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Metamath Proof Explorer

Theorem cphip0l

Description: Inner product with a zero first argument. Part of proof of Theorem 6.44 of Ponnusamy p. 361. Complex version of ip0l . (Contributed by Mario Carneiro, 16-Oct-2015)

		Ref	Expression
	Hypotheses	cphipcj.h	$⊢ \underset{˙}{,} = \cdot_{𝑖} (W)$
		cphipcj.v	$⊢ V = {Base}_{W}$
		cphip0l.z	$⊢ \underset{˙}{0} = 0_{W}$
	Assertion	cphip0l	$⊢ (W \in CPreHil \land A \in V) \to \underset{˙}{0} \underset{˙}{,} A = 0$

Proof

Step	Hyp	Ref	Expression
1		cphipcj.h	$⊢ \underset{˙}{,} = \cdot_{𝑖} (W)$
2		cphipcj.v	$⊢ V = {Base}_{W}$
3		cphip0l.z	$⊢ \underset{˙}{0} = 0_{W}$
4		cphphl	$⊢ W \in CPreHil \to W \in PreHil$
5		eqid	$⊢ Scalar (W) = Scalar (W)$
6		eqid	$⊢ 0_{Scalar (W)} = 0_{Scalar (W)}$
7	5 1 2 6 3	ip0l	$⊢ (W \in PreHil \land A \in V) \to \underset{˙}{0} \underset{˙}{,} A = 0_{Scalar (W)}$
8	4 7	sylan	$⊢ (W \in CPreHil \land A \in V) \to \underset{˙}{0} \underset{˙}{,} A = 0_{Scalar (W)}$
9		cphclm	$⊢ W \in CPreHil \to W \in CMod$
10	5	clm0	$⊢ W \in CMod \to 0 = 0_{Scalar (W)}$
11	9 10	syl	$⊢ W \in CPreHil \to 0 = 0_{Scalar (W)}$
12	11	adantr	$⊢ (W \in CPreHil \land A \in V) \to 0 = 0_{Scalar (W)}$
13	8 12	eqtr4d	$⊢ (W \in CPreHil \land A \in V) \to \underset{˙}{0} \underset{˙}{,} A = 0$