Theorem cnlmod4

Description: Lemma 4 for cnlmod . (Contributed by AV, 20-Sep-2021)

	Ref	Expression
Hypothesis	cnlmod.w	$$⊢W=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},ℂ⟩,⟨{+}_{\mathrm{ndx}},+⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),{ℂ}_{\mathrm{fld}}⟩,⟨{\cdot }_{\mathrm{ndx}},\times ⟩\right\}$$
Assertion	cnlmod4	$$⊢{\cdot }_W=\times$$

Proof

Step	Hyp	Ref	Expression
1		cnlmod.w	$$⊢W=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},ℂ⟩,⟨{+}_{\mathrm{ndx}},+⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),{ℂ}_{\mathrm{fld}}⟩,⟨{\cdot }_{\mathrm{ndx}},\times ⟩\right\}$$
2		mulex	$$⊢\times \in \mathrm{V}$$
3		qdass	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},ℂ⟩,⟨{+}_{\mathrm{ndx}},+⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),{ℂ}_{\mathrm{fld}}⟩,⟨{\cdot }_{\mathrm{ndx}},\times ⟩\right\}=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},ℂ⟩,⟨{+}_{\mathrm{ndx}},+⟩,⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),{ℂ}_{\mathrm{fld}}⟩\right\}\cup \left\{⟨{\cdot }_{\mathrm{ndx}},\times ⟩\right\}$$
4	1 3	eqtri	$$⊢W=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},ℂ⟩,⟨{+}_{\mathrm{ndx}},+⟩,⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),{ℂ}_{\mathrm{fld}}⟩\right\}\cup \left\{⟨{\cdot }_{\mathrm{ndx}},\times ⟩\right\}$$
5	4	lmodvsca	$$⊢\times \in \mathrm{V}\to \times ={\cdot }_W$$
6	5	eqcomd	$$⊢\times \in \mathrm{V}\to {\cdot }_W=\times$$
7	2 6	ax-mp	$$⊢{\cdot }_W=\times$$