Metamath Proof Explorer

Theorem clmnegsubdi2

Description: Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007) (Revised by AV, 29-Sep-2021)

Ref Expression
Hypotheses clmpm1dir.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
clmpm1dir.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
clmpm1dir.a ⊢ + = ( +g ‘ 𝑊 )
Assertion clmnegsubdi2 ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( - 1 · ( 𝐴 + ( - 1 · 𝐵 ) ) ) = ( 𝐵 + ( - 1 · 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 clmpm1dir.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 clmpm1dir.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
3 clmpm1dir.a ⊢ + = ( +g ‘ 𝑊 )
4 simp1 ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝑊 ∈ ℂMod )
5 eqid ⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
6 eqid ⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
7 5 6 clmneg1 ⊢ ( 𝑊 ∈ ℂMod → - 1 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
8 7 3ad2ant1 ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → - 1 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
9 simp2 ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 ∈ 𝑉 )
10 simpl ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) → 𝑊 ∈ ℂMod )
11 7 adantr ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) → - 1 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
12 simpr ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) → 𝐵 ∈ 𝑉 )
13 1 5 2 6 clmvscl ⊢ ( ( 𝑊 ∈ ℂMod ∧ - 1 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐵 ∈ 𝑉 ) → ( - 1 · 𝐵 ) ∈ 𝑉 )
14 10 11 12 13 syl3anc ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) → ( - 1 · 𝐵 ) ∈ 𝑉 )
15 14 3adant2 ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( - 1 · 𝐵 ) ∈ 𝑉 )
16 1 5 2 6 3 clmvsdi ⊢ ( ( 𝑊 ∈ ℂMod ∧ ( - 1 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐴 ∈ 𝑉 ∧ ( - 1 · 𝐵 ) ∈ 𝑉 ) ) → ( - 1 · ( 𝐴 + ( - 1 · 𝐵 ) ) ) = ( ( - 1 · 𝐴 ) + ( - 1 · ( - 1 · 𝐵 ) ) ) )
17 4 8 9 15 16 syl13anc ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( - 1 · ( 𝐴 + ( - 1 · 𝐵 ) ) ) = ( ( - 1 · 𝐴 ) + ( - 1 · ( - 1 · 𝐵 ) ) ) )
18 1 2 3 clmnegneg ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) → ( - 1 · ( - 1 · 𝐵 ) ) = 𝐵 )
19 18 3adant2 ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( - 1 · ( - 1 · 𝐵 ) ) = 𝐵 )
20 19 oveq2d ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( - 1 · 𝐴 ) + ( - 1 · ( - 1 · 𝐵 ) ) ) = ( ( - 1 · 𝐴 ) + 𝐵 ) )
21 clmabl ⊢ ( 𝑊 ∈ ℂMod → 𝑊 ∈ Abel )
22 21 3ad2ant1 ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝑊 ∈ Abel )
23 simpl ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ) → 𝑊 ∈ ℂMod )
24 7 adantr ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ) → - 1 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
25 simpr ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ) → 𝐴 ∈ 𝑉 )
26 1 5 2 6 clmvscl ⊢ ( ( 𝑊 ∈ ℂMod ∧ - 1 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐴 ∈ 𝑉 ) → ( - 1 · 𝐴 ) ∈ 𝑉 )
27 23 24 25 26 syl3anc ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ) → ( - 1 · 𝐴 ) ∈ 𝑉 )
28 27 3adant3 ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( - 1 · 𝐴 ) ∈ 𝑉 )
29 simp3 ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝐵 ∈ 𝑉 )
30 1 3 ablcom ⊢ ( ( 𝑊 ∈ Abel ∧ ( - 1 · 𝐴 ) ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( - 1 · 𝐴 ) + 𝐵 ) = ( 𝐵 + ( - 1 · 𝐴 ) ) )
31 22 28 29 30 syl3anc ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( - 1 · 𝐴 ) + 𝐵 ) = ( 𝐵 + ( - 1 · 𝐴 ) ) )
32 17 20 31 3eqtrd ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( - 1 · ( 𝐴 + ( - 1 · 𝐵 ) ) ) = ( 𝐵 + ( - 1 · 𝐴 ) ) )