Theorem issmgrpOLD

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

	Ref	Expression
Hypothesis	issmgrpOLD.1	⊢ 𝑋 = dom dom 𝐺
Assertion	issmgrpOLD	⊢ ( 𝐺 ∈ 𝐴 → ( 𝐺 ∈ SemiGrp ↔ ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		issmgrpOLD.1	⊢ 𝑋 = dom dom 𝐺
2		df-sgrOLD	⊢ SemiGrp = ( Magma ∩ Ass )
3	2	eleq2i	⊢ ( 𝐺 ∈ SemiGrp ↔ 𝐺 ∈ ( Magma ∩ Ass ) )
4		elin	⊢ ( 𝐺 ∈ ( Magma ∩ Ass ) ↔ ( 𝐺 ∈ Magma ∧ 𝐺 ∈ Ass ) )
5	1	ismgmOLD	⊢ ( 𝐺 ∈ 𝐴 → ( 𝐺 ∈ Magma ↔ 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ) )
6	1	isass	⊢ ( 𝐺 ∈ 𝐴 → ( 𝐺 ∈ Ass ↔ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ) )
7	5 6	anbi12d	⊢ ( 𝐺 ∈ 𝐴 → ( ( 𝐺 ∈ Magma ∧ 𝐺 ∈ Ass ) ↔ ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ) ) )
8	4 7	bitrid	⊢ ( 𝐺 ∈ 𝐴 → ( 𝐺 ∈ ( Magma ∩ Ass ) ↔ ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ) ) )
9	3 8	bitrid	⊢ ( 𝐺 ∈ 𝐴 → ( 𝐺 ∈ SemiGrp ↔ ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ) ) )