Theorem mulcnsrec 5236

Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 4284, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 5234.

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 4944.

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

Step	Hyp	Ref	Expression
1		mulcnsr 5226	. 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 2772	. . . 4 |- <.A, B>. e. V
3	2	ecid 4284	. . 3 |- [<.A, B>.]`'E = <.A, B>.
4		opex 2772	. . . 4 |- <.C, D>. e. V
5	4	ecid 4284	. . 3 |- [<.C, D>.]`'E = <.C, D>.
6	3, 5	opreq12i 3958	. 2 |- ([<.A, B>.]`'E x. [<.C, D>.]`'E) = (<.A, B>. x. <.C, D>.)
7		opex 2772	. . 3 |- <.((A .R C) +R (-1R .R (B .R D))), ((B .R C) +R (A .R D))>. e. V
8	7	ecid 4284	. 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))>.
9	1, 6, 8	3eqtr4g 1523	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)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  <.cop 2401  Ecep 2819  `'ccnv 3159  (class class class)co 3948  [cec 4243  R.cnr 4965  -1Rcm1r 4968   +R cplr 4969   .R cmr 4970   x. cmul 5211

This theorem is referenced by:  axmulcom 5248  axmulass 5250  axdistr 5251

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-eprel 2821  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188  df-opr 3950  df-oprab 3951  df-ec 4247  df-c 5212  df-mul 5218

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