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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapneg | Structured version Visualization version GIF version |
Description: Part of proof of part 12 in [Baer] p. 49 line 4. The sigma map of a negative is the negative of the sigma map. (Contributed by NM, 24-May-2015.) |
Ref | Expression |
---|---|
hdmap12b.h | β’ π» = (LHypβπΎ) |
hdmap12b.u | β’ π = ((DVecHβπΎ)βπ) |
hdmap12b.v | β’ π = (Baseβπ) |
hdmap12b.m | β’ π = (invgβπ) |
hdmap12b.c | β’ πΆ = ((LCDualβπΎ)βπ) |
hdmap12b.i | β’ πΌ = (invgβπΆ) |
hdmap12b.s | β’ π = ((HDMapβπΎ)βπ) |
hdmap12b.k | β’ (π β (πΎ β HL β§ π β π»)) |
hdmap12b.x | β’ (π β π β π) |
Ref | Expression |
---|---|
hdmapneg | β’ (π β (πβ(πβπ)) = (πΌβ(πβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap12b.h | . . . . 5 β’ π» = (LHypβπΎ) | |
2 | hdmap12b.c | . . . . 5 β’ πΆ = ((LCDualβπΎ)βπ) | |
3 | hdmap12b.k | . . . . 5 β’ (π β (πΎ β HL β§ π β π»)) | |
4 | 1, 2, 3 | lcdlmod 40101 | . . . 4 β’ (π β πΆ β LMod) |
5 | hdmap12b.u | . . . . 5 β’ π = ((DVecHβπΎ)βπ) | |
6 | hdmap12b.v | . . . . 5 β’ π = (Baseβπ) | |
7 | eqid 2733 | . . . . 5 β’ (BaseβπΆ) = (BaseβπΆ) | |
8 | hdmap12b.s | . . . . 5 β’ π = ((HDMapβπΎ)βπ) | |
9 | hdmap12b.x | . . . . 5 β’ (π β π β π) | |
10 | 1, 5, 6, 2, 7, 8, 3, 9 | hdmapcl 40339 | . . . 4 β’ (π β (πβπ) β (BaseβπΆ)) |
11 | eqid 2733 | . . . . 5 β’ (+gβπΆ) = (+gβπΆ) | |
12 | eqid 2733 | . . . . 5 β’ (0gβπΆ) = (0gβπΆ) | |
13 | hdmap12b.i | . . . . 5 β’ πΌ = (invgβπΆ) | |
14 | 7, 11, 12, 13 | lmodvnegid 20379 | . . . 4 β’ ((πΆ β LMod β§ (πβπ) β (BaseβπΆ)) β ((πβπ)(+gβπΆ)(πΌβ(πβπ))) = (0gβπΆ)) |
15 | 4, 10, 14 | syl2anc 585 | . . 3 β’ (π β ((πβπ)(+gβπΆ)(πΌβ(πβπ))) = (0gβπΆ)) |
16 | 1, 5, 3 | dvhlmod 39619 | . . . . 5 β’ (π β π β LMod) |
17 | eqid 2733 | . . . . . 6 β’ (+gβπ) = (+gβπ) | |
18 | eqid 2733 | . . . . . 6 β’ (0gβπ) = (0gβπ) | |
19 | hdmap12b.m | . . . . . 6 β’ π = (invgβπ) | |
20 | 6, 17, 18, 19 | lmodvnegid 20379 | . . . . 5 β’ ((π β LMod β§ π β π) β (π(+gβπ)(πβπ)) = (0gβπ)) |
21 | 16, 9, 20 | syl2anc 585 | . . . 4 β’ (π β (π(+gβπ)(πβπ)) = (0gβπ)) |
22 | 6, 19 | lmodvnegcl 20378 | . . . . . . 7 β’ ((π β LMod β§ π β π) β (πβπ) β π) |
23 | 16, 9, 22 | syl2anc 585 | . . . . . 6 β’ (π β (πβπ) β π) |
24 | 6, 17 | lmodvacl 20351 | . . . . . 6 β’ ((π β LMod β§ π β π β§ (πβπ) β π) β (π(+gβπ)(πβπ)) β π) |
25 | 16, 9, 23, 24 | syl3anc 1372 | . . . . 5 β’ (π β (π(+gβπ)(πβπ)) β π) |
26 | 1, 5, 6, 18, 2, 12, 8, 3, 25 | hdmapeq0 40353 | . . . 4 β’ (π β ((πβ(π(+gβπ)(πβπ))) = (0gβπΆ) β (π(+gβπ)(πβπ)) = (0gβπ))) |
27 | 21, 26 | mpbird 257 | . . 3 β’ (π β (πβ(π(+gβπ)(πβπ))) = (0gβπΆ)) |
28 | 1, 5, 6, 17, 2, 11, 8, 3, 9, 23 | hdmapadd 40352 | . . 3 β’ (π β (πβ(π(+gβπ)(πβπ))) = ((πβπ)(+gβπΆ)(πβ(πβπ)))) |
29 | 15, 27, 28 | 3eqtr2rd 2780 | . 2 β’ (π β ((πβπ)(+gβπΆ)(πβ(πβπ))) = ((πβπ)(+gβπΆ)(πΌβ(πβπ)))) |
30 | 1, 5, 6, 2, 7, 8, 3, 23 | hdmapcl 40339 | . . 3 β’ (π β (πβ(πβπ)) β (BaseβπΆ)) |
31 | 7, 13 | lmodvnegcl 20378 | . . . 4 β’ ((πΆ β LMod β§ (πβπ) β (BaseβπΆ)) β (πΌβ(πβπ)) β (BaseβπΆ)) |
32 | 4, 10, 31 | syl2anc 585 | . . 3 β’ (π β (πΌβ(πβπ)) β (BaseβπΆ)) |
33 | 7, 11 | lmodlcan 20353 | . . 3 β’ ((πΆ β LMod β§ ((πβ(πβπ)) β (BaseβπΆ) β§ (πΌβ(πβπ)) β (BaseβπΆ) β§ (πβπ) β (BaseβπΆ))) β (((πβπ)(+gβπΆ)(πβ(πβπ))) = ((πβπ)(+gβπΆ)(πΌβ(πβπ))) β (πβ(πβπ)) = (πΌβ(πβπ)))) |
34 | 4, 30, 32, 10, 33 | syl13anc 1373 | . 2 β’ (π β (((πβπ)(+gβπΆ)(πβ(πβπ))) = ((πβπ)(+gβπΆ)(πΌβ(πβπ))) β (πβ(πβπ)) = (πΌβ(πβπ)))) |
35 | 29, 34 | mpbid 231 | 1 β’ (π β (πβ(πβπ)) = (πΌβ(πβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 0gc0g 17326 invgcminusg 18754 LModclmod 20336 HLchlt 37858 LHypclh 38493 DVecHcdvh 39587 LCDualclcd 40095 HDMapchdma 40301 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-riotaBAD 37461 |
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 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-undef 8205 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-0g 17328 df-mre 17471 df-mrc 17472 df-acs 17474 df-proset 18189 df-poset 18207 df-plt 18224 df-lub 18240 df-glb 18241 df-join 18242 df-meet 18243 df-p0 18319 df-p1 18320 df-lat 18326 df-clat 18393 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-submnd 18607 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-cntz 19102 df-oppg 19129 df-lsm 19423 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-oppr 20054 df-dvdsr 20075 df-unit 20076 df-invr 20106 df-dvr 20117 df-drng 20199 df-lmod 20338 df-lss 20408 df-lsp 20448 df-lvec 20579 df-lsatoms 37484 df-lshyp 37485 df-lcv 37527 df-lfl 37566 df-lkr 37594 df-ldual 37632 df-oposet 37684 df-ol 37686 df-oml 37687 df-covers 37774 df-ats 37775 df-atl 37806 df-cvlat 37830 df-hlat 37859 df-llines 38007 df-lplanes 38008 df-lvols 38009 df-lines 38010 df-psubsp 38012 df-pmap 38013 df-padd 38305 df-lhyp 38497 df-laut 38498 df-ldil 38613 df-ltrn 38614 df-trl 38668 df-tgrp 39252 df-tendo 39264 df-edring 39266 df-dveca 39512 df-disoa 39538 df-dvech 39588 df-dib 39648 df-dic 39682 df-dih 39738 df-doch 39857 df-djh 39904 df-lcdual 40096 df-mapd 40134 df-hvmap 40266 df-hdmap1 40302 df-hdmap 40303 |
This theorem is referenced by: hdmapsub 40356 |
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