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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem18 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 41231. Baer p. 45, line 7: "Then y =/= 0..." (Contributed by NM, 20-Mar-2015.) |
Ref | Expression |
---|---|
mapdpglem.h | β’ π» = (LHypβπΎ) |
mapdpglem.m | β’ π = ((mapdβπΎ)βπ) |
mapdpglem.u | β’ π = ((DVecHβπΎ)βπ) |
mapdpglem.v | β’ π = (Baseβπ) |
mapdpglem.s | β’ β = (-gβπ) |
mapdpglem.n | β’ π = (LSpanβπ) |
mapdpglem.c | β’ πΆ = ((LCDualβπΎ)βπ) |
mapdpglem.k | β’ (π β (πΎ β HL β§ π β π»)) |
mapdpglem.x | β’ (π β π β π) |
mapdpglem.y | β’ (π β π β π) |
mapdpglem1.p | β’ β = (LSSumβπΆ) |
mapdpglem2.j | β’ π½ = (LSpanβπΆ) |
mapdpglem3.f | β’ πΉ = (BaseβπΆ) |
mapdpglem3.te | β’ (π β π‘ β ((πβ(πβ{π})) β (πβ(πβ{π})))) |
mapdpglem3.a | β’ π΄ = (Scalarβπ) |
mapdpglem3.b | β’ π΅ = (Baseβπ΄) |
mapdpglem3.t | β’ Β· = ( Β·π βπΆ) |
mapdpglem3.r | β’ π = (-gβπΆ) |
mapdpglem3.g | β’ (π β πΊ β πΉ) |
mapdpglem3.e | β’ (π β (πβ(πβ{π})) = (π½β{πΊ})) |
mapdpglem4.q | β’ π = (0gβπ) |
mapdpglem.ne | β’ (π β (πβ{π}) β (πβ{π})) |
mapdpglem4.jt | β’ (π β (πβ(πβ{(π β π)})) = (π½β{π‘})) |
mapdpglem4.z | β’ 0 = (0gβπ΄) |
mapdpglem4.g4 | β’ (π β π β π΅) |
mapdpglem4.z4 | β’ (π β π§ β (πβ(πβ{π}))) |
mapdpglem4.t4 | β’ (π β π‘ = ((π Β· πΊ)π π§)) |
mapdpglem4.xn | β’ (π β π β π) |
mapdpglem12.yn | β’ (π β π β π) |
mapdpglem17.ep | β’ πΈ = (((invrβπ΄)βπ) Β· π§) |
Ref | Expression |
---|---|
mapdpglem18 | β’ (π β πΈ β (0gβπΆ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem.h | . . . . . . 7 β’ π» = (LHypβπΎ) | |
2 | mapdpglem.u | . . . . . . 7 β’ π = ((DVecHβπΎ)βπ) | |
3 | mapdpglem.k | . . . . . . 7 β’ (π β (πΎ β HL β§ π β π»)) | |
4 | 1, 2, 3 | dvhlvec 40634 | . . . . . 6 β’ (π β π β LVec) |
5 | mapdpglem3.a | . . . . . . 7 β’ π΄ = (Scalarβπ) | |
6 | 5 | lvecdrng 20989 | . . . . . 6 β’ (π β LVec β π΄ β DivRing) |
7 | 4, 6 | syl 17 | . . . . 5 β’ (π β π΄ β DivRing) |
8 | mapdpglem4.g4 | . . . . 5 β’ (π β π β π΅) | |
9 | mapdpglem.m | . . . . . 6 β’ π = ((mapdβπΎ)βπ) | |
10 | mapdpglem.v | . . . . . 6 β’ π = (Baseβπ) | |
11 | mapdpglem.s | . . . . . 6 β’ β = (-gβπ) | |
12 | mapdpglem.n | . . . . . 6 β’ π = (LSpanβπ) | |
13 | mapdpglem.c | . . . . . 6 β’ πΆ = ((LCDualβπΎ)βπ) | |
14 | mapdpglem.x | . . . . . 6 β’ (π β π β π) | |
15 | mapdpglem.y | . . . . . 6 β’ (π β π β π) | |
16 | mapdpglem1.p | . . . . . 6 β’ β = (LSSumβπΆ) | |
17 | mapdpglem2.j | . . . . . 6 β’ π½ = (LSpanβπΆ) | |
18 | mapdpglem3.f | . . . . . 6 β’ πΉ = (BaseβπΆ) | |
19 | mapdpglem3.te | . . . . . 6 β’ (π β π‘ β ((πβ(πβ{π})) β (πβ(πβ{π})))) | |
20 | mapdpglem3.b | . . . . . 6 β’ π΅ = (Baseβπ΄) | |
21 | mapdpglem3.t | . . . . . 6 β’ Β· = ( Β·π βπΆ) | |
22 | mapdpglem3.r | . . . . . 6 β’ π = (-gβπΆ) | |
23 | mapdpglem3.g | . . . . . 6 β’ (π β πΊ β πΉ) | |
24 | mapdpglem3.e | . . . . . 6 β’ (π β (πβ(πβ{π})) = (π½β{πΊ})) | |
25 | mapdpglem4.q | . . . . . 6 β’ π = (0gβπ) | |
26 | mapdpglem.ne | . . . . . 6 β’ (π β (πβ{π}) β (πβ{π})) | |
27 | mapdpglem4.jt | . . . . . 6 β’ (π β (πβ(πβ{(π β π)})) = (π½β{π‘})) | |
28 | mapdpglem4.z | . . . . . 6 β’ 0 = (0gβπ΄) | |
29 | mapdpglem4.z4 | . . . . . 6 β’ (π β π§ β (πβ(πβ{π}))) | |
30 | mapdpglem4.t4 | . . . . . 6 β’ (π β π‘ = ((π Β· πΊ)π π§)) | |
31 | mapdpglem4.xn | . . . . . 6 β’ (π β π β π) | |
32 | 1, 9, 2, 10, 11, 12, 13, 3, 14, 15, 16, 17, 18, 19, 5, 20, 21, 22, 23, 24, 25, 26, 27, 28, 8, 29, 30, 31 | mapdpglem11 41207 | . . . . 5 β’ (π β π β 0 ) |
33 | eqid 2725 | . . . . . 6 β’ (invrβπ΄) = (invrβπ΄) | |
34 | 20, 28, 33 | drnginvrn0 20646 | . . . . 5 β’ ((π΄ β DivRing β§ π β π΅ β§ π β 0 ) β ((invrβπ΄)βπ) β 0 ) |
35 | 7, 8, 32, 34 | syl3anc 1368 | . . . 4 β’ (π β ((invrβπ΄)βπ) β 0 ) |
36 | eqid 2725 | . . . . 5 β’ (ScalarβπΆ) = (ScalarβπΆ) | |
37 | eqid 2725 | . . . . 5 β’ (0gβ(ScalarβπΆ)) = (0gβ(ScalarβπΆ)) | |
38 | 1, 2, 5, 28, 13, 36, 37, 3 | lcd0 41133 | . . . 4 β’ (π β (0gβ(ScalarβπΆ)) = 0 ) |
39 | 35, 38 | neeqtrrd 3005 | . . 3 β’ (π β ((invrβπ΄)βπ) β (0gβ(ScalarβπΆ))) |
40 | mapdpglem12.yn | . . . 4 β’ (π β π β π) | |
41 | 1, 9, 2, 10, 11, 12, 13, 3, 14, 15, 16, 17, 18, 19, 5, 20, 21, 22, 23, 24, 25, 26, 27, 28, 8, 29, 30, 31, 40 | mapdpglem16 41212 | . . 3 β’ (π β π§ β (0gβπΆ)) |
42 | eqid 2725 | . . . 4 β’ (Baseβ(ScalarβπΆ)) = (Baseβ(ScalarβπΆ)) | |
43 | eqid 2725 | . . . 4 β’ (0gβπΆ) = (0gβπΆ) | |
44 | 1, 13, 3 | lcdlvec 41116 | . . . 4 β’ (π β πΆ β LVec) |
45 | 20, 28, 33 | drnginvrcl 20645 | . . . . . 6 β’ ((π΄ β DivRing β§ π β π΅ β§ π β 0 ) β ((invrβπ΄)βπ) β π΅) |
46 | 7, 8, 32, 45 | syl3anc 1368 | . . . . 5 β’ (π β ((invrβπ΄)βπ) β π΅) |
47 | 1, 2, 5, 20, 13, 36, 42, 3 | lcdsbase 41125 | . . . . 5 β’ (π β (Baseβ(ScalarβπΆ)) = π΅) |
48 | 46, 47 | eleqtrrd 2828 | . . . 4 β’ (π β ((invrβπ΄)βπ) β (Baseβ(ScalarβπΆ))) |
49 | eqid 2725 | . . . . . . 7 β’ (LSubSpβπ) = (LSubSpβπ) | |
50 | eqid 2725 | . . . . . . 7 β’ (LSubSpβπΆ) = (LSubSpβπΆ) | |
51 | 1, 2, 3 | dvhlmod 40635 | . . . . . . . 8 β’ (π β π β LMod) |
52 | 10, 49, 12 | lspsncl 20860 | . . . . . . . 8 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
53 | 51, 15, 52 | syl2anc 582 | . . . . . . 7 β’ (π β (πβ{π}) β (LSubSpβπ)) |
54 | 1, 9, 2, 49, 13, 50, 3, 53 | mapdcl2 41181 | . . . . . 6 β’ (π β (πβ(πβ{π})) β (LSubSpβπΆ)) |
55 | 18, 50 | lssss 20819 | . . . . . 6 β’ ((πβ(πβ{π})) β (LSubSpβπΆ) β (πβ(πβ{π})) β πΉ) |
56 | 54, 55 | syl 17 | . . . . 5 β’ (π β (πβ(πβ{π})) β πΉ) |
57 | 56, 29 | sseldd 3974 | . . . 4 β’ (π β π§ β πΉ) |
58 | 18, 21, 36, 42, 37, 43, 44, 48, 57 | lvecvsn0 20996 | . . 3 β’ (π β ((((invrβπ΄)βπ) Β· π§) β (0gβπΆ) β (((invrβπ΄)βπ) β (0gβ(ScalarβπΆ)) β§ π§ β (0gβπΆ)))) |
59 | 39, 41, 58 | mpbir2and 711 | . 2 β’ (π β (((invrβπ΄)βπ) Β· π§) β (0gβπΆ)) |
60 | mapdpglem17.ep | . 2 β’ πΈ = (((invrβπ΄)βπ) Β· π§) | |
61 | 59, 60, 43 | 3netr4g 3010 | 1 β’ (π β πΈ β (0gβπΆ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2930 β wss 3941 {csn 4625 βcfv 6543 (class class class)co 7413 Basecbs 17174 Scalarcsca 17230 Β·π cvsca 17231 0gc0g 17415 -gcsg 18891 LSSumclsm 19588 invrcinvr 20325 DivRingcdr 20623 LModclmod 20742 LSubSpclss 20814 LSpanclspn 20854 LVecclvec 20986 HLchlt 38874 LHypclh 39509 DVecHcdvh 40603 LCDualclcd 41111 mapdcmpd 41149 |
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 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-riotaBAD 38477 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7866 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-0g 17417 df-mre 17560 df-mrc 17561 df-acs 17563 df-proset 18281 df-poset 18299 df-plt 18316 df-lub 18332 df-glb 18333 df-join 18334 df-meet 18335 df-p0 18411 df-p1 18412 df-lat 18418 df-clat 18485 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-cntz 19267 df-oppg 19296 df-lsm 19590 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-drng 20625 df-lmod 20744 df-lss 20815 df-lsp 20855 df-lvec 20987 df-lsatoms 38500 df-lshyp 38501 df-lcv 38543 df-lfl 38582 df-lkr 38610 df-ldual 38648 df-oposet 38700 df-ol 38702 df-oml 38703 df-covers 38790 df-ats 38791 df-atl 38822 df-cvlat 38846 df-hlat 38875 df-llines 39023 df-lplanes 39024 df-lvols 39025 df-lines 39026 df-psubsp 39028 df-pmap 39029 df-padd 39321 df-lhyp 39513 df-laut 39514 df-ldil 39629 df-ltrn 39630 df-trl 39684 df-tgrp 40268 df-tendo 40280 df-edring 40282 df-dveca 40528 df-disoa 40554 df-dvech 40604 df-dib 40664 df-dic 40698 df-dih 40754 df-doch 40873 df-djh 40920 df-lcdual 41112 df-mapd 41150 |
This theorem is referenced by: mapdpglem20 41216 |
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