Theorem issmgrpOLD 38060

Description: Obsolete version of issgrp 18645 as of 3-Feb-2020. The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
issmgrpOLD.1	⊢ 𝑋 = dom dom 𝐺

Assertion

Ref	Expression
issmgrpOLD	⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ SemiGrp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))))

Distinct variable groups:   𝑥,𝐺,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem issmgrpOLD

Step	Hyp	Ref	Expression
1		df-sgrOLD 38058	. . 3 ⊢ SemiGrp = (Magma ∩ Ass)
2	1	eleq2i 2828	. 2 ⊢ (𝐺 ∈ SemiGrp ↔ 𝐺 ∈ (Magma ∩ Ass))
3		elin 3917	. . 3 ⊢ (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass))
4		issmgrpOLD.1	. . . . 5 ⊢ 𝑋 = dom dom 𝐺
5	4	ismgmOLD 38047	. . . 4 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
6	4	isass 38043	. . . 4 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Ass ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
7	5, 6	anbi12d 632	. . 3 ⊢ (𝐺 ∈ 𝐴 → ((𝐺 ∈ Magma ∧ 𝐺 ∈ Ass) ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))))
8	3, 7	bitrid 283	. 2 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))))
9	2, 8	bitrid 283	1 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ SemiGrp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ∩ cin 3900   × cxp 5622  dom cdm 5624  ⟶wf 6488  (class class class)co 7358  Asscass 38039  Magmacmagm 38045  SemiGrpcsem 38057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-ass 38040  df-mgmOLD 38046  df-sgrOLD 38058

This theorem is referenced by:  smgrpmgm  38061  smgrpassOLD  38062  ismndo1  38070