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Metamath Proof Explorer

Theorem lmodvs1

Description: Scalar product with the ring unity. ( ax-hvmulid analog.) (Contributed by NM, 10-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014)

Ref Expression
Hypotheses lmodvs1.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
lmodvs1.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
lmodvs1.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
lmodvs1.u ⊢ 1 = ( 1r ‘ 𝐹 )
Assertion lmodvs1 ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 1 · 𝑋 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 lmodvs1.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 lmodvs1.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
3 lmodvs1.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
4 lmodvs1.u ⊢ 1 = ( 1r ‘ 𝐹 )
5 simpl ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑊 ∈ LMod )
6 eqid ⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 )
7 2 6 4 lmod1cl ⊢ ( 𝑊 ∈ LMod → 1 ∈ ( Base ‘ 𝐹 ) )
8 7 adantr ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 1 ∈ ( Base ‘ 𝐹 ) )
9 simpr ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ 𝑉 )
10 eqid ⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 )
11 eqid ⊢ ( +g ‘ 𝐹 ) = ( +g ‘ 𝐹 )
12 eqid ⊢ ( .r ‘ 𝐹 ) = ( .r ‘ 𝐹 )
13 1 10 3 2 6 11 12 4 lmodlema ⊢ ( ( 𝑊 ∈ LMod ∧ ( 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 ) ) )
14 13 simprrd ⊢ ( ( 𝑊 ∈ LMod ∧ ( 1 ∈ ( Base ‘ 𝐹 ) ∧ 1 ∈ ( Base ‘ 𝐹 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) → ( 1 · 𝑋 ) = 𝑋 )
15 5 8 8 9 9 14 syl122anc ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 1 · 𝑋 ) = 𝑋 )