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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia1dim | Structured version Visualization version GIF version |
Description: Two expressions for the 1-dimensional subspaces of partial vector space A (when πΉ is a nonzero vector i.e. non-identity translation). Remark after Lemma L in [Crawley] p. 120 line 21. (Contributed by NM, 15-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.) |
Ref | Expression |
---|---|
dia1dim.h | β’ π» = (LHypβπΎ) |
dia1dim.t | β’ π = ((LTrnβπΎ)βπ) |
dia1dim.r | β’ π = ((trLβπΎ)βπ) |
dia1dim.e | β’ πΈ = ((TEndoβπΎ)βπ) |
dia1dim.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
Ref | Expression |
---|---|
dia1dim | β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πΌβ(π βπΉ)) = {π β£ βπ β πΈ π = (π βπΉ)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πΎ β HL β§ π β π»)) | |
2 | eqid 2726 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
3 | dia1dim.h | . . . 4 β’ π» = (LHypβπΎ) | |
4 | dia1dim.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
5 | dia1dim.r | . . . 4 β’ π = ((trLβπΎ)βπ) | |
6 | 2, 3, 4, 5 | trlcl 39546 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π βπΉ) β (BaseβπΎ)) |
7 | eqid 2726 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
8 | 7, 3, 4, 5 | trlle 39566 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π βπΉ)(leβπΎ)π) |
9 | dia1dim.i | . . . 4 β’ πΌ = ((DIsoAβπΎ)βπ) | |
10 | 2, 7, 3, 4, 5, 9 | diaval 40414 | . . 3 β’ (((πΎ β HL β§ π β π») β§ ((π βπΉ) β (BaseβπΎ) β§ (π βπΉ)(leβπΎ)π)) β (πΌβ(π βπΉ)) = {π β π β£ (π βπ)(leβπΎ)(π βπΉ)}) |
11 | 1, 6, 8, 10 | syl12anc 834 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πΌβ(π βπΉ)) = {π β π β£ (π βπ)(leβπΎ)(π βπΉ)}) |
12 | dia1dim.e | . . 3 β’ πΈ = ((TEndoβπΎ)βπ) | |
13 | 7, 3, 4, 5, 12 | dva1dim 40367 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β {π β£ βπ β πΈ π = (π βπΉ)} = {π β π β£ (π βπ)(leβπΎ)(π βπΉ)}) |
14 | 11, 13 | eqtr4d 2769 | 1 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πΌβ(π βπΉ)) = {π β£ βπ β πΈ π = (π βπΉ)}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {cab 2703 βwrex 3064 {crab 3426 class class class wbr 5141 βcfv 6536 Basecbs 17151 lecple 17211 HLchlt 38731 LHypclh 39366 LTrncltrn 39483 trLctrl 39540 TEndoctendo 40134 DIsoAcdia 40410 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-riotaBAD 38334 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-undef 8256 df-map 8821 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-oposet 38557 df-ol 38559 df-oml 38560 df-covers 38647 df-ats 38648 df-atl 38679 df-cvlat 38703 df-hlat 38732 df-llines 38880 df-lplanes 38881 df-lvols 38882 df-lines 38883 df-psubsp 38885 df-pmap 38886 df-padd 39178 df-lhyp 39370 df-laut 39371 df-ldil 39486 df-ltrn 39487 df-trl 39541 df-tendo 40137 df-disoa 40411 |
This theorem is referenced by: dia1dim2 40444 dib1dim 40547 |
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