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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap11 | Structured version Visualization version GIF version |
Description: Part of proof of part 12 in [Baer] p. 49 line 4, aS=bS iff a=b in their notation (S = sigma). The sigma map is one-to-one. (Contributed by NM, 26-May-2015.) |
Ref | Expression |
---|---|
hdmap12d.h | β’ π» = (LHypβπΎ) |
hdmap12d.u | β’ π = ((DVecHβπΎ)βπ) |
hdmap12d.v | β’ π = (Baseβπ) |
hdmap12d.s | β’ π = ((HDMapβπΎ)βπ) |
hdmap12d.k | β’ (π β (πΎ β HL β§ π β π»)) |
hdmap12d.x | β’ (π β π β π) |
hdmap12d.y | β’ (π β π β π) |
Ref | Expression |
---|---|
hdmap11 | β’ (π β ((πβπ) = (πβπ) β π = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap12d.h | . . . . 5 β’ π» = (LHypβπΎ) | |
2 | hdmap12d.u | . . . . 5 β’ π = ((DVecHβπΎ)βπ) | |
3 | hdmap12d.v | . . . . 5 β’ π = (Baseβπ) | |
4 | eqid 2730 | . . . . 5 β’ (-gβπ) = (-gβπ) | |
5 | eqid 2730 | . . . . 5 β’ ((LCDualβπΎ)βπ) = ((LCDualβπΎ)βπ) | |
6 | eqid 2730 | . . . . 5 β’ (-gβ((LCDualβπΎ)βπ)) = (-gβ((LCDualβπΎ)βπ)) | |
7 | hdmap12d.s | . . . . 5 β’ π = ((HDMapβπΎ)βπ) | |
8 | hdmap12d.k | . . . . 5 β’ (π β (πΎ β HL β§ π β π»)) | |
9 | hdmap12d.x | . . . . 5 β’ (π β π β π) | |
10 | hdmap12d.y | . . . . 5 β’ (π β π β π) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | hdmapsub 41021 | . . . 4 β’ (π β (πβ(π(-gβπ)π)) = ((πβπ)(-gβ((LCDualβπΎ)βπ))(πβπ))) |
12 | 11 | eqeq1d 2732 | . . 3 β’ (π β ((πβ(π(-gβπ)π)) = (0gβ((LCDualβπΎ)βπ)) β ((πβπ)(-gβ((LCDualβπΎ)βπ))(πβπ)) = (0gβ((LCDualβπΎ)βπ)))) |
13 | eqid 2730 | . . . 4 β’ (0gβπ) = (0gβπ) | |
14 | eqid 2730 | . . . 4 β’ (0gβ((LCDualβπΎ)βπ)) = (0gβ((LCDualβπΎ)βπ)) | |
15 | 1, 2, 8 | dvhlmod 40284 | . . . . 5 β’ (π β π β LMod) |
16 | 3, 4 | lmodvsubcl 20661 | . . . . 5 β’ ((π β LMod β§ π β π β§ π β π) β (π(-gβπ)π) β π) |
17 | 15, 9, 10, 16 | syl3anc 1369 | . . . 4 β’ (π β (π(-gβπ)π) β π) |
18 | 1, 2, 3, 13, 5, 14, 7, 8, 17 | hdmapeq0 41018 | . . 3 β’ (π β ((πβ(π(-gβπ)π)) = (0gβ((LCDualβπΎ)βπ)) β (π(-gβπ)π) = (0gβπ))) |
19 | 1, 5, 8 | lcdlmod 40766 | . . . . 5 β’ (π β ((LCDualβπΎ)βπ) β LMod) |
20 | lmodgrp 20621 | . . . . 5 β’ (((LCDualβπΎ)βπ) β LMod β ((LCDualβπΎ)βπ) β Grp) | |
21 | 19, 20 | syl 17 | . . . 4 β’ (π β ((LCDualβπΎ)βπ) β Grp) |
22 | eqid 2730 | . . . . 5 β’ (Baseβ((LCDualβπΎ)βπ)) = (Baseβ((LCDualβπΎ)βπ)) | |
23 | 1, 2, 3, 5, 22, 7, 8, 9 | hdmapcl 41004 | . . . 4 β’ (π β (πβπ) β (Baseβ((LCDualβπΎ)βπ))) |
24 | 1, 2, 3, 5, 22, 7, 8, 10 | hdmapcl 41004 | . . . 4 β’ (π β (πβπ) β (Baseβ((LCDualβπΎ)βπ))) |
25 | 22, 14, 6 | grpsubeq0 18945 | . . . 4 β’ ((((LCDualβπΎ)βπ) β Grp β§ (πβπ) β (Baseβ((LCDualβπΎ)βπ)) β§ (πβπ) β (Baseβ((LCDualβπΎ)βπ))) β (((πβπ)(-gβ((LCDualβπΎ)βπ))(πβπ)) = (0gβ((LCDualβπΎ)βπ)) β (πβπ) = (πβπ))) |
26 | 21, 23, 24, 25 | syl3anc 1369 | . . 3 β’ (π β (((πβπ)(-gβ((LCDualβπΎ)βπ))(πβπ)) = (0gβ((LCDualβπΎ)βπ)) β (πβπ) = (πβπ))) |
27 | 12, 18, 26 | 3bitr3rd 309 | . 2 β’ (π β ((πβπ) = (πβπ) β (π(-gβπ)π) = (0gβπ))) |
28 | lmodgrp 20621 | . . . 4 β’ (π β LMod β π β Grp) | |
29 | 15, 28 | syl 17 | . . 3 β’ (π β π β Grp) |
30 | 3, 13, 4 | grpsubeq0 18945 | . . 3 β’ ((π β Grp β§ π β π β§ π β π) β ((π(-gβπ)π) = (0gβπ) β π = π)) |
31 | 29, 9, 10, 30 | syl3anc 1369 | . 2 β’ (π β ((π(-gβπ)π) = (0gβπ) β π = π)) |
32 | 27, 31 | bitrd 278 | 1 β’ (π β ((πβπ) = (πβπ) β π = π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1539 β wcel 2104 βcfv 6542 (class class class)co 7411 Basecbs 17148 0gc0g 17389 Grpcgrp 18855 -gcsg 18857 LModclmod 20614 HLchlt 38523 LHypclh 39158 DVecHcdvh 40252 LCDualclcd 40760 HDMapchdma 40966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-riotaBAD 38126 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 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-ot 4636 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-undef 8260 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-0g 17391 df-mre 17534 df-mrc 17535 df-acs 17537 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19039 df-cntz 19222 df-oppg 19251 df-lsm 19545 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-dvr 20292 df-drng 20502 df-lmod 20616 df-lss 20687 df-lsp 20727 df-lvec 20858 df-lsatoms 38149 df-lshyp 38150 df-lcv 38192 df-lfl 38231 df-lkr 38259 df-ldual 38297 df-oposet 38349 df-ol 38351 df-oml 38352 df-covers 38439 df-ats 38440 df-atl 38471 df-cvlat 38495 df-hlat 38524 df-llines 38672 df-lplanes 38673 df-lvols 38674 df-lines 38675 df-psubsp 38677 df-pmap 38678 df-padd 38970 df-lhyp 39162 df-laut 39163 df-ldil 39278 df-ltrn 39279 df-trl 39333 df-tgrp 39917 df-tendo 39929 df-edring 39931 df-dveca 40177 df-disoa 40203 df-dvech 40253 df-dib 40313 df-dic 40347 df-dih 40403 df-doch 40522 df-djh 40569 df-lcdual 40761 df-mapd 40799 df-hvmap 40931 df-hdmap1 40967 df-hdmap 40968 |
This theorem is referenced by: hdmapf1oN 41039 hgmap11 41076 |
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