MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulcnsrec Structured version   Visualization version   GIF version

Theorem mulcnsrec 11182
Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 8821, which shows that the coset of the converse membership relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 11180.

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 10882. (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 11174 . 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 5475 . . . 4 𝐴, 𝐵⟩ ∈ V
32ecid 8821 . . 3 [⟨𝐴, 𝐵⟩] E = ⟨𝐴, 𝐵
4 opex 5475 . . . 4 𝐶, 𝐷⟩ ∈ V
54ecid 8821 . . 3 [⟨𝐶, 𝐷⟩] E = ⟨𝐶, 𝐷
63, 5oveq12i 7443 . 2 ([⟨𝐴, 𝐵⟩] E · [⟨𝐶, 𝐷⟩] E ) = (⟨𝐴, 𝐵⟩ · ⟨𝐶, 𝐷⟩)
7 opex 5475 . . 3 ⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩ ∈ V
87ecid 8821 . 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 2800 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 1537  wcel 2106  cop 4637   E cep 5588  ccnv 5688  (class class class)co 7431  [cec 8742  Rcnr 10903  -1Rcm1r 10906   +R cplr 10907   ·R cmr 10908   · cmul 11158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-ec 8746  df-c 11159  df-mul 11165
This theorem is referenced by:  axmulcom  11193  axmulass  11195  axdistr  11196
  Copyright terms: Public domain W3C validator