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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia2dimlem4 | Structured version Visualization version GIF version |
Description: Lemma for dia2dim 39543. 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 39185 | . . 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 1372 | . 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 496 | . . . 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 38611 | . . . 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 1376 | . . 3 β’ (π β ((π· β πΊ)βπ) = (π·β(πΊβπ))) |
15 | dia2dimlem4.gv | . . . 4 β’ (π β (πΊβπ) = π) | |
16 | 15 | fveq2d 6847 | . . 3 β’ (π β (π·β(πΊβπ)) = (π·βπ)) |
17 | dia2dimlem4.dv | . . 3 β’ (π β (π·βπ) = (πΉβπ)) | |
18 | 14, 16, 17 | 3eqtrd 2781 | . 2 β’ (π β ((π· β πΊ)βπ) = (πΉβπ)) |
19 | 11, 12, 4, 5 | cdlemd 38673 | . 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 1385 | 1 β’ (π β (π· β πΊ) = πΉ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 class class class wbr 5106 β ccom 5638 βcfv 6497 lecple 17141 Atomscatm 37728 HLchlt 37815 LHypclh 38450 LTrncltrn 38567 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-riotaBAD 37418 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 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-iun 4957 df-iin 4958 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-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-undef 8205 df-map 8768 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-clat 18389 df-oposet 37641 df-ol 37643 df-oml 37644 df-covers 37731 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-llines 37964 df-lplanes 37965 df-lvols 37966 df-lines 37967 df-psubsp 37969 df-pmap 37970 df-padd 38262 df-lhyp 38454 df-laut 38455 df-ldil 38570 df-ltrn 38571 df-trl 38625 |
This theorem is referenced by: dia2dimlem5 39534 |
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