Theorem phlvsca

Description: The scalar product operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013) (Revised by Mario Carneiro, 29-Aug-2015)

	Ref	Expression
Hypothesis	phlfn.h	$$⊢H=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),T⟩\right\}\cup \left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\underset{\dot{}}{,}⟩\right\}$$
Assertion	phlvsca	$$⊢\underset{\dot{}}{·}\in X\to \underset{\dot{}}{·}={\cdot }_H$$

Proof

Step	Hyp	Ref	Expression
1		phlfn.h	$$⊢H=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),T⟩\right\}\cup \left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\underset{\dot{}}{,}⟩\right\}$$
2	1	phlstr	$$⊢H\mathrm{Struct}⟨1,8⟩$$
3		vscaid	$$⊢{\cdot }_{𝑠}=\mathrm{Slot}{\cdot }_{\mathrm{ndx}}$$
4		snsspr1	$$⊢\left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\subseteq \left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\underset{\dot{}}{,}⟩\right\}$$
5		ssun2	$$⊢\left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\underset{\dot{}}{,}⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),T⟩\right\}\cup \left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\underset{\dot{}}{,}⟩\right\}$$
6	5 1	sseqtrri	$$⊢\left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\underset{\dot{}}{,}⟩\right\}\subseteq H$$
7	4 6	sstri	$$⊢\left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\subseteq H$$
8	2 3 7	strfv	$$⊢\underset{\dot{}}{·}\in X\to \underset{\dot{}}{·}={\cdot }_H$$