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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia2dimlem4 | Structured version Visualization version GIF version |
Description: Lemma for dia2dim 40474. Show that the composition (sum) of translations (vectors) πΊ and π· equals πΉ. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.) |
Ref | Expression |
---|---|
dia2dimlem4.l | β’ β€ = (leβπΎ) |
dia2dimlem4.a | β’ π΄ = (AtomsβπΎ) |
dia2dimlem4.h | β’ π» = (LHypβπΎ) |
dia2dimlem4.t | β’ π = ((LTrnβπΎ)βπ) |
dia2dimlem4.k | β’ (π β (πΎ β HL β§ π β π»)) |
dia2dimlem4.p | β’ (π β (π β π΄ β§ Β¬ π β€ π)) |
dia2dimlem4.f | β’ (π β πΉ β π) |
dia2dimlem4.g | β’ (π β πΊ β π) |
dia2dimlem4.gv | β’ (π β (πΊβπ) = π) |
dia2dimlem4.d | β’ (π β π· β π) |
dia2dimlem4.dv | β’ (π β (π·βπ) = (πΉβπ)) |
Ref | Expression |
---|---|
dia2dimlem4 | β’ (π β (π· β πΊ) = πΉ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dia2dimlem4.k | . 2 β’ (π β (πΎ β HL β§ π β π»)) | |
2 | dia2dimlem4.d | . . 3 β’ (π β π· β π) | |
3 | dia2dimlem4.g | . . 3 β’ (π β πΊ β π) | |
4 | dia2dimlem4.h | . . . 4 β’ π» = (LHypβπΎ) | |
5 | dia2dimlem4.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
6 | 4, 5 | ltrnco 40116 | . . 3 β’ (((πΎ β HL β§ π β π») β§ π· β π β§ πΊ β π) β (π· β πΊ) β π) |
7 | 1, 2, 3, 6 | syl3anc 1369 | . 2 β’ (π β (π· β πΊ) β π) |
8 | dia2dimlem4.f | . 2 β’ (π β πΉ β π) | |
9 | dia2dimlem4.p | . 2 β’ (π β (π β π΄ β§ Β¬ π β€ π)) | |
10 | 9 | simpld 494 | . . . 4 β’ (π β π β π΄) |
11 | dia2dimlem4.l | . . . . 5 β’ β€ = (leβπΎ) | |
12 | dia2dimlem4.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
13 | 11, 12, 4, 5 | ltrncoval 39542 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π· β π β§ πΊ β π) β§ π β π΄) β ((π· β πΊ)βπ) = (π·β(πΊβπ))) |
14 | 1, 2, 3, 10, 13 | syl121anc 1373 | . . 3 β’ (π β ((π· β πΊ)βπ) = (π·β(πΊβπ))) |
15 | dia2dimlem4.gv | . . . 4 β’ (π β (πΊβπ) = π) | |
16 | 15 | fveq2d 6895 | . . 3 β’ (π β (π·β(πΊβπ)) = (π·βπ)) |
17 | dia2dimlem4.dv | . . 3 β’ (π β (π·βπ) = (πΉβπ)) | |
18 | 14, 16, 17 | 3eqtrd 2771 | . 2 β’ (π β ((π· β πΊ)βπ) = (πΉβπ)) |
19 | 11, 12, 4, 5 | cdlemd 39604 | . 2 β’ ((((πΎ β HL β§ π β π») β§ (π· β πΊ) β π β§ πΉ β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ ((π· β πΊ)βπ) = (πΉβπ)) β (π· β πΊ) = πΉ) |
20 | 1, 7, 8, 9, 18, 19 | syl311anc 1382 | 1 β’ (π β (π· β πΊ) = πΉ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 class class class wbr 5142 β ccom 5676 βcfv 6542 lecple 17225 Atomscatm 38659 HLchlt 38746 LHypclh 39381 LTrncltrn 39498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-riotaBAD 38349 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7985 df-2nd 7986 df-undef 8270 df-map 8836 df-proset 18272 df-poset 18290 df-plt 18307 df-lub 18323 df-glb 18324 df-join 18325 df-meet 18326 df-p0 18402 df-p1 18403 df-lat 18409 df-clat 18476 df-oposet 38572 df-ol 38574 df-oml 38575 df-covers 38662 df-ats 38663 df-atl 38694 df-cvlat 38718 df-hlat 38747 df-llines 38895 df-lplanes 38896 df-lvols 38897 df-lines 38898 df-psubsp 38900 df-pmap 38901 df-padd 39193 df-lhyp 39385 df-laut 39386 df-ldil 39501 df-ltrn 39502 df-trl 39556 |
This theorem is referenced by: dia2dimlem5 40465 |
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