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Metamath Proof Explorer

Theorem kbfval

Description: The outer product of two vectors, expressed as | A >. <. B | in Dirac notation. See df-kb . (Contributed by NM, 15-May-2006) (Revised by Mario Carneiro, 23-Aug-2014) (New usage is discouraged.)

		Ref	Expression
	Assertion	kbfval	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ketbra 𝐵 ) = ( 𝑥 ∈ ℋ ↦ ( ( 𝑥 ·_ih 𝐵 ) ·_ℎ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		oveq2	⊢ ( 𝑦 = 𝐴 → ( ( 𝑥 ·_ih 𝑧 ) ·_ℎ 𝑦 ) = ( ( 𝑥 ·_ih 𝑧 ) ·_ℎ 𝐴 ) )
2	1	mpteq2dv	⊢ ( 𝑦 = 𝐴 → ( 𝑥 ∈ ℋ ↦ ( ( 𝑥 ·_ih 𝑧 ) ·_ℎ 𝑦 ) ) = ( 𝑥 ∈ ℋ ↦ ( ( 𝑥 ·_ih 𝑧 ) ·_ℎ 𝐴 ) ) )
3		oveq2	⊢ ( 𝑧 = 𝐵 → ( 𝑥 ·_ih 𝑧 ) = ( 𝑥 ·_ih 𝐵 ) )
4	3	oveq1d	⊢ ( 𝑧 = 𝐵 → ( ( 𝑥 ·_ih 𝑧 ) ·_ℎ 𝐴 ) = ( ( 𝑥 ·_ih 𝐵 ) ·_ℎ 𝐴 ) )
5	4	mpteq2dv	⊢ ( 𝑧 = 𝐵 → ( 𝑥 ∈ ℋ ↦ ( ( 𝑥 ·_ih 𝑧 ) ·_ℎ 𝐴 ) ) = ( 𝑥 ∈ ℋ ↦ ( ( 𝑥 ·_ih 𝐵 ) ·_ℎ 𝐴 ) ) )
6		df-kb	⊢ ketbra = ( 𝑦 ∈ ℋ , 𝑧 ∈ ℋ ↦ ( 𝑥 ∈ ℋ ↦ ( ( 𝑥 ·_ih 𝑧 ) ·_ℎ 𝑦 ) ) )
7		ax-hilex	⊢ ℋ ∈ V
8	7	mptex	⊢ ( 𝑥 ∈ ℋ ↦ ( ( 𝑥 ·_ih 𝐵 ) ·_ℎ 𝐴 ) ) ∈ V
9	2 5 6 8	ovmpo	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ketbra 𝐵 ) = ( 𝑥 ∈ ℋ ↦ ( ( 𝑥 ·_ih 𝐵 ) ·_ℎ 𝐴 ) ) )