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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem14 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 40219. (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) |
mapdpglem12.g0 | β’ (π β π§ = (0gβπΆ)) |
Ref | Expression |
---|---|
mapdpglem14 | β’ (π β π β (πβ{π})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem.h | . . . 4 β’ π» = (LHypβπΎ) | |
2 | mapdpglem.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
3 | mapdpglem.k | . . . 4 β’ (π β (πΎ β HL β§ π β π»)) | |
4 | 1, 2, 3 | dvhlmod 39623 | . . 3 β’ (π β π β LMod) |
5 | mapdpglem.y | . . 3 β’ (π β π β π) | |
6 | mapdpglem.x | . . 3 β’ (π β π β π) | |
7 | mapdpglem.v | . . . 4 β’ π = (Baseβπ) | |
8 | eqid 2733 | . . . 4 β’ (+gβπ) = (+gβπ) | |
9 | mapdpglem.s | . . . 4 β’ β = (-gβπ) | |
10 | 7, 8, 9 | lmodvnpcan 20420 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β ((π β π)(+gβπ)π) = π) |
11 | 4, 5, 6, 10 | syl3anc 1372 | . 2 β’ (π β ((π β π)(+gβπ)π) = π) |
12 | eqid 2733 | . . . . 5 β’ (LSubSpβπ) = (LSubSpβπ) | |
13 | mapdpglem.n | . . . . 5 β’ π = (LSpanβπ) | |
14 | 7, 12, 13 | lspsncl 20482 | . . . 4 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
15 | 4, 6, 14 | syl2anc 585 | . . 3 β’ (π β (πβ{π}) β (LSubSpβπ)) |
16 | lmodgrp 20372 | . . . . . 6 β’ (π β LMod β π β Grp) | |
17 | 4, 16 | syl 17 | . . . . 5 β’ (π β π β Grp) |
18 | eqid 2733 | . . . . . 6 β’ (invgβπ) = (invgβπ) | |
19 | 7, 9, 18 | grpinvsub 18837 | . . . . 5 β’ ((π β Grp β§ π β π β§ π β π) β ((invgβπ)β(π β π)) = (π β π)) |
20 | 17, 6, 5, 19 | syl3anc 1372 | . . . 4 β’ (π β ((invgβπ)β(π β π)) = (π β π)) |
21 | mapdpglem.m | . . . . . . 7 β’ π = ((mapdβπΎ)βπ) | |
22 | mapdpglem.c | . . . . . . 7 β’ πΆ = ((LCDualβπΎ)βπ) | |
23 | mapdpglem1.p | . . . . . . 7 β’ β = (LSSumβπΆ) | |
24 | mapdpglem2.j | . . . . . . 7 β’ π½ = (LSpanβπΆ) | |
25 | mapdpglem3.f | . . . . . . 7 β’ πΉ = (BaseβπΆ) | |
26 | mapdpglem3.te | . . . . . . 7 β’ (π β π‘ β ((πβ(πβ{π})) β (πβ(πβ{π})))) | |
27 | mapdpglem3.a | . . . . . . 7 β’ π΄ = (Scalarβπ) | |
28 | mapdpglem3.b | . . . . . . 7 β’ π΅ = (Baseβπ΄) | |
29 | mapdpglem3.t | . . . . . . 7 β’ Β· = ( Β·π βπΆ) | |
30 | mapdpglem3.r | . . . . . . 7 β’ π = (-gβπΆ) | |
31 | mapdpglem3.g | . . . . . . 7 β’ (π β πΊ β πΉ) | |
32 | mapdpglem3.e | . . . . . . 7 β’ (π β (πβ(πβ{π})) = (π½β{πΊ})) | |
33 | mapdpglem4.q | . . . . . . 7 β’ π = (0gβπ) | |
34 | mapdpglem.ne | . . . . . . 7 β’ (π β (πβ{π}) β (πβ{π})) | |
35 | mapdpglem4.jt | . . . . . . 7 β’ (π β (πβ(πβ{(π β π)})) = (π½β{π‘})) | |
36 | mapdpglem4.z | . . . . . . 7 β’ 0 = (0gβπ΄) | |
37 | mapdpglem4.g4 | . . . . . . 7 β’ (π β π β π΅) | |
38 | mapdpglem4.z4 | . . . . . . 7 β’ (π β π§ β (πβ(πβ{π}))) | |
39 | mapdpglem4.t4 | . . . . . . 7 β’ (π β π‘ = ((π Β· πΊ)π π§)) | |
40 | mapdpglem4.xn | . . . . . . 7 β’ (π β π β π) | |
41 | mapdpglem12.yn | . . . . . . 7 β’ (π β π β π) | |
42 | mapdpglem12.g0 | . . . . . . 7 β’ (π β π§ = (0gβπΆ)) | |
43 | 1, 21, 2, 7, 9, 13, 22, 3, 6, 5, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42 | mapdpglem13 40197 | . . . . . 6 β’ (π β (πβ{(π β π)}) β (πβ{π})) |
44 | 7, 9 | lmodvsubcl 20411 | . . . . . . . 8 β’ ((π β LMod β§ π β π β§ π β π) β (π β π) β π) |
45 | 4, 6, 5, 44 | syl3anc 1372 | . . . . . . 7 β’ (π β (π β π) β π) |
46 | 7, 13 | lspsnid 20498 | . . . . . . 7 β’ ((π β LMod β§ (π β π) β π) β (π β π) β (πβ{(π β π)})) |
47 | 4, 45, 46 | syl2anc 585 | . . . . . 6 β’ (π β (π β π) β (πβ{(π β π)})) |
48 | 43, 47 | sseldd 3949 | . . . . 5 β’ (π β (π β π) β (πβ{π})) |
49 | 12, 18 | lssvnegcl 20461 | . . . . 5 β’ ((π β LMod β§ (πβ{π}) β (LSubSpβπ) β§ (π β π) β (πβ{π})) β ((invgβπ)β(π β π)) β (πβ{π})) |
50 | 4, 15, 48, 49 | syl3anc 1372 | . . . 4 β’ (π β ((invgβπ)β(π β π)) β (πβ{π})) |
51 | 20, 50 | eqeltrrd 2835 | . . 3 β’ (π β (π β π) β (πβ{π})) |
52 | 7, 13 | lspsnid 20498 | . . . 4 β’ ((π β LMod β§ π β π) β π β (πβ{π})) |
53 | 4, 6, 52 | syl2anc 585 | . . 3 β’ (π β π β (πβ{π})) |
54 | 8, 12 | lssvacl 20459 | . . 3 β’ (((π β LMod β§ (πβ{π}) β (LSubSpβπ)) β§ ((π β π) β (πβ{π}) β§ π β (πβ{π}))) β ((π β π)(+gβπ)π) β (πβ{π})) |
55 | 4, 15, 51, 53, 54 | syl22anc 838 | . 2 β’ (π β ((π β π)(+gβπ)π) β (πβ{π})) |
56 | 11, 55 | eqeltrrd 2835 | 1 β’ (π β π β (πβ{π})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 {csn 4590 βcfv 6500 (class class class)co 7361 Basecbs 17091 +gcplusg 17141 Scalarcsca 17144 Β·π cvsca 17145 0gc0g 17329 Grpcgrp 18756 invgcminusg 18757 -gcsg 18758 LSSumclsm 19424 LModclmod 20365 LSubSpclss 20436 LSpanclspn 20476 HLchlt 37862 LHypclh 38497 DVecHcdvh 39591 LCDualclcd 40099 mapdcmpd 40137 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-riotaBAD 37465 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-tpos 8161 df-undef 8208 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-0g 17331 df-mre 17474 df-mrc 17475 df-acs 17477 df-proset 18192 df-poset 18210 df-plt 18227 df-lub 18243 df-glb 18244 df-join 18245 df-meet 18246 df-p0 18322 df-p1 18323 df-lat 18329 df-clat 18396 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-grp 18759 df-minusg 18760 df-sbg 18761 df-subg 18933 df-cntz 19105 df-oppg 19132 df-lsm 19426 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-dvr 20120 df-drng 20221 df-lmod 20367 df-lss 20437 df-lsp 20477 df-lvec 20608 df-lsatoms 37488 df-lshyp 37489 df-lcv 37531 df-lfl 37570 df-lkr 37598 df-ldual 37636 df-oposet 37688 df-ol 37690 df-oml 37691 df-covers 37778 df-ats 37779 df-atl 37810 df-cvlat 37834 df-hlat 37863 df-llines 38011 df-lplanes 38012 df-lvols 38013 df-lines 38014 df-psubsp 38016 df-pmap 38017 df-padd 38309 df-lhyp 38501 df-laut 38502 df-ldil 38617 df-ltrn 38618 df-trl 38672 df-tgrp 39256 df-tendo 39268 df-edring 39270 df-dveca 39516 df-disoa 39542 df-dvech 39592 df-dib 39652 df-dic 39686 df-dih 39742 df-doch 39861 df-djh 39908 df-lcdual 40100 df-mapd 40138 |
This theorem is referenced by: mapdpglem15 40199 |
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