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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 3982 | . . . 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 4043 | . . . 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 1439 | . . 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 2724 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
6 | eqid 2724 | . . . 4 β’ (Β·πβπ) = (Β·πβπ) | |
7 | eqid 2724 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
8 | eqid 2724 | . . . 4 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
9 | ocv2ss.o | . . . 4 β’ β₯ = (ocvβπ) | |
10 | 5, 6, 7, 8, 9 | elocv 21550 | . . 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 21550 | . . 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 3981 | 1 β’ (π β π β ( β₯ βπ) β ( β₯ βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3053 β wss 3941 βcfv 6534 (class class class)co 7402 Basecbs 17149 Scalarcsca 17205 Β·πcip 17207 0gc0g 17390 ocvcocv 21542 |
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 21545 |
This theorem is referenced by: ocvsscon 21557 ocvlsp 21558 ocvcss 21569 cssmre 21575 mrccss 21576 clsocv 25122 |
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