Theorem mulcnsrec 11157

Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 8784, which shows that the coset of the converse membership relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 11155.

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 10857. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)

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 11149	. 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 5443	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
3	2	ecid 8784	. . 3 ⊢ [⟨𝐴, 𝐵⟩]◡ E = ⟨𝐴, 𝐵⟩
4		opex 5443	. . . 4 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
5	4	ecid 8784	. . 3 ⊢ [⟨𝐶, 𝐷⟩]◡ E = ⟨𝐶, 𝐷⟩
6	3, 5	oveq12i 7429	. 2 ⊢ ([⟨𝐴, 𝐵⟩]◡ E · [⟨𝐶, 𝐷⟩]◡ E ) = (⟨𝐴, 𝐵⟩ · ⟨𝐶, 𝐷⟩)
7		opex 5443	. . 3 ⊢ ⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩ ∈ V
8	7	ecid 8784	. 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 2822	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 )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ⟨cop 4593   E cep 5558  ^◡ccnv 5658  (class class class)co 7417  [cec 8698  Rcnr 10878  -1_Rcm1r 10881   +_R cplr 10882   ·_R cmr 10883   · cmul 11133

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-eprel 5559  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420  df-oprab 7421  df-ec 8702  df-c 11134  df-mul 11140

This theorem is used by:  axmulcom  11168  axmulass  11170  axdistr  11171