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Mirrors > Home > HSE Home > Th. List > braval | Structured version Visualization version GIF version |
Description: A bra-ket juxtaposition, expressed as 〈𝐴 ∣ 𝐵〉 in Dirac notation, equals the inner product of the vectors. Based on definition of bra in [Prugovecki] p. 186. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
braval | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·ih 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brafval 28802 | . . 3 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))) | |
2 | 1 | fveq1d 6193 | . 2 ⊢ (𝐴 ∈ ℋ → ((bra‘𝐴)‘𝐵) = ((𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))‘𝐵)) |
3 | oveq1 6657 | . . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ·ih 𝐴) = (𝐵 ·ih 𝐴)) | |
4 | eqid 2622 | . . 3 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)) | |
5 | ovex 6678 | . . 3 ⊢ (𝐵 ·ih 𝐴) ∈ V | |
6 | 3, 4, 5 | fvmpt 6282 | . 2 ⊢ (𝐵 ∈ ℋ → ((𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))‘𝐵) = (𝐵 ·ih 𝐴)) |
7 | 2, 6 | sylan9eq 2676 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·ih 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℋchil 27776 ·ih csp 27779 bracbr 27813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-hilex 27856 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-bra 28709 |
This theorem is referenced by: braadd 28804 bramul 28805 brafnmul 28810 branmfn 28964 rnbra 28966 bra11 28967 cnvbraval 28969 kbass1 28975 kbass2 28976 kbass6 28980 |
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