Theorem imasvsca

Description: The scalar multiplication operation of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015) (Revised by Thierry Arnoux, 16-Jun-2019)

	Ref	Expression
Hypotheses	imasbas.u	$$⊢\phi \to U=F{“}_{𝑠}R$$
	imasbas.v	$$⊢\phi \to V={\mathrm{Base}}_R$$
	imasbas.f	$$⊢\phi \to F:V\underset{onto}{⟶}B$$
	imasbas.r	$$⊢\phi \to R\in Z$$
	imassca.g	$$⊢G=\mathrm{Scalar}\left(R\right)$$
	imasvsca.k	$$⊢K={\mathrm{Base}}_G$$
	imasvsca.q	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
	imasvsca.s	$$⊢\underset{\dot{}}{\bullet }={\cdot }_U$$
Assertion	imasvsca	$$⊢\phi \to \underset{\dot{}}{\bullet }=\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		imasbas.u	$$⊢\phi \to U=F{“}_{𝑠}R$$
2		imasbas.v	$$⊢\phi \to V={\mathrm{Base}}_R$$
3		imasbas.f	$$⊢\phi \to F:V\underset{onto}{⟶}B$$
4		imasbas.r	$$⊢\phi \to R\in Z$$
5		imassca.g	$$⊢G=\mathrm{Scalar}\left(R\right)$$
6		imasvsca.k	$$⊢K={\mathrm{Base}}_G$$
7		imasvsca.q	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
8		imasvsca.s	$$⊢\underset{\dot{}}{\bullet }={\cdot }_U$$
9		eqid	$$⊢{+}_R={+}_R$$
10		eqid	$$⊢{\cdot }_R={\cdot }_R$$
11		eqid	$$⊢\mathrm{Scalar}\left(R\right)=\mathrm{Scalar}\left(R\right)$$
12	5	fveq2i	$$⊢{\mathrm{Base}}_G={\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)}$$
13	6 12	eqtri	$$⊢K={\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)}$$
14		eqid	$$⊢{\cdot }_{𝑖}\left(R\right)={\cdot }_{𝑖}\left(R\right)$$
15		eqid	$$⊢\mathrm{TopOpen}\left(R\right)=\mathrm{TopOpen}\left(R\right)$$
16		eqid	$$⊢\mathrm{dist}\left(R\right)=\mathrm{dist}\left(R\right)$$
17		eqid	$$⊢{\le }_R={\le }_R$$
18		eqid	$$⊢{+}_U={+}_U$$
19	1 2 3 4 9 18	imasplusg	$$⊢\phi \to {+}_U=\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{+}_{R}q\right)⟩\right\}$$
20		eqid	$$⊢{\cdot }_U={\cdot }_U$$
21	1 2 3 4 10 20	imasmulr	$$⊢\phi \to {\cdot }_U=\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{\cdot }_{R}q\right)⟩\right\}$$
22		eqidd	$$⊢\phi \to \bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)=\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)$$
23		eqidd	$$⊢\phi \to \bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}=\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}$$
24		eqidd	$$⊢\phi \to \mathrm{TopOpen}\left(R\right)\mathrm{qTop}F=\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F$$
25		eqid	$$⊢\mathrm{dist}\left(U\right)=\mathrm{dist}\left(U\right)$$
26	1 2 3 4 16 25	imasds	$$⊢\phi \to \mathrm{dist}\left(U\right)=\left(x\in B,y\in B⟼inf\left(\bigcup _{u\in \mathbb{N}}\mathrm{ran}\left(z\in \left\{w\in \left({\left(V\times V\right)}^{\left(1\dots u\right)}\right)|\left(F\left(1^{st}\left(w\left(1\right)\right)\right)=x\wedge F\left(2^{nd}\left(w\left(u\right)\right)\right)=y\wedge \forall v\in \left(1\dots u-1\right)F\left(2^{nd}\left(w\left(v\right)\right)\right)=F\left(1^{st}\left(w\left(v+1\right)\right)\right)\right)\right\}⟼{\sum }_{{ℝ}_{𝑠}^{*}}\left(\mathrm{dist}\left(R\right)\circ z\right)\right),{ℝ}^{*},<\right)\right)$$
27		eqidd	$$⊢\phi \to \left(F\circ {\le }_R\right)\circ F^{-1}=\left(F\circ {\le }_R\right)\circ F^{-1}$$
28	1 2 9 10 11 13 7 14 15 16 17 19 21 22 23 24 26 27 3 4	imasval	$$⊢\phi \to U=\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}$$
29		eqid	$$⊢\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}=\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}$$
30	29	imasvalstr	$$⊢\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}\right)\mathrm{Struct}⟨1,12⟩$$
31		vscaid	$$⊢{\cdot }_{𝑠}=\mathrm{Slot}{\cdot }_{\mathrm{ndx}}$$
32		snsstp2	$$⊢\left\{⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)⟩\right\}\subseteq \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}$$
33		ssun2	$$⊢\left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}$$
34		ssun1	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\subseteq \left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}$$
35	33 34	sstri	$$⊢\left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\subseteq \left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}$$
36	32 35	sstri	$$⊢\left\{⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)⟩\right\}\subseteq \left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}$$
37		fvex	$$⊢{\mathrm{Base}}_R\in \mathrm{V}$$
38	2 37	eqeltrdi	$$⊢\phi \to V\in \mathrm{V}$$
39	6	fvexi	$$⊢K\in \mathrm{V}$$
40		snex	$$⊢\left\{F\left(q\right)\right\}\in \mathrm{V}$$
41	39 40	mpoex	$$⊢\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)\in \mathrm{V}$$
42	41	rgenw	$$⊢\forall q\in V\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)\in \mathrm{V}$$
43		iunexg	$$⊢\left(V\in \mathrm{V}\wedge \forall q\in V\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)\in \mathrm{V}\right)\to \bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)\in \mathrm{V}$$
44	38 42 43	sylancl	$$⊢\phi \to \bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)\in \mathrm{V}$$
45	28 30 31 36 44 8	strfv3	$$⊢\phi \to \underset{\dot{}}{\bullet }=\bigcup _{q\in V}\left(p\in K,x\in \left\{F\left(q\right)\right\}⟼F\left(p\underset{\dot{}}{·}q\right)\right)$$