Definition df-asslaw

Description: The associative law for binary operations, see definitions of laws A1. and M1. in section 1.1 of Hall p. 1, or definition 5 in BourbakiAlg1 p. 4: the value of a binary operation applied the value of the binary operation applied to two operands and a third operand equals the value of the binary operation applied to the first operand and the value of the binary operation applied to the second and third operand. By this definition, the associative law is expressed as binary relation: a binary operation is related to a set by assLaw if the associative law holds for this binary operation regarding this set. Note that the binary operation needs neither to be closed nor to be a function. (Contributed by FL, 1-Nov-2009) (Revised by AV, 13-Jan-2020)

	Ref	Expression
Assertion	df-asslaw	⊢ assLaw = { ⟨ 𝑜 , 𝑚 ⟩ ∣ ∀ 𝑥 ∈ 𝑚 ∀ 𝑦 ∈ 𝑚 ∀ 𝑧 ∈ 𝑚 ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 ) = ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) ) }

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		casslaw	⊢ assLaw
1		vo	⊢ 𝑜
2		vm	⊢ 𝑚
3		vx	⊢ 𝑥
4	2	cv	⊢ 𝑚
5		vy	⊢ 𝑦
6		vz	⊢ 𝑧
7	3	cv	⊢ 𝑥
8	1	cv	⊢ 𝑜
9	5	cv	⊢ 𝑦
10	7 9 8	co	⊢ ( 𝑥 𝑜 𝑦 )
11	6	cv	⊢ 𝑧
12	10 11 8	co	⊢ ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 )
13	9 11 8	co	⊢ ( 𝑦 𝑜 𝑧 )
14	7 13 8	co	⊢ ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) )
15	12 14	wceq	⊢ ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 ) = ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) )
16	15 6 4	wral	⊢ ∀ 𝑧 ∈ 𝑚 ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 ) = ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) )
17	16 5 4	wral	⊢ ∀ 𝑦 ∈ 𝑚 ∀ 𝑧 ∈ 𝑚 ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 ) = ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) )
18	17 3 4	wral	⊢ ∀ 𝑥 ∈ 𝑚 ∀ 𝑦 ∈ 𝑚 ∀ 𝑧 ∈ 𝑚 ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 ) = ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) )
19	18 1 2	copab	⊢ { ⟨ 𝑜 , 𝑚 ⟩ ∣ ∀ 𝑥 ∈ 𝑚 ∀ 𝑦 ∈ 𝑚 ∀ 𝑧 ∈ 𝑚 ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 ) = ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) ) }
20	0 19	wceq	⊢ assLaw = { ⟨ 𝑜 , 𝑚 ⟩ ∣ ∀ 𝑥 ∈ 𝑚 ∀ 𝑦 ∈ 𝑚 ∀ 𝑧 ∈ 𝑚 ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 ) = ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) ) }