Theorem asclpropd

Description: If two structures have the same components (properties), one is an associative algebra iff the other one is. The last hypotheses on 1r can be discharged either by letting W = _V (if strong equality is known on .s ) or assuming K is a ring. (Contributed by Mario Carneiro, 5-Jul-2015)

	Ref	Expression
Hypotheses	asclpropd.f	⊢ 𝐹 = ( Scalar ‘ 𝐾 )
	asclpropd.g	⊢ 𝐺 = ( Scalar ‘ 𝐿 )
	asclpropd.1	⊢ ( 𝜑 → 𝑃 = ( Base ‘ 𝐹 ) )
	asclpropd.2	⊢ ( 𝜑 → 𝑃 = ( Base ‘ 𝐺 ) )
	asclpropd.3	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑊 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( ·𝑠 ‘ 𝐿 ) 𝑦 ) )
	asclpropd.4	⊢ ( 𝜑 → ( 1r ‘ 𝐾 ) = ( 1r ‘ 𝐿 ) )
	asclpropd.5	⊢ ( 𝜑 → ( 1r ‘ 𝐾 ) ∈ 𝑊 )
Assertion	asclpropd	⊢ ( 𝜑 → ( algSc ‘ 𝐾 ) = ( algSc ‘ 𝐿 ) )

Proof

Step	Hyp	Ref	Expression
1		asclpropd.f	⊢ 𝐹 = ( Scalar ‘ 𝐾 )
2		asclpropd.g	⊢ 𝐺 = ( Scalar ‘ 𝐿 )
3		asclpropd.1	⊢ ( 𝜑 → 𝑃 = ( Base ‘ 𝐹 ) )
4		asclpropd.2	⊢ ( 𝜑 → 𝑃 = ( Base ‘ 𝐺 ) )
5		asclpropd.3	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑊 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( ·𝑠 ‘ 𝐿 ) 𝑦 ) )
6		asclpropd.4	⊢ ( 𝜑 → ( 1r ‘ 𝐾 ) = ( 1r ‘ 𝐿 ) )
7		asclpropd.5	⊢ ( 𝜑 → ( 1r ‘ 𝐾 ) ∈ 𝑊 )
8	5	oveqrspc2v	⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑃 ∧ ( 1r ‘ 𝐾 ) ∈ 𝑊 ) ) → ( 𝑧 ( ·𝑠 ‘ 𝐾 ) ( 1r ‘ 𝐾 ) ) = ( 𝑧 ( ·𝑠 ‘ 𝐿 ) ( 1r ‘ 𝐾 ) ) )
9	8	anassrs	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑃 ) ∧ ( 1r ‘ 𝐾 ) ∈ 𝑊 ) → ( 𝑧 ( ·𝑠 ‘ 𝐾 ) ( 1r ‘ 𝐾 ) ) = ( 𝑧 ( ·𝑠 ‘ 𝐿 ) ( 1r ‘ 𝐾 ) ) )
10	7 9	mpidan	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑃 ) → ( 𝑧 ( ·𝑠 ‘ 𝐾 ) ( 1r ‘ 𝐾 ) ) = ( 𝑧 ( ·𝑠 ‘ 𝐿 ) ( 1r ‘ 𝐾 ) ) )
11	6	oveq2d	⊢ ( 𝜑 → ( 𝑧 ( ·𝑠 ‘ 𝐿 ) ( 1r ‘ 𝐾 ) ) = ( 𝑧 ( ·𝑠 ‘ 𝐿 ) ( 1r ‘ 𝐿 ) ) )
12	11	adantr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑃 ) → ( 𝑧 ( ·𝑠 ‘ 𝐿 ) ( 1r ‘ 𝐾 ) ) = ( 𝑧 ( ·𝑠 ‘ 𝐿 ) ( 1r ‘ 𝐿 ) ) )
13	10 12	eqtrd	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑃 ) → ( 𝑧 ( ·𝑠 ‘ 𝐾 ) ( 1r ‘ 𝐾 ) ) = ( 𝑧 ( ·𝑠 ‘ 𝐿 ) ( 1r ‘ 𝐿 ) ) )
14	13	mpteq2dva	⊢ ( 𝜑 → ( 𝑧 ∈ 𝑃 ↦ ( 𝑧 ( ·𝑠 ‘ 𝐾 ) ( 1r ‘ 𝐾 ) ) ) = ( 𝑧 ∈ 𝑃 ↦ ( 𝑧 ( ·𝑠 ‘ 𝐿 ) ( 1r ‘ 𝐿 ) ) ) )
15	3	mpteq1d	⊢ ( 𝜑 → ( 𝑧 ∈ 𝑃 ↦ ( 𝑧 ( ·𝑠 ‘ 𝐾 ) ( 1r ‘ 𝐾 ) ) ) = ( 𝑧 ∈ ( Base ‘ 𝐹 ) ↦ ( 𝑧 ( ·𝑠 ‘ 𝐾 ) ( 1r ‘ 𝐾 ) ) ) )
16	4	mpteq1d	⊢ ( 𝜑 → ( 𝑧 ∈ 𝑃 ↦ ( 𝑧 ( ·𝑠 ‘ 𝐿 ) ( 1r ‘ 𝐿 ) ) ) = ( 𝑧 ∈ ( Base ‘ 𝐺 ) ↦ ( 𝑧 ( ·𝑠 ‘ 𝐿 ) ( 1r ‘ 𝐿 ) ) ) )
17	14 15 16	3eqtr3d	⊢ ( 𝜑 → ( 𝑧 ∈ ( Base ‘ 𝐹 ) ↦ ( 𝑧 ( ·𝑠 ‘ 𝐾 ) ( 1r ‘ 𝐾 ) ) ) = ( 𝑧 ∈ ( Base ‘ 𝐺 ) ↦ ( 𝑧 ( ·𝑠 ‘ 𝐿 ) ( 1r ‘ 𝐿 ) ) ) )
18		eqid	⊢ ( algSc ‘ 𝐾 ) = ( algSc ‘ 𝐾 )
19		eqid	⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 )
20		eqid	⊢ ( ·𝑠 ‘ 𝐾 ) = ( ·𝑠 ‘ 𝐾 )
21		eqid	⊢ ( 1r ‘ 𝐾 ) = ( 1r ‘ 𝐾 )
22	18 1 19 20 21	asclfval	⊢ ( algSc ‘ 𝐾 ) = ( 𝑧 ∈ ( Base ‘ 𝐹 ) ↦ ( 𝑧 ( ·𝑠 ‘ 𝐾 ) ( 1r ‘ 𝐾 ) ) )
23		eqid	⊢ ( algSc ‘ 𝐿 ) = ( algSc ‘ 𝐿 )
24		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
25		eqid	⊢ ( ·𝑠 ‘ 𝐿 ) = ( ·𝑠 ‘ 𝐿 )
26		eqid	⊢ ( 1r ‘ 𝐿 ) = ( 1r ‘ 𝐿 )
27	23 2 24 25 26	asclfval	⊢ ( algSc ‘ 𝐿 ) = ( 𝑧 ∈ ( Base ‘ 𝐺 ) ↦ ( 𝑧 ( ·𝑠 ‘ 𝐿 ) ( 1r ‘ 𝐿 ) ) )
28	17 22 27	3eqtr4g	⊢ ( 𝜑 → ( algSc ‘ 𝐾 ) = ( algSc ‘ 𝐿 ) )