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	$$⊢V={\mathrm{Base}}_W$$
	lmod4.p	$$⊢\underset{\dot{}}{+}={+}_W$$
	lmodvaddsub4.m	$$⊢\underset{\dot{}}{-}={-}_W$$
Assertion	lmodvaddsub4	$$⊢\left(W\in \mathrm{LMod}\wedge \left(A\in V\wedge B\in V\right)\wedge \left(C\in V\wedge D\in V\right)\right)\to \left(A\underset{\dot{}}{+}B=C\underset{\dot{}}{+}D\leftrightarrow A\underset{\dot{}}{-}C=D\underset{\dot{}}{-}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		lmod4.v	$$⊢V={\mathrm{Base}}_W$$
2		lmod4.p	$$⊢\underset{\dot{}}{+}={+}_W$$
3		lmodvaddsub4.m	$$⊢\underset{\dot{}}{-}={-}_W$$
4		lmodabl	$$⊢W\in \mathrm{LMod}\to W\in \mathrm{Abel}$$
5	1 2 3	abladdsub4	$$⊢\left(W\in \mathrm{Abel}\wedge \left(A\in V\wedge B\in V\right)\wedge \left(C\in V\wedge D\in V\right)\right)\to \left(A\underset{\dot{}}{+}B=C\underset{\dot{}}{+}D\leftrightarrow A\underset{\dot{}}{-}C=D\underset{\dot{}}{-}B\right)$$
6	4 5	syl3an1	$$⊢\left(W\in \mathrm{LMod}\wedge \left(A\in V\wedge B\in V\right)\wedge \left(C\in V\wedge D\in V\right)\right)\to \left(A\underset{\dot{}}{+}B=C\underset{\dot{}}{+}D\leftrightarrow A\underset{\dot{}}{-}C=D\underset{\dot{}}{-}B\right)$$