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Mirrors > Home > HSE Home > Th. List > brafval | Structured version Visualization version GIF version |
Description: The bra of a vector, expressed as β¨π΄ β£ in Dirac notation. See df-bra 31537. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
brafval | β’ (π΄ β β β (braβπ΄) = (π₯ β β β¦ (π₯ Β·ih π΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7420 | . . 3 β’ (π¦ = π΄ β (π₯ Β·ih π¦) = (π₯ Β·ih π΄)) | |
2 | 1 | mpteq2dv 5250 | . 2 β’ (π¦ = π΄ β (π₯ β β β¦ (π₯ Β·ih π¦)) = (π₯ β β β¦ (π₯ Β·ih π΄))) |
3 | df-bra 31537 | . 2 β’ bra = (π¦ β β β¦ (π₯ β β β¦ (π₯ Β·ih π¦))) | |
4 | ax-hilex 30686 | . . 3 β’ β β V | |
5 | 4 | mptex 7227 | . 2 β’ (π₯ β β β¦ (π₯ Β·ih π΄)) β V |
6 | 2, 3, 5 | fvmpt 6998 | 1 β’ (π΄ β β β (braβπ΄) = (π₯ β β β¦ (π₯ Β·ih π΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 β¦ cmpt 5231 βcfv 6543 (class class class)co 7412 βchba 30606 Β·ih csp 30609 bracbr 30643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-hilex 30686 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-bra 31537 |
This theorem is referenced by: braval 31631 brafn 31634 bra0 31637 brafnmul 31638 |
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