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Metamath Proof Explorer

Theorem lmodvaddsub4

Description: Vector addition/subtraction law. (Contributed by NM, 31-Mar-2014) (Revised by Mario Carneiro, 19-Jun-2014)

Ref Expression
Hypotheses lmod4.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
lmod4.p ⊢ + = ( +g ‘ 𝑊 )
lmodvaddsub4.m ⊢ − = ( -g ‘ 𝑊 )
Assertion lmodvaddsub4 ( ( 𝑊 ∈ LMod ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) → ( ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) ↔ ( 𝐴 − 𝐶 ) = ( 𝐷 − 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 lmod4.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 lmod4.p ⊢ + = ( +g ‘ 𝑊 )
3 lmodvaddsub4.m ⊢ − = ( -g ‘ 𝑊 )
4 lmodabl ⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Abel )
5 1 2 3 abladdsub4 ⊢ ( ( 𝑊 ∈ Abel ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) → ( ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) ↔ ( 𝐴 − 𝐶 ) = ( 𝐷 − 𝐵 ) ) )
6 4 5 syl3an1 ⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) → ( ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) ↔ ( 𝐴 − 𝐶 ) = ( 𝐷 − 𝐵 ) ) )