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Mirrors > Home > MPE Home > Th. List > ocv0 | Structured version Visualization version GIF version |
Description: The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
ocvz.v | β’ π = (Baseβπ) |
ocvz.o | β’ β₯ = (ocvβπ) |
Ref | Expression |
---|---|
ocv0 | β’ ( β₯ ββ ) = π |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4393 | . . 3 β’ β β π | |
2 | ocvz.v | . . . 4 β’ π = (Baseβπ) | |
3 | eqid 2728 | . . . 4 β’ (Β·πβπ) = (Β·πβπ) | |
4 | eqid 2728 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
5 | eqid 2728 | . . . 4 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
6 | ocvz.o | . . . 4 β’ β₯ = (ocvβπ) | |
7 | 2, 3, 4, 5, 6 | ocvval 21593 | . . 3 β’ (β β π β ( β₯ ββ ) = {π₯ β π β£ βπ¦ β β (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ))}) |
8 | 1, 7 | ax-mp 5 | . 2 β’ ( β₯ ββ ) = {π₯ β π β£ βπ¦ β β (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ))} |
9 | ral0 4509 | . . . 4 β’ βπ¦ β β (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ)) | |
10 | 9 | rgenw 3061 | . . 3 β’ βπ₯ β π βπ¦ β β (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ)) |
11 | rabid2 3460 | . . 3 β’ (π = {π₯ β π β£ βπ¦ β β (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ))} β βπ₯ β π βπ¦ β β (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ))) | |
12 | 10, 11 | mpbir 230 | . 2 β’ π = {π₯ β π β£ βπ¦ β β (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ))} |
13 | 8, 12 | eqtr4i 2759 | 1 β’ ( β₯ ββ ) = π |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 βwral 3057 {crab 3428 β wss 3945 β c0 4319 βcfv 6543 (class class class)co 7415 Basecbs 17174 Scalarcsca 17230 Β·πcip 17232 0gc0g 17415 ocvcocv 21586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7418 df-ocv 21589 |
This theorem is referenced by: ocvz 21604 css1 21616 |
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