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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 481 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πΎ β HL β§ π β π»)) | |
2 | eqid 2727 | . . . 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 39641 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π βπΉ) β (BaseβπΎ)) |
7 | eqid 2727 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
8 | 7, 3, 4, 5 | trlle 39661 | . . 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 40509 | . . 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 835 | . 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 40462 | . 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 2770 | 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 394 = wceq 1533 β wcel 2098 {cab 2704 βwrex 3066 {crab 3428 class class class wbr 5150 βcfv 6551 Basecbs 17185 lecple 17245 HLchlt 38826 LHypclh 39461 LTrncltrn 39578 trLctrl 39635 TEndoctendo 40229 DIsoAcdia 40505 |
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 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-riotaBAD 38429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 7997 df-2nd 7998 df-undef 8283 df-map 8851 df-proset 18292 df-poset 18310 df-plt 18327 df-lub 18343 df-glb 18344 df-join 18345 df-meet 18346 df-p0 18422 df-p1 18423 df-lat 18429 df-clat 18496 df-oposet 38652 df-ol 38654 df-oml 38655 df-covers 38742 df-ats 38743 df-atl 38774 df-cvlat 38798 df-hlat 38827 df-llines 38975 df-lplanes 38976 df-lvols 38977 df-lines 38978 df-psubsp 38980 df-pmap 38981 df-padd 39273 df-lhyp 39465 df-laut 39466 df-ldil 39581 df-ltrn 39582 df-trl 39636 df-tendo 40232 df-disoa 40506 |
This theorem is referenced by: dia1dim2 40539 dib1dim 40642 |
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