kbfval - Hilbert Space Explorer

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Theorem kbfval 31205

Description: The outer product of two vectors, expressed as ∣ 𝐴⟩⟨𝐵 ∣ in Dirac notation. See df-kb 31104. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

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

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

Proof of Theorem kbfval Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7417	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ·_ih 𝑧) ·_ℎ 𝑦) = ((𝑥 ·_ih 𝑧) ·_ℎ 𝐴))
2	1	mpteq2dv 5251	. 2 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih 𝑧) ·_ℎ 𝑦)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih 𝑧) ·_ℎ 𝐴)))
3		oveq2 7417	. . . 4 ⊢ (𝑧 = 𝐵 → (𝑥 ·_ih 𝑧) = (𝑥 ·_ih 𝐵))
4	3	oveq1d 7424	. . 3 ⊢ (𝑧 = 𝐵 → ((𝑥 ·_ih 𝑧) ·_ℎ 𝐴) = ((𝑥 ·_ih 𝐵) ·_ℎ 𝐴))
5	4	mpteq2dv 5251	. 2 ⊢ (𝑧 = 𝐵 → (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih 𝑧) ·_ℎ 𝐴)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih 𝐵) ·_ℎ 𝐴)))
6		df-kb 31104	. 2 ⊢ ketbra = (𝑦 ∈ ℋ, 𝑧 ∈ ℋ ↦ (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih 𝑧) ·_ℎ 𝑦)))
7		ax-hilex 30252	. . 3 ⊢ ℋ ∈ V
8	7	mptex 7225	. 2 ⊢ (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih 𝐵) ·_ℎ 𝐴)) ∈ V
9	2, 5, 6, 8	ovmpo 7568	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih 𝐵) ·_ℎ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ↦ cmpt 5232 (class class class)co 7409 ℋchba 30172 ·_ℎ csm 30174 ·_ih csp 30175 ketbra ck 30210

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-hilex 30252

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-kb 31104

This theorem is referenced by: kbop 31206 kbval 31207 kbmul 31208

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