Theorem mulcnsrec 6416

Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 5359, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 6414.

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 6124.

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 6406	. 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 3527	. . . 4 |- <.A, B>. e. _V
3	2	ecid 5359	. . 3 |- [<.A, B>.]`' _E = <.A, B>.
4		opex 3527	. . . 4 |- <.C, D>. e. _V
5	4	ecid 5359	. . 3 |- [<.C, D>.]`' _E = <.C, D>.
6	3, 5	opreq12i 4894	. 2 |- ([<.A, B>.]`' _E x. [<.C, D>.]`' _E ) = (<.A, B>. x. <.C, D>.)
7		opex 3527	. . 3 |- <.((A .R C) +R (-1R .R (B .R D))), ((B .R C) +R (A .R D))>. e. _V
8	7	ecid 5359	. 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 1953	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 240   = wceq 1298   e. wcel 1300  <.cop 3046   _E cep 3581  `'ccnv 3985  (class class class)co 4884  [cec 5316  R.cnr 6145  -1Rcm1r 6148   +R cplr 6149   .R cmr 6150   x. cmul 6391

This theorem is referenced by:  axmulcom 6429  axmulass 6431  axdistr 6432

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-eprel 3583  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-opr 4886  df-oprab 4887  df-ec 5320  df-c 6392  df-mul 6398

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