Theorem smgrpmgm 38617

Description: Obsolete theorem, use sgrpmgm 18829 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 38616	. . 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 1570   ∈ wcel 2145  ∀wral 3076   × cxp 5653  dom cdm 5655  ⟶wf 6529  (class class class)co 7414  SemiGrpcsem 38613

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-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

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-sb 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7417  df-ass 38596  df-mgmOLD 38602  df-sgrOLD 38614

This theorem is used by:  ismndo1  38626