Theorem ldualvsass

Description: Associative law for scalar product operation. (Contributed by NM, 20-Oct-2014)

	Ref	Expression
Hypotheses	ldualvsass.f	$$⊢F=\mathrm{LFnl}\left(W\right)$$
	ldualvsass.r	$$⊢R=\mathrm{Scalar}\left(W\right)$$
	ldualvsass.k	$$⊢K={\mathrm{Base}}_R$$
	ldualvsass.t	$$⊢\underset{\dot{}}{\times }={\cdot }_R$$
	ldualvsass.d	$$⊢D=\mathrm{LDual}\left(W\right)$$
	ldualvsass.s	$$⊢\underset{\dot{}}{·}={\cdot }_D$$
	ldualvsass.w	$$⊢\phi \to W\in \mathrm{LMod}$$
	ldualvsass.x	$$⊢\phi \to X\in K$$
	ldualvsass.y	$$⊢\phi \to Y\in K$$
	ldualvsass.g	$$⊢\phi \to G\in F$$
Assertion	ldualvsass	$$⊢\phi \to \left(Y\underset{\dot{}}{\times }X\right)\underset{\dot{}}{·}G=X\underset{\dot{}}{·}\left(Y\underset{\dot{}}{·}G\right)$$

Proof

Step	Hyp	Ref	Expression
1		ldualvsass.f	$$⊢F=\mathrm{LFnl}\left(W\right)$$
2		ldualvsass.r	$$⊢R=\mathrm{Scalar}\left(W\right)$$
3		ldualvsass.k	$$⊢K={\mathrm{Base}}_R$$
4		ldualvsass.t	$$⊢\underset{\dot{}}{\times }={\cdot }_R$$
5		ldualvsass.d	$$⊢D=\mathrm{LDual}\left(W\right)$$
6		ldualvsass.s	$$⊢\underset{\dot{}}{·}={\cdot }_D$$
7		ldualvsass.w	$$⊢\phi \to W\in \mathrm{LMod}$$
8		ldualvsass.x	$$⊢\phi \to X\in K$$
9		ldualvsass.y	$$⊢\phi \to Y\in K$$
10		ldualvsass.g	$$⊢\phi \to G\in F$$
11		eqid	$$⊢{\mathrm{Base}}_W={\mathrm{Base}}_W$$
12	11 2 3 4 1 7 9 8 10	lflvsass	$$⊢\phi \to G{\underset{\dot{}}{\times }}_f\left({\mathrm{Base}}_W\times \left\{Y\underset{\dot{}}{\times }X\right\}\right)=\left(G{\underset{\dot{}}{\times }}_f\left({\mathrm{Base}}_W\times \left\{Y\right\}\right)\right){\underset{\dot{}}{\times }}_f\left({\mathrm{Base}}_W\times \left\{X\right\}\right)$$
13	2	lmodring	$$⊢W\in \mathrm{LMod}\to R\in \mathrm{Ring}$$
14	7 13	syl	$$⊢\phi \to R\in \mathrm{Ring}$$
15	3 4	ringcl	$$⊢\left(R\in \mathrm{Ring}\wedge Y\in K\wedge X\in K\right)\to Y\underset{\dot{}}{\times }X\in K$$
16	14 9 8 15	syl3anc	$$⊢\phi \to Y\underset{\dot{}}{\times }X\in K$$
17	1 11 2 3 4 5 6 7 16 10	ldualvs	$$⊢\phi \to \left(Y\underset{\dot{}}{\times }X\right)\underset{\dot{}}{·}G=G{\underset{\dot{}}{\times }}_f\left({\mathrm{Base}}_W\times \left\{Y\underset{\dot{}}{\times }X\right\}\right)$$
18	11 2 3 4 1 7 10 9	lflvscl	$$⊢\phi \to G{\underset{\dot{}}{\times }}_f\left({\mathrm{Base}}_W\times \left\{Y\right\}\right)\in F$$
19	1 11 2 3 4 5 6 7 8 18	ldualvs	$$⊢\phi \to X\underset{\dot{}}{·}\left(G{\underset{\dot{}}{\times }}_f\left({\mathrm{Base}}_W\times \left\{Y\right\}\right)\right)=\left(G{\underset{\dot{}}{\times }}_f\left({\mathrm{Base}}_W\times \left\{Y\right\}\right)\right){\underset{\dot{}}{\times }}_f\left({\mathrm{Base}}_W\times \left\{X\right\}\right)$$
20	12 17 19	3eqtr4d	$$⊢\phi \to \left(Y\underset{\dot{}}{\times }X\right)\underset{\dot{}}{·}G=X\underset{\dot{}}{·}\left(G{\underset{\dot{}}{\times }}_f\left({\mathrm{Base}}_W\times \left\{Y\right\}\right)\right)$$
21	1 11 2 3 4 5 6 7 9 10	ldualvs	$$⊢\phi \to Y\underset{\dot{}}{·}G=G{\underset{\dot{}}{\times }}_f\left({\mathrm{Base}}_W\times \left\{Y\right\}\right)$$
22	21	oveq2d	$$⊢\phi \to X\underset{\dot{}}{·}\left(Y\underset{\dot{}}{·}G\right)=X\underset{\dot{}}{·}\left(G{\underset{\dot{}}{\times }}_f\left({\mathrm{Base}}_W\times \left\{Y\right\}\right)\right)$$
23	20 22	eqtr4d	$$⊢\phi \to \left(Y\underset{\dot{}}{\times }X\right)\underset{\dot{}}{·}G=X\underset{\dot{}}{·}\left(Y\underset{\dot{}}{·}G\right)$$