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Theorem mulcnsrec 10003
Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 7855, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 10001.

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 9704. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)

Assertion
Ref Expression
mulcnsrec (((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → ([⟨𝐴, 𝐵⟩] E · [⟨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 9995 . 2 (((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → (⟨𝐴, 𝐵⟩ · ⟨𝐶, 𝐷⟩) = ⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩)
2 opex 4962 . . . 4 𝐴, 𝐵⟩ ∈ V
32ecid 7855 . . 3 [⟨𝐴, 𝐵⟩] E = ⟨𝐴, 𝐵
4 opex 4962 . . . 4 𝐶, 𝐷⟩ ∈ V
54ecid 7855 . . 3 [⟨𝐶, 𝐷⟩] E = ⟨𝐶, 𝐷
63, 5oveq12i 6702 . 2 ([⟨𝐴, 𝐵⟩] E · [⟨𝐶, 𝐷⟩] E ) = (⟨𝐴, 𝐵⟩ · ⟨𝐶, 𝐷⟩)
7 opex 4962 . . 3 ⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩ ∈ V
87ecid 7855 . 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 2710 1 (((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → ([⟨𝐴, 𝐵⟩] E · [⟨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 383   = wceq 1523  wcel 2030  cop 4216   E cep 5057  ccnv 5142  (class class class)co 6690  [cec 7785  Rcnr 9725  -1Rcm1r 9728   +R cplr 9729   ·R cmr 9730   · cmul 9979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-eprel 5058  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-ec 7789  df-c 9980  df-mul 9986
This theorem is referenced by:  axmulcom  10014  axmulass  10016  axdistr  10017
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