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Mirrors > Home > MPE Home > Th. List > ocv2ss | Structured version Visualization version GIF version |
Description: Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
ocv2ss.o | β’ β₯ = (ocvβπ) |
Ref | Expression |
---|---|
ocv2ss | β’ (π β π β ( β₯ βπ) β ( β₯ βπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr2 3987 | . . . 4 β’ (π β π β (π β (Baseβπ) β π β (Baseβπ))) | |
2 | idd 24 | . . . 4 β’ (π β π β (π₯ β (Baseβπ) β π₯ β (Baseβπ))) | |
3 | ssralv 4048 | . . . 4 β’ (π β π β (βπ¦ β π (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ)) β βπ¦ β π (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ)))) | |
4 | 1, 2, 3 | 3anim123d 1440 | . . 3 β’ (π β π β ((π β (Baseβπ) β§ π₯ β (Baseβπ) β§ βπ¦ β π (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ))) β (π β (Baseβπ) β§ π₯ β (Baseβπ) β§ βπ¦ β π (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ))))) |
5 | eqid 2728 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
6 | eqid 2728 | . . . 4 β’ (Β·πβπ) = (Β·πβπ) | |
7 | eqid 2728 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
8 | eqid 2728 | . . . 4 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
9 | ocv2ss.o | . . . 4 β’ β₯ = (ocvβπ) | |
10 | 5, 6, 7, 8, 9 | elocv 21600 | . . 3 β’ (π₯ β ( β₯ βπ) β (π β (Baseβπ) β§ π₯ β (Baseβπ) β§ βπ¦ β π (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ)))) |
11 | 5, 6, 7, 8, 9 | elocv 21600 | . . 3 β’ (π₯ β ( β₯ βπ) β (π β (Baseβπ) β§ π₯ β (Baseβπ) β§ βπ¦ β π (π₯(Β·πβπ)π¦) = (0gβ(Scalarβπ)))) |
12 | 4, 10, 11 | 3imtr4g 296 | . 2 β’ (π β π β (π₯ β ( β₯ βπ) β π₯ β ( β₯ βπ))) |
13 | 12 | ssrdv 3986 | 1 β’ (π β π β ( β₯ βπ) β ( β₯ βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 βwral 3058 β wss 3947 βcfv 6548 (class class class)co 7420 Basecbs 17180 Scalarcsca 17236 Β·πcip 17238 0gc0g 17421 ocvcocv 21592 |
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 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
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 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-ocv 21595 |
This theorem is referenced by: ocvsscon 21607 ocvlsp 21608 ocvcss 21619 cssmre 21625 mrccss 21626 clsocv 25191 |
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