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Theorem mulcnsrec 11055
Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 8717, which shows that the coset of the converse membership relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 11053.

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 10755. (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 11047 . 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 5412 . . . 4 𝐴, 𝐵⟩ ∈ V
32ecid 8717 . . 3 [⟨𝐴, 𝐵⟩] E = ⟨𝐴, 𝐵
4 opex 5412 . . . 4 𝐶, 𝐷⟩ ∈ V
54ecid 8717 . . 3 [⟨𝐶, 𝐷⟩] E = ⟨𝐶, 𝐷
63, 5oveq12i 7370 . 2 ([⟨𝐴, 𝐵⟩] E · [⟨𝐶, 𝐷⟩] E ) = (⟨𝐴, 𝐵⟩ · ⟨𝐶, 𝐷⟩)
7 opex 5412 . . 3 ⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩ ∈ V
87ecid 8717 . 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 2796 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 395   = wceq 1541  wcel 2113  cop 4586   E cep 5523  ccnv 5623  (class class class)co 7358  [cec 8633  Rcnr 10776  -1Rcm1r 10779   +R cplr 10780   ·R cmr 10781   · cmul 11031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-eprel 5524  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-ec 8637  df-c 11032  df-mul 11038
This theorem is referenced by:  axmulcom  11066  axmulass  11068  axdistr  11069
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