Theorem imasip

Description: The inner product of an image structure. (Contributed 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	$$⊢\phi \to R\in Z$$
	imasip.i	$$⊢\underset{\dot{}}{,}={\cdot }_{𝑖}\left(R\right)$$
	imasip.w	$$⊢I={\cdot }_{𝑖}\left(U\right)$$
Assertion	imasip	$$⊢\phi \to I=\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}$$

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		imasip.i	$$⊢\underset{\dot{}}{,}={\cdot }_{𝑖}\left(R\right)$$
6		imasip.w	$$⊢I={\cdot }_{𝑖}\left(U\right)$$
7		eqid	$$⊢{+}_R={+}_R$$
8		eqid	$$⊢{\cdot }_R={\cdot }_R$$
9		eqid	$$⊢\mathrm{Scalar}\left(R\right)=\mathrm{Scalar}\left(R\right)$$
10		eqid	$$⊢{\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)}={\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)}$$
11		eqid	$$⊢{\cdot }_R={\cdot }_R$$
12		eqid	$$⊢\mathrm{TopOpen}\left(R\right)=\mathrm{TopOpen}\left(R\right)$$
13		eqid	$$⊢\mathrm{dist}\left(R\right)=\mathrm{dist}\left(R\right)$$
14		eqid	$$⊢{\le }_R={\le }_R$$
15		eqid	$$⊢{+}_U={+}_U$$
16	1 2 3 4 7 15	imasplusg	$$⊢\phi \to {+}_U=\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{+}_{R}q\right)⟩\right\}$$
17		eqid	$$⊢{\cdot }_U={\cdot }_U$$
18	1 2 3 4 8 17	imasmulr	$$⊢\phi \to {\cdot }_U=\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{\cdot }_{R}q\right)⟩\right\}$$
19		eqid	$$⊢{\cdot }_U={\cdot }_U$$
20	1 2 3 4 9 10 11 19	imasvsca	$$⊢\phi \to {\cdot }_U=\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)$$
21		eqidd	$$⊢\phi \to \bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}=\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}$$
22		eqidd	$$⊢\phi \to \mathrm{TopOpen}\left(R\right)\mathrm{qTop}F=\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F$$
23		eqid	$$⊢\mathrm{dist}\left(U\right)=\mathrm{dist}\left(U\right)$$
24	1 2 3 4 13 23	imasds	$$⊢\phi \to \mathrm{dist}\left(U\right)=\left(x\in B,y\in B⟼sup\left(\bigcup _{u\in \mathbb{N}}\mathrm{ran}\left(z\in \left\{w\in \left({\left(V\times V\right)}^{\left(1\dots u\right)}\right)|\left(F\left(1^{st}\left(w\left(1\right)\right)\right)=x\wedge F\left(2^{nd}\left(w\left(u\right)\right)\right)=y\wedge \forall v\in \left(1\dots u-1\right)F\left(2^{nd}\left(w\left(v\right)\right)\right)=F\left(1^{st}\left(w\left(v+1\right)\right)\right)\right)\right\}⟼{\sum }_{{ℝ}_{𝑠}^{*}}\left(\mathrm{dist}\left(R\right)\circ z\right)\right),{ℝ}^{*},<\right)\right)$$
25		eqidd	$$⊢\phi \to \left(F\circ {\le }_R\right)\circ F^{-1}=\left(F\circ {\le }_R\right)\circ F^{-1}$$
26	1 2 7 8 9 10 11 5 12 13 14 16 18 20 21 22 24 25 3 4	imasval	$$⊢\phi \to U=\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}$$
27		eqid	$$⊢\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}=\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}$$
28	27	imasvalstr	$$⊢\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}\right)\mathrm{Struct}⟨1,12⟩$$
29		ipid	$$⊢{\cdot }_{𝑖}=\mathrm{Slot}{\cdot }_{𝑖}\left(\mathrm{ndx}\right)$$
30		snsstp3	$$⊢\left\{⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}⟩\right\}\subseteq \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}⟩\right\}$$
31		ssun2	$$⊢\left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}⟩\right\}$$
32	30 31	sstri	$$⊢\left\{⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}⟩\right\}$$
33		ssun1	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}⟩\right\}\subseteq \left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}$$
34	32 33	sstri	$$⊢\left\{⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}⟩\right\}\subseteq \left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},{+}_U⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_U⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{dist}\left(U\right)⟩\right\}$$
35		fvex	$$⊢{\mathrm{Base}}_R\in \mathrm{V}$$
36	2 35	eqeltrdi	$$⊢\phi \to V\in \mathrm{V}$$
37		snex	$$⊢\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}\in \mathrm{V}$$
38	37	rgenw	$$⊢\forall q\in V\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}\in \mathrm{V}$$
39		iunexg	$$⊢\left(V\in \mathrm{V}\wedge \forall q\in V\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}\in \mathrm{V}\right)\to \bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}\in \mathrm{V}$$
40	36 38 39	sylancl	$$⊢\phi \to \bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}\in \mathrm{V}$$
41	40	ralrimivw	$$⊢\phi \to \forall p\in V\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}\in \mathrm{V}$$
42		iunexg	$$⊢\left(V\in \mathrm{V}\wedge \forall p\in V\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}\in \mathrm{V}\right)\to \bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}\in \mathrm{V}$$
43	36 41 42	syl2anc	$$⊢\phi \to \bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}\in \mathrm{V}$$
44	26 28 29 34 43 6	strfv3	$$⊢\phi \to I=\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p\underset{\dot{}}{,}q⟩\right\}$$