Theorem imassca

Description: The scalar field 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	$$⊢G=\mathrm{Scalar}\left(R\right)$$
Assertion	imassca	$$⊢\phi \to G=\mathrm{Scalar}\left(U\right)$$

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	5	fvexi	$$⊢G\in \mathrm{V}$$
7		eqid	$$⊢\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_G,x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}=\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_G,x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}$$
8	7	imasvalstr	$$⊢\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_G,x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}\right)\mathrm{Struct}⟨1,12⟩$$
9		scaid	$$⊢\mathrm{Scalar}=\mathrm{Slot}\mathrm{Scalar}\left(\mathrm{ndx}\right)$$
10		snsstp1	$$⊢\left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩\right\}\subseteq \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_G,x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}$$
11		ssun2	$$⊢\left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_G,x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_G,x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}$$
12	10 11	sstri	$$⊢\left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_G,x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}$$
13		ssun1	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_G,x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\subseteq \left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_G,x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}$$
14	12 13	sstri	$$⊢\left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩\right\}\subseteq \left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_G,x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}$$
15	8 9 14	strfv	$$⊢G\in \mathrm{V}\to G=\mathrm{Scalar}\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_G,x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}\right)$$
16	6 15	ax-mp	$$⊢G=\mathrm{Scalar}\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_G,x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}\right)$$
17		eqid	$$⊢{+}_R={+}_R$$
18		eqid	$$⊢{\cdot }_R={\cdot }_R$$
19		eqid	$$⊢{\mathrm{Base}}_G={\mathrm{Base}}_G$$
20		eqid	$$⊢{\cdot }_R={\cdot }_R$$
21		eqid	$$⊢{\cdot }_{𝑖}\left(R\right)={\cdot }_{𝑖}\left(R\right)$$
22		eqid	$$⊢\mathrm{TopOpen}\left(R\right)=\mathrm{TopOpen}\left(R\right)$$
23		eqid	$$⊢\mathrm{dist}\left(R\right)=\mathrm{dist}\left(R\right)$$
24		eqid	$$⊢{\le }_R={\le }_R$$
25		eqid	$$⊢{+}_U={+}_U$$
26	1 2 3 4 17 25	imasplusg	$$⊢\phi \to {+}_U=\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{+}_{R}q\right)⟩\right\}$$
27		eqid	$$⊢{\cdot }_U={\cdot }_U$$
28	1 2 3 4 18 27	imasmulr	$$⊢\phi \to {\cdot }_U=\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{\cdot }_{R}q\right)⟩\right\}$$
29		eqidd	$$⊢\phi \to \bigcup _{q\in V}\left(p\in {\mathrm{Base}}_G,x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)=\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_G,x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)$$
30		eqidd	$$⊢\phi \to \bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}=\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}$$
31		eqidd	$$⊢\phi \to \mathrm{TopOpen}\left(R\right)\mathrm{qTop}F=\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F$$
32		eqid	$$⊢\mathrm{dist}\left(U\right)=\mathrm{dist}\left(U\right)$$
33	1 2 3 4 23 32	imasds	$$⊢\phi \to \mathrm{dist}\left(U\right)=\left(x\in B,y\in B⟼inf\left(\bigcup _{n\in \mathbb{N}}\mathrm{ran}\left(g\in \left\{h\in \left({\left(V\times V\right)}^{\left(1\dots n\right)}\right)|\left(F\left(1^{st}\left(h\left(1\right)\right)\right)=x\wedge F\left(2^{nd}\left(h\left(n\right)\right)\right)=y\wedge \forall i\in \left(1\dots n-1\right)F\left(2^{nd}\left(h\left(i\right)\right)\right)=F\left(1^{st}\left(h\left(i+1\right)\right)\right)\right)\right\}⟼{\sum }_{{ℝ}_{𝑠}^{*}}\left(\mathrm{dist}\left(R\right)\circ g\right)\right),{ℝ}^{*},<\right)\right)$$
34		eqidd	$$⊢\phi \to \left(F\circ {\le }_R\right)\circ F^{-1}=\left(F\circ {\le }_R\right)\circ F^{-1}$$
35	1 2 17 18 5 19 20 21 22 23 24 26 28 29 30 31 33 34 3 4	imasval	$$⊢\phi \to U=\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_G,x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}$$
36	35	fveq2d	$$⊢\phi \to \mathrm{Scalar}\left(U\right)=\mathrm{Scalar}\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),G⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_G,x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}\right)$$
37	16 36	eqtr4id	$$⊢\phi \to G=\mathrm{Scalar}\left(U\right)$$