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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem11 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 40572. (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) |
Ref | Expression |
---|---|
mapdpglem11 | β’ (π β π β 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem.ne | . 2 β’ (π β (πβ{π}) β (πβ{π})) | |
2 | mapdpglem.h | . . . . 5 β’ π» = (LHypβπΎ) | |
3 | mapdpglem.m | . . . . 5 β’ π = ((mapdβπΎ)βπ) | |
4 | mapdpglem.u | . . . . 5 β’ π = ((DVecHβπΎ)βπ) | |
5 | mapdpglem.v | . . . . 5 β’ π = (Baseβπ) | |
6 | mapdpglem.s | . . . . 5 β’ β = (-gβπ) | |
7 | mapdpglem.n | . . . . 5 β’ π = (LSpanβπ) | |
8 | mapdpglem.c | . . . . 5 β’ πΆ = ((LCDualβπΎ)βπ) | |
9 | mapdpglem.k | . . . . . 6 β’ (π β (πΎ β HL β§ π β π»)) | |
10 | 9 | adantr 481 | . . . . 5 β’ ((π β§ π = 0 ) β (πΎ β HL β§ π β π»)) |
11 | mapdpglem.x | . . . . . 6 β’ (π β π β π) | |
12 | 11 | adantr 481 | . . . . 5 β’ ((π β§ π = 0 ) β π β π) |
13 | mapdpglem.y | . . . . . 6 β’ (π β π β π) | |
14 | 13 | adantr 481 | . . . . 5 β’ ((π β§ π = 0 ) β π β π) |
15 | mapdpglem1.p | . . . . 5 β’ β = (LSSumβπΆ) | |
16 | mapdpglem2.j | . . . . 5 β’ π½ = (LSpanβπΆ) | |
17 | mapdpglem3.f | . . . . 5 β’ πΉ = (BaseβπΆ) | |
18 | mapdpglem3.te | . . . . . 6 β’ (π β π‘ β ((πβ(πβ{π})) β (πβ(πβ{π})))) | |
19 | 18 | adantr 481 | . . . . 5 β’ ((π β§ π = 0 ) β π‘ β ((πβ(πβ{π})) β (πβ(πβ{π})))) |
20 | mapdpglem3.a | . . . . 5 β’ π΄ = (Scalarβπ) | |
21 | mapdpglem3.b | . . . . 5 β’ π΅ = (Baseβπ΄) | |
22 | mapdpglem3.t | . . . . 5 β’ Β· = ( Β·π βπΆ) | |
23 | mapdpglem3.r | . . . . 5 β’ π = (-gβπΆ) | |
24 | mapdpglem3.g | . . . . . 6 β’ (π β πΊ β πΉ) | |
25 | 24 | adantr 481 | . . . . 5 β’ ((π β§ π = 0 ) β πΊ β πΉ) |
26 | mapdpglem3.e | . . . . . 6 β’ (π β (πβ(πβ{π})) = (π½β{πΊ})) | |
27 | 26 | adantr 481 | . . . . 5 β’ ((π β§ π = 0 ) β (πβ(πβ{π})) = (π½β{πΊ})) |
28 | mapdpglem4.q | . . . . 5 β’ π = (0gβπ) | |
29 | 1 | adantr 481 | . . . . 5 β’ ((π β§ π = 0 ) β (πβ{π}) β (πβ{π})) |
30 | mapdpglem4.jt | . . . . . 6 β’ (π β (πβ(πβ{(π β π)})) = (π½β{π‘})) | |
31 | 30 | adantr 481 | . . . . 5 β’ ((π β§ π = 0 ) β (πβ(πβ{(π β π)})) = (π½β{π‘})) |
32 | mapdpglem4.z | . . . . 5 β’ 0 = (0gβπ΄) | |
33 | mapdpglem4.g4 | . . . . . 6 β’ (π β π β π΅) | |
34 | 33 | adantr 481 | . . . . 5 β’ ((π β§ π = 0 ) β π β π΅) |
35 | mapdpglem4.z4 | . . . . . 6 β’ (π β π§ β (πβ(πβ{π}))) | |
36 | 35 | adantr 481 | . . . . 5 β’ ((π β§ π = 0 ) β π§ β (πβ(πβ{π}))) |
37 | mapdpglem4.t4 | . . . . . 6 β’ (π β π‘ = ((π Β· πΊ)π π§)) | |
38 | 37 | adantr 481 | . . . . 5 β’ ((π β§ π = 0 ) β π‘ = ((π Β· πΊ)π π§)) |
39 | mapdpglem4.xn | . . . . . 6 β’ (π β π β π) | |
40 | 39 | adantr 481 | . . . . 5 β’ ((π β§ π = 0 ) β π β π) |
41 | simpr 485 | . . . . 5 β’ ((π β§ π = 0 ) β π = 0 ) | |
42 | 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 27, 28, 29, 31, 32, 34, 36, 38, 40, 41 | mapdpglem10 40547 | . . . 4 β’ ((π β§ π = 0 ) β (πβ{π}) = (πβ{π})) |
43 | 42 | ex 413 | . . 3 β’ (π β (π = 0 β (πβ{π}) = (πβ{π}))) |
44 | 43 | necon3d 2961 | . 2 β’ (π β ((πβ{π}) β (πβ{π}) β π β 0 )) |
45 | 1, 44 | mpd 15 | 1 β’ (π β π β 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β wne 2940 {csn 4628 βcfv 6543 (class class class)co 7408 Basecbs 17143 Scalarcsca 17199 Β·π cvsca 17200 0gc0g 17384 -gcsg 18820 LSSumclsm 19501 LSpanclspn 20581 HLchlt 38215 LHypclh 38850 DVecHcdvh 39944 LCDualclcd 40452 mapdcmpd 40490 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-riotaBAD 37818 |
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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-tpos 8210 df-undef 8257 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-0g 17386 df-mre 17529 df-mrc 17530 df-acs 17532 df-proset 18247 df-poset 18265 df-plt 18282 df-lub 18298 df-glb 18299 df-join 18300 df-meet 18301 df-p0 18377 df-p1 18378 df-lat 18384 df-clat 18451 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-submnd 18671 df-grp 18821 df-minusg 18822 df-sbg 18823 df-subg 19002 df-cntz 19180 df-oppg 19209 df-lsm 19503 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-oppr 20149 df-dvdsr 20170 df-unit 20171 df-invr 20201 df-dvr 20214 df-drng 20358 df-lmod 20472 df-lss 20542 df-lsp 20582 df-lvec 20713 df-lsatoms 37841 df-lshyp 37842 df-lcv 37884 df-lfl 37923 df-lkr 37951 df-ldual 37989 df-oposet 38041 df-ol 38043 df-oml 38044 df-covers 38131 df-ats 38132 df-atl 38163 df-cvlat 38187 df-hlat 38216 df-llines 38364 df-lplanes 38365 df-lvols 38366 df-lines 38367 df-psubsp 38369 df-pmap 38370 df-padd 38662 df-lhyp 38854 df-laut 38855 df-ldil 38970 df-ltrn 38971 df-trl 39025 df-tgrp 39609 df-tendo 39621 df-edring 39623 df-dveca 39869 df-disoa 39895 df-dvech 39945 df-dib 40005 df-dic 40039 df-dih 40095 df-doch 40214 df-djh 40261 df-lcdual 40453 df-mapd 40491 |
This theorem is referenced by: mapdpglem17N 40554 mapdpglem18 40555 mapdpglem19 40556 mapdpglem21 40558 mapdpglem22 40559 |
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