Theorem smgrpmgm 38941

Description: Obsolete theorem, use sgrpmgm 18928 instead. A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification 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.

Step	Hyp	Ref	Expression
1		smgrpmgm.1	. . . 4 ⊢ 𝑋 = dom dom 𝐺
2	1	issmgrpOLD 38940	. . 3 ⊢ (𝐺 ∈ SemiGrp → (𝐺 ∈ SemiGrp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))))
3		simpl 488	. . 3 ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
4	2, 3	biimtrdi 256	. 2 ⊢ (𝐺 ∈ SemiGrp → (𝐺 ∈ SemiGrp → 𝐺:(𝑋 × 𝑋)⟶𝑋))
5	4	pm2.43i 53	1 ⊢ (𝐺 ∈ SemiGrp → 𝐺:(𝑋 × 𝑋)⟶𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ∀wral 3080   × cxp 5653  dom cdm 5655  ⟶wf 6543  (class class class)co 7430  SemiGrpcsem 38937

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-fv 6555  df-ov 7433  df-ass 38920  df-mgmOLD 38926  df-sgrOLD 38938

This theorem is used by:  ismndo1  38950