Theorem slmdvs1

Description: Scalar product with ring unit. ( ax-hvmulid analog.) (Contributed by NM, 10-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

	Ref	Expression
Hypotheses	slmdvs1.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	slmdvs1.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	slmdvs1.s	⊢ · = ( ·𝑠 ‘ 𝑊 )
	slmdvs1.u	⊢ 1 = ( 1r ‘ 𝐹 )
Assertion	slmdvs1	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → ( 1 · 𝑋 ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		slmdvs1.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		slmdvs1.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
3		slmdvs1.s	⊢ · = ( ·𝑠 ‘ 𝑊 )
4		slmdvs1.u	⊢ 1 = ( 1r ‘ 𝐹 )
5		simpl	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → 𝑊 ∈ SLMod )
6		eqid	⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 )
7	2 6 4	slmd1cl	⊢ ( 𝑊 ∈ SLMod → 1 ∈ ( Base ‘ 𝐹 ) )
8	7	adantr	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → 1 ∈ ( Base ‘ 𝐹 ) )
9		simpr	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ 𝑉 )
10		eqid	⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 )
11		eqid	⊢ ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝑊 )
12		eqid	⊢ ( +g ‘ 𝐹 ) = ( +g ‘ 𝐹 )
13		eqid	⊢ ( .r ‘ 𝐹 ) = ( .r ‘ 𝐹 )
14		eqid	⊢ ( 0g ‘ 𝐹 ) = ( 0g ‘ 𝐹 )
15	1 10 3 11 2 6 12 13 4 14	slmdlema	⊢ ( ( 𝑊 ∈ SLMod ∧ ( 1 ∈ ( Base ‘ 𝐹 ) ∧ 1 ∈ ( Base ‘ 𝐹 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( ( 1 · 𝑋 ) ∈ 𝑉 ∧ ( 1 · ( 𝑋 ( +g ‘ 𝑊 ) 𝑋 ) ) = ( ( 1 · 𝑋 ) ( +g ‘ 𝑊 ) ( 1 · 𝑋 ) ) ∧ ( ( 1 ( +g ‘ 𝐹 ) 1 ) · 𝑋 ) = ( ( 1 · 𝑋 ) ( +g ‘ 𝑊 ) ( 1 · 𝑋 ) ) ) ∧ ( ( ( 1 ( .r ‘ 𝐹 ) 1 ) · 𝑋 ) = ( 1 · ( 1 · 𝑋 ) ) ∧ ( 1 · 𝑋 ) = 𝑋 ∧ ( ( 0g ‘ 𝐹 ) · 𝑋 ) = ( 0g ‘ 𝑊 ) ) ) )
16	15	simprd	⊢ ( ( 𝑊 ∈ SLMod ∧ ( 1 ∈ ( Base ‘ 𝐹 ) ∧ 1 ∈ ( Base ‘ 𝐹 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( ( 1 ( .r ‘ 𝐹 ) 1 ) · 𝑋 ) = ( 1 · ( 1 · 𝑋 ) ) ∧ ( 1 · 𝑋 ) = 𝑋 ∧ ( ( 0g ‘ 𝐹 ) · 𝑋 ) = ( 0g ‘ 𝑊 ) ) )
17	16	simp2d	⊢ ( ( 𝑊 ∈ SLMod ∧ ( 1 ∈ ( Base ‘ 𝐹 ) ∧ 1 ∈ ( Base ‘ 𝐹 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) → ( 1 · 𝑋 ) = 𝑋 )
18	5 8 8 9 9 17	syl122anc	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → ( 1 · 𝑋 ) = 𝑋 )