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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 4396 | . . 3 β’ β β π | |
2 | ocvz.v | . . . 4 β’ π = (Baseβπ) | |
3 | eqid 2732 | . . . 4 β’ (Β·πβπ) = (Β·πβπ) | |
4 | eqid 2732 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
5 | eqid 2732 | . . . 4 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
6 | ocvz.o | . . . 4 β’ β₯ = (ocvβπ) | |
7 | 2, 3, 4, 5, 6 | ocvval 21219 | . . 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 4512 | . . . 4 β’ βπ¦ β β (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ)) | |
10 | 9 | rgenw 3065 | . . 3 β’ βπ₯ β π βπ¦ β β (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ)) |
11 | rabid2 3464 | . . 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 2763 | 1 β’ ( β₯ ββ ) = π |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 βwral 3061 {crab 3432 β wss 3948 β c0 4322 βcfv 6543 (class class class)co 7408 Basecbs 17143 Scalarcsca 17199 Β·πcip 17201 0gc0g 17384 ocvcocv 21212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 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-fv 6551 df-ov 7411 df-ocv 21215 |
This theorem is referenced by: ocvz 21230 css1 21242 |
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