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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 40565. 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 39968 | . . . . . 6 β’ (π β π β LVec) |
5 | mapdpglem3.a | . . . . . . 7 β’ π΄ = (Scalarβπ) | |
6 | 5 | lvecdrng 20708 | . . . . . 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 40541 | . . . . 5 β’ (π β π β 0 ) |
33 | eqid 2732 | . . . . . 6 β’ (invrβπ΄) = (invrβπ΄) | |
34 | 20, 28, 33 | drnginvrn0 20330 | . . . . 5 β’ ((π΄ β DivRing β§ π β π΅ β§ π β 0 ) β ((invrβπ΄)βπ) β 0 ) |
35 | 7, 8, 32, 34 | syl3anc 1371 | . . . 4 β’ (π β ((invrβπ΄)βπ) β 0 ) |
36 | eqid 2732 | . . . . 5 β’ (ScalarβπΆ) = (ScalarβπΆ) | |
37 | eqid 2732 | . . . . 5 β’ (0gβ(ScalarβπΆ)) = (0gβ(ScalarβπΆ)) | |
38 | 1, 2, 5, 28, 13, 36, 37, 3 | lcd0 40467 | . . . 4 β’ (π β (0gβ(ScalarβπΆ)) = 0 ) |
39 | 35, 38 | neeqtrrd 3015 | . . 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 40546 | . . 3 β’ (π β π§ β (0gβπΆ)) |
42 | eqid 2732 | . . . 4 β’ (Baseβ(ScalarβπΆ)) = (Baseβ(ScalarβπΆ)) | |
43 | eqid 2732 | . . . 4 β’ (0gβπΆ) = (0gβπΆ) | |
44 | 1, 13, 3 | lcdlvec 40450 | . . . 4 β’ (π β πΆ β LVec) |
45 | 20, 28, 33 | drnginvrcl 20329 | . . . . . 6 β’ ((π΄ β DivRing β§ π β π΅ β§ π β 0 ) β ((invrβπ΄)βπ) β π΅) |
46 | 7, 8, 32, 45 | syl3anc 1371 | . . . . 5 β’ (π β ((invrβπ΄)βπ) β π΅) |
47 | 1, 2, 5, 20, 13, 36, 42, 3 | lcdsbase 40459 | . . . . 5 β’ (π β (Baseβ(ScalarβπΆ)) = π΅) |
48 | 46, 47 | eleqtrrd 2836 | . . . 4 β’ (π β ((invrβπ΄)βπ) β (Baseβ(ScalarβπΆ))) |
49 | eqid 2732 | . . . . . . 7 β’ (LSubSpβπ) = (LSubSpβπ) | |
50 | eqid 2732 | . . . . . . 7 β’ (LSubSpβπΆ) = (LSubSpβπΆ) | |
51 | 1, 2, 3 | dvhlmod 39969 | . . . . . . . 8 β’ (π β π β LMod) |
52 | 10, 49, 12 | lspsncl 20580 | . . . . . . . 8 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
53 | 51, 15, 52 | syl2anc 584 | . . . . . . 7 β’ (π β (πβ{π}) β (LSubSpβπ)) |
54 | 1, 9, 2, 49, 13, 50, 3, 53 | mapdcl2 40515 | . . . . . 6 β’ (π β (πβ(πβ{π})) β (LSubSpβπΆ)) |
55 | 18, 50 | lssss 20539 | . . . . . 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 3982 | . . . 4 β’ (π β π§ β πΉ) |
58 | 18, 21, 36, 42, 37, 43, 44, 48, 57 | lvecvsn0 20714 | . . 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 3020 | 1 β’ (π β πΈ β (0gβπΆ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β wne 2940 β wss 3947 {csn 4627 βcfv 6540 (class class class)co 7405 Basecbs 17140 Scalarcsca 17196 Β·π cvsca 17197 0gc0g 17381 -gcsg 18817 LSSumclsm 19496 invrcinvr 20193 DivRingcdr 20307 LModclmod 20463 LSubSpclss 20534 LSpanclspn 20574 LVecclvec 20705 HLchlt 38208 LHypclh 38843 DVecHcdvh 39937 LCDualclcd 40445 mapdcmpd 40483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-riotaBAD 37811 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-0g 17383 df-mre 17526 df-mrc 17527 df-acs 17529 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-cntz 19175 df-oppg 19204 df-lsm 19498 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-drng 20309 df-lmod 20465 df-lss 20535 df-lsp 20575 df-lvec 20706 df-lsatoms 37834 df-lshyp 37835 df-lcv 37877 df-lfl 37916 df-lkr 37944 df-ldual 37982 df-oposet 38034 df-ol 38036 df-oml 38037 df-covers 38124 df-ats 38125 df-atl 38156 df-cvlat 38180 df-hlat 38209 df-llines 38357 df-lplanes 38358 df-lvols 38359 df-lines 38360 df-psubsp 38362 df-pmap 38363 df-padd 38655 df-lhyp 38847 df-laut 38848 df-ldil 38963 df-ltrn 38964 df-trl 39018 df-tgrp 39602 df-tendo 39614 df-edring 39616 df-dveca 39862 df-disoa 39888 df-dvech 39938 df-dib 39998 df-dic 40032 df-dih 40088 df-doch 40207 df-djh 40254 df-lcdual 40446 df-mapd 40484 |
This theorem is referenced by: mapdpglem20 40550 |
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