Theorem clmabs

Description: Norm in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Hypotheses	clm0.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
	clmsub.k	$$⊢K={\mathrm{Base}}_F$$
Assertion	clmabs	$$⊢\left(W\in \mathrm{CMod}\wedge A\in K\right)\to \left|A\right|=\mathrm{norm}\left(F\right)\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		clm0.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		clmsub.k	$$⊢K={\mathrm{Base}}_F$$
3	1 2	clmsca	$$⊢W\in \mathrm{CMod}\to F={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K$$
4	3	fveq2d	$$⊢W\in \mathrm{CMod}\to \mathrm{norm}\left(F\right)=\mathrm{norm}\left({ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\right)$$
5	4	adantr	$$⊢\left(W\in \mathrm{CMod}\wedge A\in K\right)\to \mathrm{norm}\left(F\right)=\mathrm{norm}\left({ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\right)$$
6	5	fveq1d	$$⊢\left(W\in \mathrm{CMod}\wedge A\in K\right)\to \mathrm{norm}\left(F\right)\left(A\right)=\mathrm{norm}\left({ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\right)\left(A\right)$$
7	1 2	clmsubrg	$$⊢W\in \mathrm{CMod}\to K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)$$
8		subrgsubg	$$⊢K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\to K\in \mathrm{SubGrp}\left({ℂ}_{\mathrm{fld}}\right)$$
9	7 8	syl	$$⊢W\in \mathrm{CMod}\to K\in \mathrm{SubGrp}\left({ℂ}_{\mathrm{fld}}\right)$$
10		eqid	$$⊢{ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K$$
11		cnfldnm	$$⊢\mathrm{abs}=\mathrm{norm}\left({ℂ}_{\mathrm{fld}}\right)$$
12		eqid	$$⊢\mathrm{norm}\left({ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\right)=\mathrm{norm}\left({ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\right)$$
13	10 11 12	subgnm2	$$⊢\left(K\in \mathrm{SubGrp}\left({ℂ}_{\mathrm{fld}}\right)\wedge A\in K\right)\to \mathrm{norm}\left({ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\right)\left(A\right)=\left|A\right|$$
14	9 13	sylan	$$⊢\left(W\in \mathrm{CMod}\wedge A\in K\right)\to \mathrm{norm}\left({ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\right)\left(A\right)=\left|A\right|$$
15	6 14	eqtr2d	$$⊢\left(W\in \mathrm{CMod}\wedge A\in K\right)\to \left|A\right|=\mathrm{norm}\left(F\right)\left(A\right)$$