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Theorem smgrpmgm 37525
Description: A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
Hypothesis
Ref Expression
smgrpmgm.1 𝑋 = dom dom 𝐺
Assertion
Ref Expression
smgrpmgm (𝐺 ∈ SemiGrp → 𝐺:(𝑋 × 𝑋)⟶𝑋)

Proof of Theorem smgrpmgm
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smgrpmgm.1 . . . 4 𝑋 = dom dom 𝐺
21issmgrpOLD 37524 . . 3 (𝐺 ∈ SemiGrp → (𝐺 ∈ SemiGrp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))))
3 simpl 481 . . 3 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
42, 3biimtrdi 252 . 2 (𝐺 ∈ SemiGrp → (𝐺 ∈ SemiGrp → 𝐺:(𝑋 × 𝑋)⟶𝑋))
54pm2.43i 52 1 (𝐺 ∈ SemiGrp → 𝐺:(𝑋 × 𝑋)⟶𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3050   × cxp 5679  dom cdm 5681  wf 6549  (class class class)co 7423  SemiGrpcsem 37521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7426  df-ass 37504  df-mgmOLD 37510  df-sgrOLD 37522
This theorem is referenced by:  ismndo1  37534
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