Theorem isclm

Description: A subcomplex module is a left module over a subring of the field of complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Hypotheses	isclm.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
	isclm.k	$$⊢K={\mathrm{Base}}_F$$
Assertion	isclm	$$⊢W\in \mathrm{CMod}\leftrightarrow \left(W\in \mathrm{LMod}\wedge F={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\wedge K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		isclm.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		isclm.k	$$⊢K={\mathrm{Base}}_F$$
3		fvexd	$$⊢w=W\to \mathrm{Scalar}\left(w\right)\in \mathrm{V}$$
4		fvexd	$$⊢\left(w=W\wedge f=\mathrm{Scalar}\left(w\right)\right)\to {\mathrm{Base}}_f\in \mathrm{V}$$
5		id	$$⊢f=\mathrm{Scalar}\left(w\right)\to f=\mathrm{Scalar}\left(w\right)$$
6		fveq2	$$⊢w=W\to \mathrm{Scalar}\left(w\right)=\mathrm{Scalar}\left(W\right)$$
7	6 1	eqtr4di	$$⊢w=W\to \mathrm{Scalar}\left(w\right)=F$$
8	5 7	sylan9eqr	$$⊢\left(w=W\wedge f=\mathrm{Scalar}\left(w\right)\right)\to f=F$$
9	8	adantr	$$⊢\left(\left(w=W\wedge f=\mathrm{Scalar}\left(w\right)\right)\wedge k={\mathrm{Base}}_f\right)\to f=F$$
10		id	$$⊢k={\mathrm{Base}}_f\to k={\mathrm{Base}}_f$$
11	8	fveq2d	$$⊢\left(w=W\wedge f=\mathrm{Scalar}\left(w\right)\right)\to {\mathrm{Base}}_f={\mathrm{Base}}_F$$
12	11 2	eqtr4di	$$⊢\left(w=W\wedge f=\mathrm{Scalar}\left(w\right)\right)\to {\mathrm{Base}}_f=K$$
13	10 12	sylan9eqr	$$⊢\left(\left(w=W\wedge f=\mathrm{Scalar}\left(w\right)\right)\wedge k={\mathrm{Base}}_f\right)\to k=K$$
14	13	oveq2d	$$⊢\left(\left(w=W\wedge f=\mathrm{Scalar}\left(w\right)\right)\wedge k={\mathrm{Base}}_f\right)\to {ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}k={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K$$
15	9 14	eqeq12d	$$⊢\left(\left(w=W\wedge f=\mathrm{Scalar}\left(w\right)\right)\wedge k={\mathrm{Base}}_f\right)\to \left(f={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}k\leftrightarrow F={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\right)$$
16	13	eleq1d	$$⊢\left(\left(w=W\wedge f=\mathrm{Scalar}\left(w\right)\right)\wedge k={\mathrm{Base}}_f\right)\to \left(k\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\leftrightarrow K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\right)$$
17	15 16	anbi12d	$$⊢\left(\left(w=W\wedge f=\mathrm{Scalar}\left(w\right)\right)\wedge k={\mathrm{Base}}_f\right)\to \left(\left(f={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}k\wedge k\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\right)\leftrightarrow \left(F={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\wedge K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\right)\right)$$
18	4 17	sbcied	$$⊢\left(w=W\wedge f=\mathrm{Scalar}\left(w\right)\right)\to \left(\underset{\dot{}}{[}{\mathrm{Base}}_f/k\underset{\dot{}}{]}\left(f={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}k\wedge k\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\right)\leftrightarrow \left(F={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\wedge K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\right)\right)$$
19	3 18	sbcied	$$⊢w=W\to \left(\underset{\dot{}}{[}\mathrm{Scalar}\left(w\right)/f\underset{\dot{}}{]}\underset{\dot{}}{[}{\mathrm{Base}}_f/k\underset{\dot{}}{]}\left(f={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}k\wedge k\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\right)\leftrightarrow \left(F={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\wedge K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\right)\right)$$
20		df-clm	$$⊢\mathrm{CMod}=\left\{w\in \mathrm{LMod}|\underset{\dot{}}{[}\mathrm{Scalar}\left(w\right)/f\underset{\dot{}}{]}\underset{\dot{}}{[}{\mathrm{Base}}_f/k\underset{\dot{}}{]}\left(f={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}k\wedge k\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\right)\right\}$$
21	19 20	elrab2	$$⊢W\in \mathrm{CMod}\leftrightarrow \left(W\in \mathrm{LMod}\wedge \left(F={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\wedge K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\right)\right)$$
22		3anass	$$⊢\left(W\in \mathrm{LMod}\wedge F={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\wedge K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\right)\leftrightarrow \left(W\in \mathrm{LMod}\wedge \left(F={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\wedge K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\right)\right)$$
23	21 22	bitr4i	$$⊢W\in \mathrm{CMod}\leftrightarrow \left(W\in \mathrm{LMod}\wedge F={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\wedge K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\right)$$