .R  C
)  +R  ( -1R 
.R  ( B  .R  D ) ) ) ,  ( ( B  .R  C )  +R  ( A  .R  D
) ) >.  e.  _V
87ecid 7439 . 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 2488 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872   <.cop 4004    _E cep 4762   `'ccnv 4852  (class class class)co 6305   [cec 7372   R.cnr 9297   -1Rcm1r 9300    +R cplr 9301    .R cmr 9302    x. cmul 9551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-eprel 4764  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-ec 7376  df-c 9552  df-mul 9558
This theorem is referenced by:  axmulcom  9586  axmulass  9588  axdistr  9589
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