Theorem ismndo1

Description: The predicate "is a monoid". (Contributed by FL, 2-Nov-2009) (Revised by Mario Carneiro, 22-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Hypothesis	ismndo1.1	⊢ 𝑋 = dom dom 𝐺
Assertion	ismndo1	⊢ ( 𝐺 ∈ 𝐴 → ( 𝐺 ∈ MndOp ↔ ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ∧ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ismndo1.1	⊢ 𝑋 = dom dom 𝐺
2	1	ismndo	⊢ ( 𝐺 ∈ 𝐴 → ( 𝐺 ∈ MndOp ↔ ( 𝐺 ∈ SemiGrp ∧ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) ) )
3	1	smgrpmgm	⊢ ( 𝐺 ∈ SemiGrp → 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 )
4	3	ad2antrl	⊢ ( ( 𝐺 ∈ 𝐴 ∧ ( 𝐺 ∈ SemiGrp ∧ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) ) → 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 )
5	1	smgrpassOLD	⊢ ( 𝐺 ∈ SemiGrp → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) )
6	5	ad2antrl	⊢ ( ( 𝐺 ∈ 𝐴 ∧ ( 𝐺 ∈ SemiGrp ∧ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) ) → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) )
7		simprr	⊢ ( ( 𝐺 ∈ 𝐴 ∧ ( 𝐺 ∈ SemiGrp ∧ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) ) → ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) )
8	4 6 7	3jca	⊢ ( ( 𝐺 ∈ 𝐴 ∧ ( 𝐺 ∈ SemiGrp ∧ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) ) → ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ∧ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) )
9		3simpa	⊢ ( ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ∧ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) → ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ) )
10	1	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 ) ) ) ) )
11	9 10	syl5ibr	⊢ ( 𝐺 ∈ 𝐴 → ( ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ∧ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) → 𝐺 ∈ SemiGrp ) )
12	11	imp	⊢ ( ( 𝐺 ∈ 𝐴 ∧ ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ∧ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) ) → 𝐺 ∈ SemiGrp )
13		simpr3	⊢ ( ( 𝐺 ∈ 𝐴 ∧ ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ∧ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) ) → ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) )
14	12 13	jca	⊢ ( ( 𝐺 ∈ 𝐴 ∧ ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ∧ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) ) → ( 𝐺 ∈ SemiGrp ∧ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) )
15	8 14	impbida	⊢ ( 𝐺 ∈ 𝐴 → ( ( 𝐺 ∈ SemiGrp ∧ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) ↔ ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ∧ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) ) )
16	2 15	bitrd	⊢ ( 𝐺 ∈ 𝐴 → ( 𝐺 ∈ MndOp ↔ ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ∧ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) ) )