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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 2731 | . . . . 5 β’ (-gβπ) = (-gβπ) | |
5 | eqid 2731 | . . . . 5 β’ ((LCDualβπΎ)βπ) = ((LCDualβπΎ)βπ) | |
6 | eqid 2731 | . . . . 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 40416 | . . . 4 β’ (π β (πβ(π(-gβπ)π)) = ((πβπ)(-gβ((LCDualβπΎ)βπ))(πβπ))) |
12 | 11 | eqeq1d 2733 | . . 3 β’ (π β ((πβ(π(-gβπ)π)) = (0gβ((LCDualβπΎ)βπ)) β ((πβπ)(-gβ((LCDualβπΎ)βπ))(πβπ)) = (0gβ((LCDualβπΎ)βπ)))) |
13 | eqid 2731 | . . . 4 β’ (0gβπ) = (0gβπ) | |
14 | eqid 2731 | . . . 4 β’ (0gβ((LCDualβπΎ)βπ)) = (0gβ((LCDualβπΎ)βπ)) | |
15 | 1, 2, 8 | dvhlmod 39679 | . . . . 5 β’ (π β π β LMod) |
16 | 3, 4 | lmodvsubcl 20439 | . . . . 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 1371 | . . . 4 β’ (π β (π(-gβπ)π) β π) |
18 | 1, 2, 3, 13, 5, 14, 7, 8, 17 | hdmapeq0 40413 | . . 3 β’ (π β ((πβ(π(-gβπ)π)) = (0gβ((LCDualβπΎ)βπ)) β (π(-gβπ)π) = (0gβπ))) |
19 | 1, 5, 8 | lcdlmod 40161 | . . . . 5 β’ (π β ((LCDualβπΎ)βπ) β LMod) |
20 | lmodgrp 20400 | . . . . 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 2731 | . . . . 5 β’ (Baseβ((LCDualβπΎ)βπ)) = (Baseβ((LCDualβπΎ)βπ)) | |
23 | 1, 2, 3, 5, 22, 7, 8, 9 | hdmapcl 40399 | . . . 4 β’ (π β (πβπ) β (Baseβ((LCDualβπΎ)βπ))) |
24 | 1, 2, 3, 5, 22, 7, 8, 10 | hdmapcl 40399 | . . . 4 β’ (π β (πβπ) β (Baseβ((LCDualβπΎ)βπ))) |
25 | 22, 14, 6 | grpsubeq0 18862 | . . . 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 1371 | . . 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 20400 | . . . 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 18862 | . . 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 1371 | . 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 396 = wceq 1541 β wcel 2106 βcfv 6516 (class class class)co 7377 Basecbs 17109 0gc0g 17350 Grpcgrp 18777 -gcsg 18779 LModclmod 20393 HLchlt 37918 LHypclh 38553 DVecHcdvh 39647 LCDualclcd 40155 HDMapchdma 40361 |
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 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-riotaBAD 37521 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-ot 4615 df-uni 4886 df-int 4928 df-iun 4976 df-iin 4977 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-of 7637 df-om 7823 df-1st 7941 df-2nd 7942 df-tpos 8177 df-undef 8224 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8670 df-map 8789 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-nn 12178 df-2 12240 df-3 12241 df-4 12242 df-5 12243 df-6 12244 df-n0 12438 df-z 12524 df-uz 12788 df-fz 13450 df-struct 17045 df-sets 17062 df-slot 17080 df-ndx 17092 df-base 17110 df-ress 17139 df-plusg 17175 df-mulr 17176 df-sca 17178 df-vsca 17179 df-0g 17352 df-mre 17495 df-mrc 17496 df-acs 17498 df-proset 18213 df-poset 18231 df-plt 18248 df-lub 18264 df-glb 18265 df-join 18266 df-meet 18267 df-p0 18343 df-p1 18344 df-lat 18350 df-clat 18417 df-mgm 18526 df-sgrp 18575 df-mnd 18586 df-submnd 18631 df-grp 18780 df-minusg 18781 df-sbg 18782 df-subg 18954 df-cntz 19126 df-oppg 19153 df-lsm 19447 df-cmn 19593 df-abl 19594 df-mgp 19926 df-ur 19943 df-ring 19995 df-oppr 20078 df-dvdsr 20099 df-unit 20100 df-invr 20130 df-dvr 20141 df-drng 20242 df-lmod 20395 df-lss 20465 df-lsp 20505 df-lvec 20636 df-lsatoms 37544 df-lshyp 37545 df-lcv 37587 df-lfl 37626 df-lkr 37654 df-ldual 37692 df-oposet 37744 df-ol 37746 df-oml 37747 df-covers 37834 df-ats 37835 df-atl 37866 df-cvlat 37890 df-hlat 37919 df-llines 38067 df-lplanes 38068 df-lvols 38069 df-lines 38070 df-psubsp 38072 df-pmap 38073 df-padd 38365 df-lhyp 38557 df-laut 38558 df-ldil 38673 df-ltrn 38674 df-trl 38728 df-tgrp 39312 df-tendo 39324 df-edring 39326 df-dveca 39572 df-disoa 39598 df-dvech 39648 df-dib 39708 df-dic 39742 df-dih 39798 df-doch 39917 df-djh 39964 df-lcdual 40156 df-mapd 40194 df-hvmap 40326 df-hdmap1 40362 df-hdmap 40363 |
This theorem is referenced by: hdmapf1oN 40434 hgmap11 40471 |
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