Theorem mulcnsrec 4058

Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws.

Assertion

Ref	Expression
mulcnsrec	⊢ (((A ∈ R ∧ B ∈ R) ∧ (C ∈ R ∧ D ∈ R)) → ([⟨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

Step	Hyp	Ref	Expression
1		mulcnsr 4048	. 2 ⊢ (((A ∈ R ∧ B ∈ R) ∧ (C ∈ R ∧ D ∈ R)) → (⟨A, B⟩ · ⟨C, D⟩) = ⟨((A ·R C) +R (-1R ·R (B ·R D))), ((B ·R C) +R (A ·R D))⟩)
2		opex 1893	. . . 4 ⊢ ⟨A, B⟩ ∈ V
3	2	ecid 3236	. . 3 ⊢ [⟨A, B⟩]◡E = ⟨A, B⟩
4		opex 1893	. . . 4 ⊢ ⟨C, D⟩ ∈ V
5	4	ecid 3236	. . 3 ⊢ [⟨C, D⟩]◡E = ⟨C, D⟩
6	3, 5	opreq12i 3011	. 2 ⊢ ([⟨A, B⟩]◡E · [⟨C, D⟩]◡E) = (⟨A, B⟩ · ⟨C, D⟩)
7		opex 1893	. . 3 ⊢ ⟨((A ·R C) +R (-1R ·R (B ·R D))), ((B ·R C) +R (A ·R D))⟩ ∈ V
8	7	ecid 3236	. 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 1147	1 ⊢ (((A ∈ R ∧ B ∈ R) ∧ (C ∈ R ∧ D ∈ R)) → ([⟨A, B⟩]◡E · [⟨C, D⟩]◡E) = [⟨((A ·R C) +R (-1R ·R (B ·R D))), ((B ·R C) +R (A ·R D))⟩]◡E)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  Ecep 2056  ^◡ccnv 2409  (class class class)co 3001  [cec 3198  Rcnr 3787  -1_Rcm1r 3790   +_R cplr 3791   ·_R cmr 3792   · cmulc 4032

This theorem is referenced by:  axmulcom 4071  axmulass 4073  axdistr 4074

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-opr 3003  df-oprab 3004  df-ec 3202  df-c 4034  df-mul 4040

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