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Mirrors > Home > MPE Home > Th. List > obsocv | Structured version Visualization version GIF version |
Description: An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
obsocv.z | β’ 0 = (0gβπ) |
obsocv.o | β’ β₯ = (ocvβπ) |
Ref | Expression |
---|---|
obsocv | β’ (π΅ β (OBasisβπ) β ( β₯ βπ΅) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | eqid 2733 | . . . 4 β’ (Β·πβπ) = (Β·πβπ) | |
3 | eqid 2733 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
4 | eqid 2733 | . . . 4 β’ (1rβ(Scalarβπ)) = (1rβ(Scalarβπ)) | |
5 | eqid 2733 | . . . 4 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
6 | obsocv.o | . . . 4 β’ β₯ = (ocvβπ) | |
7 | obsocv.z | . . . 4 β’ 0 = (0gβπ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isobs 21142 | . . 3 β’ (π΅ β (OBasisβπ) β (π β PreHil β§ π΅ β (Baseβπ) β§ (βπ₯ β π΅ βπ¦ β π΅ (π₯(Β·πβπ)π¦) = if(π₯ = π¦, (1rβ(Scalarβπ)), (0gβ(Scalarβπ))) β§ ( β₯ βπ΅) = { 0 }))) |
9 | 8 | simp3bi 1148 | . 2 β’ (π΅ β (OBasisβπ) β (βπ₯ β π΅ βπ¦ β π΅ (π₯(Β·πβπ)π¦) = if(π₯ = π¦, (1rβ(Scalarβπ)), (0gβ(Scalarβπ))) β§ ( β₯ βπ΅) = { 0 })) |
10 | 9 | simprd 497 | 1 β’ (π΅ β (OBasisβπ) β ( β₯ βπ΅) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 β wss 3911 ifcif 4487 {csn 4587 βcfv 6497 (class class class)co 7358 Basecbs 17088 Scalarcsca 17141 Β·πcip 17143 0gc0g 17326 1rcur 19918 PreHilcphl 21044 ocvcocv 21080 OBasiscobs 21124 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 |
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 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-obs 21127 |
This theorem is referenced by: obs2ocv 21149 obs2ss 21151 |
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