Theorem ipsvsca

Description: The scalar product operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 29-Aug-2015) (Revised by Thierry Arnoux, 16-Jun-2019)

	Ref	Expression
Hypothesis	ipspart.a	$$⊢A=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),I⟩\right\}$$
Assertion	ipsvsca	$$⊢\underset{\dot{}}{·}\in V\to \underset{\dot{}}{·}={\cdot }_A$$

Proof

Step	Hyp	Ref	Expression
1		ipspart.a	$$⊢A=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),I⟩\right\}$$
2	1	ipsstr	$$⊢A\mathrm{Struct}⟨1,8⟩$$
3		vscaid	$$⊢{\cdot }_{𝑠}=\mathrm{Slot}{\cdot }_{\mathrm{ndx}}$$
4		snsstp2	$$⊢\left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\subseteq \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),I⟩\right\}$$
5		ssun2	$$⊢\left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),I⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),I⟩\right\}$$
6	5 1	sseqtrri	$$⊢\left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),I⟩\right\}\subseteq A$$
7	4 6	sstri	$$⊢\left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\subseteq A$$
8	2 3 7	strfv	$$⊢\underset{\dot{}}{·}\in V\to \underset{\dot{}}{·}={\cdot }_A$$