Theorem sravscaOLD

Description: Obsolete proof of sravsca as of 12-Nov-2024. The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 4-Oct-2015) (Revised by Thierry Arnoux, 16-Jun-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	srapart.a	$$⊢\phi \to A=\mathrm{subringAlg}\left(W\right)\left(S\right)$$
	srapart.s	$$⊢\phi \to S\subseteq {\mathrm{Base}}_W$$
Assertion	sravscaOLD	$$⊢\phi \to {\cdot }_W={\cdot }_A$$

Proof

Step	Hyp	Ref	Expression
1		srapart.a	$$⊢\phi \to A=\mathrm{subringAlg}\left(W\right)\left(S\right)$$
2		srapart.s	$$⊢\phi \to S\subseteq {\mathrm{Base}}_W$$
3		ovex	$$⊢W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩\in \mathrm{V}$$
4		fvex	$$⊢{\cdot }_W\in \mathrm{V}$$
5		vscaid	$$⊢{\cdot }_{𝑠}=\mathrm{Slot}{\cdot }_{\mathrm{ndx}}$$
6	5	setsid	$$⊢\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩\in \mathrm{V}\wedge {\cdot }_W\in \mathrm{V}\right)\to {\cdot }_W={\cdot }_{\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)}$$
7	3 4 6	mp2an	$$⊢{\cdot }_W={\cdot }_{\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)}$$
8		6re	$$⊢6\in ℝ$$
9		6lt8	$$⊢6<8$$
10	8 9	ltneii	$$⊢6\ne 8$$
11		vscandx	$$⊢{\cdot }_{\mathrm{ndx}}=6$$
12		ipndx	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)=8$$
13	11 12	neeq12i	$$⊢{\cdot }_{\mathrm{ndx}}\ne {\cdot }_{𝑖}\left(\mathrm{ndx}\right)\leftrightarrow 6\ne 8$$
14	10 13	mpbir	$$⊢{\cdot }_{\mathrm{ndx}}\ne {\cdot }_{𝑖}\left(\mathrm{ndx}\right)$$
15	5 14	setsnid	$$⊢{\cdot }_{\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)}={\cdot }_{\left(\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩\right)}$$
16	7 15	eqtri	$$⊢{\cdot }_W={\cdot }_{\left(\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩\right)}$$
17	1	adantl	$$⊢\left(W\in \mathrm{V}\wedge \phi \right)\to A=\mathrm{subringAlg}\left(W\right)\left(S\right)$$
18		sraval	$$⊢\left(W\in \mathrm{V}\wedge S\subseteq {\mathrm{Base}}_W\right)\to \mathrm{subringAlg}\left(W\right)\left(S\right)=\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩$$
19	2 18	sylan2	$$⊢\left(W\in \mathrm{V}\wedge \phi \right)\to \mathrm{subringAlg}\left(W\right)\left(S\right)=\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩$$
20	17 19	eqtrd	$$⊢\left(W\in \mathrm{V}\wedge \phi \right)\to A=\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩$$
21	20	fveq2d	$$⊢\left(W\in \mathrm{V}\wedge \phi \right)\to {\cdot }_A={\cdot }_{\left(\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩\right)}$$
22	16 21	eqtr4id	$$⊢\left(W\in \mathrm{V}\wedge \phi \right)\to {\cdot }_W={\cdot }_A$$
23	5	str0	$$⊢\varnothing ={\cdot }_{\varnothing }$$
24		fvprc	$$⊢\neg W\in \mathrm{V}\to {\cdot }_W=\varnothing$$
25	24	adantr	$$⊢\left(\neg W\in \mathrm{V}\wedge \phi \right)\to {\cdot }_W=\varnothing$$
26		fv2prc	$$⊢\neg W\in \mathrm{V}\to \mathrm{subringAlg}\left(W\right)\left(S\right)=\varnothing$$
27	1 26	sylan9eqr	$$⊢\left(\neg W\in \mathrm{V}\wedge \phi \right)\to A=\varnothing$$
28	27	fveq2d	$$⊢\left(\neg W\in \mathrm{V}\wedge \phi \right)\to {\cdot }_A={\cdot }_{\varnothing }$$
29	23 25 28	3eqtr4a	$$⊢\left(\neg W\in \mathrm{V}\wedge \phi \right)\to {\cdot }_W={\cdot }_A$$
30	22 29	pm2.61ian	$$⊢\phi \to {\cdot }_W={\cdot }_A$$