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Mirrors > Home > MPE Home > Th. List > obsipid | Structured version Visualization version GIF version |
Description: A basis element has length one. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
obsipid.h | β’ , = (Β·πβπ) |
obsipid.f | β’ πΉ = (Scalarβπ) |
obsipid.u | β’ 1 = (1rβπΉ) |
Ref | Expression |
---|---|
obsipid | β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β (π΄ , π΄) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | obsipid.h | . . . 4 β’ , = (Β·πβπ) | |
3 | obsipid.f | . . . 4 β’ πΉ = (Scalarβπ) | |
4 | obsipid.u | . . . 4 β’ 1 = (1rβπΉ) | |
5 | eqid 2733 | . . . 4 β’ (0gβπΉ) = (0gβπΉ) | |
6 | 1, 2, 3, 4, 5 | obsip 21143 | . . 3 β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅ β§ π΄ β π΅) β (π΄ , π΄) = if(π΄ = π΄, 1 , (0gβπΉ))) |
7 | 6 | 3anidm23 1422 | . 2 β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β (π΄ , π΄) = if(π΄ = π΄, 1 , (0gβπΉ))) |
8 | eqid 2733 | . . 3 β’ π΄ = π΄ | |
9 | 8 | iftruei 4494 | . 2 β’ if(π΄ = π΄, 1 , (0gβπΉ)) = 1 |
10 | 7, 9 | eqtrdi 2789 | 1 β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β (π΄ , π΄) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 ifcif 4487 βcfv 6497 (class class class)co 7358 Basecbs 17088 Scalarcsca 17141 Β·πcip 17143 0gc0g 17326 1rcur 19918 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: obsne0 21147 |
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