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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 31870. (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 7439 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ·ih 𝑧) ·ℎ 𝑦) = ((𝑥 ·ih 𝑧) ·ℎ 𝐴)) | |
| 2 | 1 | mpteq2dv 5244 | . 2 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) ·ℎ 𝑦)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) ·ℎ 𝐴))) | 
| 3 | oveq2 7439 | . . . 4 ⊢ (𝑧 = 𝐵 → (𝑥 ·ih 𝑧) = (𝑥 ·ih 𝐵)) | |
| 4 | 3 | oveq1d 7446 | . . 3 ⊢ (𝑧 = 𝐵 → ((𝑥 ·ih 𝑧) ·ℎ 𝐴) = ((𝑥 ·ih 𝐵) ·ℎ 𝐴)) | 
| 5 | 4 | mpteq2dv 5244 | . 2 ⊢ (𝑧 = 𝐵 → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) ·ℎ 𝐴)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴))) | 
| 6 | df-kb 31870 | . 2 ⊢ ketbra = (𝑦 ∈ ℋ, 𝑧 ∈ ℋ ↦ (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) ·ℎ 𝑦))) | |
| 7 | ax-hilex 31018 | . . 3 ⊢ ℋ ∈ V | |
| 8 | 7 | mptex 7243 | . 2 ⊢ (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴)) ∈ V | 
| 9 | 2, 5, 6, 8 | ovmpo 7593 | 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 395 = wceq 1540 ∈ wcel 2108 ↦ cmpt 5225 (class class class)co 7431 ℋchba 30938 ·ℎ csm 30940 ·ih csp 30941 ketbra ck 30976 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-hilex 31018 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-kb 31870 | 
| This theorem is referenced by: kbop 31972 kbval 31973 kbmul 31974 | 
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