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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 | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap12b.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap12b.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap12b.m | ⊢ 𝑀 = (invg‘𝑈) |
hdmap12b.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap12b.i | ⊢ 𝐼 = (invg‘𝐶) |
hdmap12b.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap12b.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap12b.x | ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
Ref | Expression |
---|---|
hdmapneg | ⊢ (𝜑 → (𝑆‘(𝑀‘𝑇)) = (𝐼‘(𝑆‘𝑇))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap12b.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap12b.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
3 | hdmap12b.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | lcdlmod 37605 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
5 | hdmap12b.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | hdmap12b.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
7 | eqid 2797 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
8 | hdmap12b.s | . . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
9 | hdmap12b.x | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
10 | 1, 5, 6, 2, 7, 8, 3, 9 | hdmapcl 37843 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑇) ∈ (Base‘𝐶)) |
11 | eqid 2797 | . . . . 5 ⊢ (+g‘𝐶) = (+g‘𝐶) | |
12 | eqid 2797 | . . . . 5 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
13 | hdmap12b.i | . . . . 5 ⊢ 𝐼 = (invg‘𝐶) | |
14 | 7, 11, 12, 13 | lmodvnegid 19220 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑇) ∈ (Base‘𝐶)) → ((𝑆‘𝑇)(+g‘𝐶)(𝐼‘(𝑆‘𝑇))) = (0g‘𝐶)) |
15 | 4, 10, 14 | syl2anc 580 | . . 3 ⊢ (𝜑 → ((𝑆‘𝑇)(+g‘𝐶)(𝐼‘(𝑆‘𝑇))) = (0g‘𝐶)) |
16 | 1, 5, 3 | dvhlmod 37123 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
17 | eqid 2797 | . . . . . 6 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
18 | eqid 2797 | . . . . . 6 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
19 | hdmap12b.m | . . . . . 6 ⊢ 𝑀 = (invg‘𝑈) | |
20 | 6, 17, 18, 19 | lmodvnegid 19220 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑇 ∈ 𝑉) → (𝑇(+g‘𝑈)(𝑀‘𝑇)) = (0g‘𝑈)) |
21 | 16, 9, 20 | syl2anc 580 | . . . 4 ⊢ (𝜑 → (𝑇(+g‘𝑈)(𝑀‘𝑇)) = (0g‘𝑈)) |
22 | 6, 19 | lmodvnegcl 19219 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑇 ∈ 𝑉) → (𝑀‘𝑇) ∈ 𝑉) |
23 | 16, 9, 22 | syl2anc 580 | . . . . . 6 ⊢ (𝜑 → (𝑀‘𝑇) ∈ 𝑉) |
24 | 6, 17 | lmodvacl 19192 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑇 ∈ 𝑉 ∧ (𝑀‘𝑇) ∈ 𝑉) → (𝑇(+g‘𝑈)(𝑀‘𝑇)) ∈ 𝑉) |
25 | 16, 9, 23, 24 | syl3anc 1491 | . . . . 5 ⊢ (𝜑 → (𝑇(+g‘𝑈)(𝑀‘𝑇)) ∈ 𝑉) |
26 | 1, 5, 6, 18, 2, 12, 8, 3, 25 | hdmapeq0 37857 | . . . 4 ⊢ (𝜑 → ((𝑆‘(𝑇(+g‘𝑈)(𝑀‘𝑇))) = (0g‘𝐶) ↔ (𝑇(+g‘𝑈)(𝑀‘𝑇)) = (0g‘𝑈))) |
27 | 21, 26 | mpbird 249 | . . 3 ⊢ (𝜑 → (𝑆‘(𝑇(+g‘𝑈)(𝑀‘𝑇))) = (0g‘𝐶)) |
28 | 1, 5, 6, 17, 2, 11, 8, 3, 9, 23 | hdmapadd 37856 | . . 3 ⊢ (𝜑 → (𝑆‘(𝑇(+g‘𝑈)(𝑀‘𝑇))) = ((𝑆‘𝑇)(+g‘𝐶)(𝑆‘(𝑀‘𝑇)))) |
29 | 15, 27, 28 | 3eqtr2rd 2838 | . 2 ⊢ (𝜑 → ((𝑆‘𝑇)(+g‘𝐶)(𝑆‘(𝑀‘𝑇))) = ((𝑆‘𝑇)(+g‘𝐶)(𝐼‘(𝑆‘𝑇)))) |
30 | 1, 5, 6, 2, 7, 8, 3, 23 | hdmapcl 37843 | . . 3 ⊢ (𝜑 → (𝑆‘(𝑀‘𝑇)) ∈ (Base‘𝐶)) |
31 | 7, 13 | lmodvnegcl 19219 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑇) ∈ (Base‘𝐶)) → (𝐼‘(𝑆‘𝑇)) ∈ (Base‘𝐶)) |
32 | 4, 10, 31 | syl2anc 580 | . . 3 ⊢ (𝜑 → (𝐼‘(𝑆‘𝑇)) ∈ (Base‘𝐶)) |
33 | 7, 11 | lmodlcan 19194 | . . 3 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘(𝑀‘𝑇)) ∈ (Base‘𝐶) ∧ (𝐼‘(𝑆‘𝑇)) ∈ (Base‘𝐶) ∧ (𝑆‘𝑇) ∈ (Base‘𝐶))) → (((𝑆‘𝑇)(+g‘𝐶)(𝑆‘(𝑀‘𝑇))) = ((𝑆‘𝑇)(+g‘𝐶)(𝐼‘(𝑆‘𝑇))) ↔ (𝑆‘(𝑀‘𝑇)) = (𝐼‘(𝑆‘𝑇)))) |
34 | 4, 30, 32, 10, 33 | syl13anc 1492 | . 2 ⊢ (𝜑 → (((𝑆‘𝑇)(+g‘𝐶)(𝑆‘(𝑀‘𝑇))) = ((𝑆‘𝑇)(+g‘𝐶)(𝐼‘(𝑆‘𝑇))) ↔ (𝑆‘(𝑀‘𝑇)) = (𝐼‘(𝑆‘𝑇)))) |
35 | 29, 34 | mpbid 224 | 1 ⊢ (𝜑 → (𝑆‘(𝑀‘𝑇)) = (𝐼‘(𝑆‘𝑇))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ‘cfv 6099 (class class class)co 6876 Basecbs 16181 +gcplusg 16264 0gc0g 16412 invgcminusg 17736 LModclmod 19178 HLchlt 35363 LHypclh 35997 DVecHcdvh 37091 LCDualclcd 37599 HDMapchdma 37805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-riotaBAD 34966 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-ot 4375 df-uni 4627 df-int 4666 df-iun 4710 df-iin 4711 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-of 7129 df-om 7298 df-1st 7399 df-2nd 7400 df-tpos 7588 df-undef 7635 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-map 8095 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-n0 11577 df-z 11663 df-uz 11927 df-fz 12577 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-sca 16280 df-vsca 16281 df-0g 16414 df-mre 16558 df-mrc 16559 df-acs 16561 df-proset 17240 df-poset 17258 df-plt 17270 df-lub 17286 df-glb 17287 df-join 17288 df-meet 17289 df-p0 17351 df-p1 17352 df-lat 17358 df-clat 17420 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-submnd 17648 df-grp 17738 df-minusg 17739 df-sbg 17740 df-subg 17901 df-cntz 18059 df-oppg 18085 df-lsm 18361 df-cmn 18507 df-abl 18508 df-mgp 18803 df-ur 18815 df-ring 18862 df-oppr 18936 df-dvdsr 18954 df-unit 18955 df-invr 18985 df-dvr 18996 df-drng 19064 df-lmod 19180 df-lss 19248 df-lsp 19290 df-lvec 19421 df-lsatoms 34989 df-lshyp 34990 df-lcv 35032 df-lfl 35071 df-lkr 35099 df-ldual 35137 df-oposet 35189 df-ol 35191 df-oml 35192 df-covers 35279 df-ats 35280 df-atl 35311 df-cvlat 35335 df-hlat 35364 df-llines 35511 df-lplanes 35512 df-lvols 35513 df-lines 35514 df-psubsp 35516 df-pmap 35517 df-padd 35809 df-lhyp 36001 df-laut 36002 df-ldil 36117 df-ltrn 36118 df-trl 36172 df-tgrp 36756 df-tendo 36768 df-edring 36770 df-dveca 37016 df-disoa 37042 df-dvech 37092 df-dib 37152 df-dic 37186 df-dih 37242 df-doch 37361 df-djh 37408 df-lcdual 37600 df-mapd 37638 df-hvmap 37770 df-hdmap1 37806 df-hdmap 37807 |
This theorem is referenced by: hdmapsub 37860 |
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