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Mirrors > Home > MPE Home > Th. List > ocvi | Structured version Visualization version GIF version |
Description: Property of a member of the orthocomplement of a subset. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
ocvfval.v | β’ π = (Baseβπ) |
ocvfval.i | β’ , = (Β·πβπ) |
ocvfval.f | β’ πΉ = (Scalarβπ) |
ocvfval.z | β’ 0 = (0gβπΉ) |
ocvfval.o | β’ β₯ = (ocvβπ) |
Ref | Expression |
---|---|
ocvi | β’ ((π΄ β ( β₯ βπ) β§ π΅ β π) β (π΄ , π΅) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ocvfval.v | . . . 4 β’ π = (Baseβπ) | |
2 | ocvfval.i | . . . 4 β’ , = (Β·πβπ) | |
3 | ocvfval.f | . . . 4 β’ πΉ = (Scalarβπ) | |
4 | ocvfval.z | . . . 4 β’ 0 = (0gβπΉ) | |
5 | ocvfval.o | . . . 4 β’ β₯ = (ocvβπ) | |
6 | 1, 2, 3, 4, 5 | elocv 21095 | . . 3 β’ (π΄ β ( β₯ βπ) β (π β π β§ π΄ β π β§ βπ₯ β π (π΄ , π₯) = 0 )) |
7 | 6 | simp3bi 1148 | . 2 β’ (π΄ β ( β₯ βπ) β βπ₯ β π (π΄ , π₯) = 0 ) |
8 | oveq2 7369 | . . . 4 β’ (π₯ = π΅ β (π΄ , π₯) = (π΄ , π΅)) | |
9 | 8 | eqeq1d 2735 | . . 3 β’ (π₯ = π΅ β ((π΄ , π₯) = 0 β (π΄ , π΅) = 0 )) |
10 | 9 | rspccva 3582 | . 2 β’ ((βπ₯ β π (π΄ , π₯) = 0 β§ π΅ β π) β (π΄ , π΅) = 0 ) |
11 | 7, 10 | sylan 581 | 1 β’ ((π΄ β ( β₯ βπ) β§ π΅ β π) β (π΄ , π΅) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 β wss 3914 βcfv 6500 (class class class)co 7361 Basecbs 17091 Scalarcsca 17144 Β·πcip 17146 0gc0g 17329 ocvcocv 21087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 df-ocv 21090 |
This theorem is referenced by: ocvocv 21098 ocvlss 21099 ocvin 21101 lsmcss 21119 clsocv 24637 |
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