Theorem mulcnsrec 5247

Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 4293, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 5245.

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 4955.

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 5237	. 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 2778	. . . 4 |- <.A, B>. e. V
3	2	ecid 4293	. . 3 |- [<.A, B>.]`'E = <.A, B>.
4		opex 2778	. . . 4 |- <.C, D>. e. V
5	4	ecid 4293	. . 3 |- [<.C, D>.]`'E = <.C, D>.
6	3, 5	opreq12i 3968	. 2 |- ([<.A, B>.]`'E x. [<.C, D>.]`'E) = (<.A, B>. x. <.C, D>.)
7		opex 2778	. . 3 |- <.((A .R C) +R (-1R .R (B .R D))), ((B .R C) +R (A .R D))>. e. V
8	7	ecid 4293	. 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 1529	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 955   e. wcel 957  <.cop 2408  Ecep 2826  `'ccnv 3165  (class class class)co 3958  [cec 4252  R.cnr 4976  -1Rcm1r 4979   +R cplr 4980   .R cmr 4981   x. cmul 5222

This theorem is referenced by:  axmulcom 5259  axmulass 5261  axdistr 5262

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-eprel 2828  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fv 3194  df-opr 3960  df-oprab 3961  df-ec 4256  df-c 5223  df-mul 5229

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