Theorem sraval

Description: Lemma for srabase through sravsca . (Contributed by Mario Carneiro, 27-Nov-2014) (Revised by Thierry Arnoux, 16-Jun-2019)

	Ref	Expression
Assertion	sraval	$$⊢\left(W\in V\wedge S\subseteq {\mathrm{Base}}_W\right)\to \mathrm{subringAlg}\left(W\right)\left(S\right)=\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩$$

Proof

Step	Hyp	Ref	Expression
1		elex	$$⊢W\in V\to W\in \mathrm{V}$$
2	1	adantr	$$⊢\left(W\in V\wedge S\subseteq {\mathrm{Base}}_W\right)\to W\in \mathrm{V}$$
3		fveq2	$$⊢w=W\to {\mathrm{Base}}_w={\mathrm{Base}}_W$$
4	3	pweqd	$$⊢w=W\to 𝒫{\mathrm{Base}}_w=𝒫{\mathrm{Base}}_W$$
5		id	$$⊢w=W\to w=W$$
6		oveq1	$$⊢w=W\to w{\upharpoonright }_{𝑠}s=W{\upharpoonright }_{𝑠}s$$
7	6	opeq2d	$$⊢w=W\to ⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),w{\upharpoonright }_{𝑠}s⟩=⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}s⟩$$
8	5 7	oveq12d	$$⊢w=W\to w\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),w{\upharpoonright }_{𝑠}s⟩=W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}s⟩$$
9		fveq2	$$⊢w=W\to {\cdot }_w={\cdot }_W$$
10	9	opeq2d	$$⊢w=W\to ⟨{\cdot }_{\mathrm{ndx}},{\cdot }_w⟩=⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩$$
11	8 10	oveq12d	$$⊢w=W\to \left(w\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),w{\upharpoonright }_{𝑠}s⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_w⟩=\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}s⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩$$
12	9	opeq2d	$$⊢w=W\to ⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_w⟩=⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩$$
13	11 12	oveq12d	$$⊢w=W\to \left(\left(w\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),w{\upharpoonright }_{𝑠}s⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_w⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_w⟩=\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}s⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩$$
14	4 13	mpteq12dv	$$⊢w=W\to \left(s\in 𝒫{\mathrm{Base}}_w⟼\left(\left(w\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),w{\upharpoonright }_{𝑠}s⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_w⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_w⟩\right)=\left(s\in 𝒫{\mathrm{Base}}_W⟼\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}s⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩\right)$$
15		df-sra	$$⊢\mathrm{subringAlg}=\left(w\in \mathrm{V}⟼\left(s\in 𝒫{\mathrm{Base}}_w⟼\left(\left(w\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),w{\upharpoonright }_{𝑠}s⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_w⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_w⟩\right)\right)$$
16		fvex	$$⊢{\mathrm{Base}}_W\in \mathrm{V}$$
17	16	pwex	$$⊢𝒫{\mathrm{Base}}_W\in \mathrm{V}$$
18	17	mptex	$$⊢\left(s\in 𝒫{\mathrm{Base}}_W⟼\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}s⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩\right)\in \mathrm{V}$$
19	14 15 18	fvmpt	$$⊢W\in \mathrm{V}\to \mathrm{subringAlg}\left(W\right)=\left(s\in 𝒫{\mathrm{Base}}_W⟼\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}s⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩\right)$$
20	2 19	syl	$$⊢\left(W\in V\wedge S\subseteq {\mathrm{Base}}_W\right)\to \mathrm{subringAlg}\left(W\right)=\left(s\in 𝒫{\mathrm{Base}}_W⟼\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}s⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩\right)$$
21		simpr	$$⊢\left(\left(W\in V\wedge S\subseteq {\mathrm{Base}}_W\right)\wedge s=S\right)\to s=S$$
22	21	oveq2d	$$⊢\left(\left(W\in V\wedge S\subseteq {\mathrm{Base}}_W\right)\wedge s=S\right)\to W{\upharpoonright }_{𝑠}s=W{\upharpoonright }_{𝑠}S$$
23	22	opeq2d	$$⊢\left(\left(W\in V\wedge S\subseteq {\mathrm{Base}}_W\right)\wedge s=S\right)\to ⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}s⟩=⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩$$
24	23	oveq2d	$$⊢\left(\left(W\in V\wedge S\subseteq {\mathrm{Base}}_W\right)\wedge s=S\right)\to W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}s⟩=W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩$$
25	24	oveq1d	$$⊢\left(\left(W\in V\wedge S\subseteq {\mathrm{Base}}_W\right)\wedge s=S\right)\to \left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}s⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩=\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩$$
26	25	oveq1d	$$⊢\left(\left(W\in V\wedge S\subseteq {\mathrm{Base}}_W\right)\wedge s=S\right)\to \left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}s⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩=\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩$$
27		simpr	$$⊢\left(W\in V\wedge S\subseteq {\mathrm{Base}}_W\right)\to S\subseteq {\mathrm{Base}}_W$$
28	16	elpw2	$$⊢S\in 𝒫{\mathrm{Base}}_W\leftrightarrow S\subseteq {\mathrm{Base}}_W$$
29	27 28	sylibr	$$⊢\left(W\in V\wedge S\subseteq {\mathrm{Base}}_W\right)\to S\in 𝒫{\mathrm{Base}}_W$$
30		ovexd	$$⊢\left(W\in V\wedge S\subseteq {\mathrm{Base}}_W\right)\to \left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩\in \mathrm{V}$$
31	20 26 29 30	fvmptd	$$⊢\left(W\in V\wedge S\subseteq {\mathrm{Base}}_W\right)\to \mathrm{subringAlg}\left(W\right)\left(S\right)=\left(\left(W\mathrm{sSet}⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),W{\upharpoonright }_{𝑠}S⟩\right)\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},{\cdot }_W⟩\right)\mathrm{sSet}⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),{\cdot }_W⟩$$