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Metamath Proof Explorer

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 ) ) ) ) )