Theorem mulcnsrec 5264

Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 4300, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 5262.

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 4972.

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 5254	. 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 2782	. . . 4 |- <.A, B>. e. V
3	2	ecid 4300	. . 3 |- [<.A, B>.]`'E = <.A, B>.
4		opex 2782	. . . 4 |- <.C, D>. e. V
5	4	ecid 4300	. . 3 |- [<.C, D>.]`'E = <.C, D>.
6	3, 5	opreq12i 3973	. 2 |- ([<.A, B>.]`'E x. [<.C, D>.]`'E) = (<.A, B>. x. <.C, D>.)
7		opex 2782	. . 3 |- <.((A .R C) +R (-1R .R (B .R D))), ((B .R C) +R (A .R D))>. e. V
8	7	ecid 4300	. 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 1531	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 956   e. wcel 958  <.cop 2411  Ecep 2830  `'ccnv 3169  (class class class)co 3963  [cec 4259  R.cnr 4993  -1Rcm1r 4996   +R cplr 4997   .R cmr 4998   x. cmul 5239

This theorem is referenced by:  axmulcom 5276  axmulass 5278  axdistr 5279

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-eprel 2832  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-opr 3965  df-oprab 3966  df-ec 4263  df-c 5240  df-mul 5246

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