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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 4389 | . . 3 β’ β β π | |
2 | ocvz.v | . . . 4 β’ π = (Baseβπ) | |
3 | eqid 2724 | . . . 4 β’ (Β·πβπ) = (Β·πβπ) | |
4 | eqid 2724 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
5 | eqid 2724 | . . . 4 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
6 | ocvz.o | . . . 4 β’ β₯ = (ocvβπ) | |
7 | 2, 3, 4, 5, 6 | ocvval 21530 | . . 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 4505 | . . . 4 β’ βπ¦ β β (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ)) | |
10 | 9 | rgenw 3057 | . . 3 β’ βπ₯ β π βπ¦ β β (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ)) |
11 | rabid2 3456 | . . 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 2755 | 1 β’ ( β₯ ββ ) = π |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 βwral 3053 {crab 3424 β wss 3941 β c0 4315 βcfv 6534 (class class class)co 7402 Basecbs 17145 Scalarcsca 17201 Β·πcip 17203 0gc0g 17386 ocvcocv 21523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-ov 7405 df-ocv 21526 |
This theorem is referenced by: ocvz 21541 css1 21553 |
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