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Mirrors > Home > HSE Home > Th. List > kbfval | Structured version Visualization version GIF version |
Description: The outer product of two vectors, expressed as ∣ 𝐴〉 〈𝐵 ∣ in Dirac notation. See df-kb 28094. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
kbfval | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6557 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ·ih 𝑧) ·ℎ 𝑦) = ((𝑥 ·ih 𝑧) ·ℎ 𝐴)) | |
2 | 1 | mpteq2dv 4673 | . 2 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) ·ℎ 𝑦)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) ·ℎ 𝐴))) |
3 | oveq2 6557 | . . . 4 ⊢ (𝑧 = 𝐵 → (𝑥 ·ih 𝑧) = (𝑥 ·ih 𝐵)) | |
4 | 3 | oveq1d 6564 | . . 3 ⊢ (𝑧 = 𝐵 → ((𝑥 ·ih 𝑧) ·ℎ 𝐴) = ((𝑥 ·ih 𝐵) ·ℎ 𝐴)) |
5 | 4 | mpteq2dv 4673 | . 2 ⊢ (𝑧 = 𝐵 → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) ·ℎ 𝐴)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴))) |
6 | df-kb 28094 | . 2 ⊢ ketbra = (𝑦 ∈ ℋ, 𝑧 ∈ ℋ ↦ (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) ·ℎ 𝑦))) | |
7 | ax-hilex 27240 | . . 3 ⊢ ℋ ∈ V | |
8 | 7 | mptex 6390 | . 2 ⊢ (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴)) ∈ V |
9 | 2, 5, 6, 8 | ovmpt2 6694 | 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 383 = wceq 1475 ∈ wcel 1977 ↦ cmpt 4643 (class class class)co 6549 ℋchil 27160 ·ℎ csm 27162 ·ih csp 27163 ketbra ck 27198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-hilex 27240 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-kb 28094 |
This theorem is referenced by: kbop 28196 kbval 28197 kbmul 28198 |
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