Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clmgmOLD Structured version   Visualization version   GIF version

Theorem clmgmOLD 37838
Description: Obsolete version of mgmcl 18669 as of 3-Feb-2020. Closure of a magma. (Contributed by FL, 14-Sep-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
clmgmOLD.1 𝑋 = dom dom 𝐺
Assertion
Ref Expression
clmgmOLD ((𝐺 ∈ Magma ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Proof of Theorem clmgmOLD
StepHypRef Expression
1 clmgmOLD.1 . . . . 5 𝑋 = dom dom 𝐺
21ismgmOLD 37837 . . . 4 (𝐺 ∈ Magma → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
3 fovcdm 7603 . . . . 5 ((𝐺:(𝑋 × 𝑋)⟶𝑋𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
433exp 1118 . . . 4 (𝐺:(𝑋 × 𝑋)⟶𝑋 → (𝐴𝑋 → (𝐵𝑋 → (𝐴𝐺𝐵) ∈ 𝑋)))
52, 4biimtrdi 253 . . 3 (𝐺 ∈ Magma → (𝐺 ∈ Magma → (𝐴𝑋 → (𝐵𝑋 → (𝐴𝐺𝐵) ∈ 𝑋))))
65pm2.43i 52 . 2 (𝐺 ∈ Magma → (𝐴𝑋 → (𝐵𝑋 → (𝐴𝐺𝐵) ∈ 𝑋)))
763imp 1110 1 ((𝐺 ∈ Magma ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106   × cxp 5687  dom cdm 5689  wf 6559  (class class class)co 7431  Magmacmagm 37835
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  ax-un 7754
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-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-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-mgmOLD 37836
This theorem is referenced by:  exidcl  37863
  Copyright terms: Public domain W3C validator