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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 | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap12d.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap12d.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap12d.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap12d.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap12d.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
hdmap12d.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
hdmap11 | ⊢ (𝜑 → ((𝑆‘𝑋) = (𝑆‘𝑌) ↔ 𝑋 = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap12d.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap12d.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap12d.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
4 | eqid 2731 | . . . . 5 ⊢ (-g‘𝑈) = (-g‘𝑈) | |
5 | eqid 2731 | . . . . 5 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊) | |
6 | eqid 2731 | . . . . 5 ⊢ (-g‘((LCDual‘𝐾)‘𝑊)) = (-g‘((LCDual‘𝐾)‘𝑊)) | |
7 | hdmap12d.s | . . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
8 | hdmap12d.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | hdmap12d.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
10 | hdmap12d.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | hdmapsub 40523 | . . . 4 ⊢ (𝜑 → (𝑆‘(𝑋(-g‘𝑈)𝑌)) = ((𝑆‘𝑋)(-g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝑌))) |
12 | 11 | eqeq1d 2733 | . . 3 ⊢ (𝜑 → ((𝑆‘(𝑋(-g‘𝑈)𝑌)) = (0g‘((LCDual‘𝐾)‘𝑊)) ↔ ((𝑆‘𝑋)(-g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝑌)) = (0g‘((LCDual‘𝐾)‘𝑊)))) |
13 | eqid 2731 | . . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
14 | eqid 2731 | . . . 4 ⊢ (0g‘((LCDual‘𝐾)‘𝑊)) = (0g‘((LCDual‘𝐾)‘𝑊)) | |
15 | 1, 2, 8 | dvhlmod 39786 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
16 | 3, 4 | lmodvsubcl 20466 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋(-g‘𝑈)𝑌) ∈ 𝑉) |
17 | 15, 9, 10, 16 | syl3anc 1371 | . . . 4 ⊢ (𝜑 → (𝑋(-g‘𝑈)𝑌) ∈ 𝑉) |
18 | 1, 2, 3, 13, 5, 14, 7, 8, 17 | hdmapeq0 40520 | . . 3 ⊢ (𝜑 → ((𝑆‘(𝑋(-g‘𝑈)𝑌)) = (0g‘((LCDual‘𝐾)‘𝑊)) ↔ (𝑋(-g‘𝑈)𝑌) = (0g‘𝑈))) |
19 | 1, 5, 8 | lcdlmod 40268 | . . . . 5 ⊢ (𝜑 → ((LCDual‘𝐾)‘𝑊) ∈ LMod) |
20 | lmodgrp 20427 | . . . . 5 ⊢ (((LCDual‘𝐾)‘𝑊) ∈ LMod → ((LCDual‘𝐾)‘𝑊) ∈ Grp) | |
21 | 19, 20 | syl 17 | . . . 4 ⊢ (𝜑 → ((LCDual‘𝐾)‘𝑊) ∈ Grp) |
22 | eqid 2731 | . . . . 5 ⊢ (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊)) | |
23 | 1, 2, 3, 5, 22, 7, 8, 9 | hdmapcl 40506 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘((LCDual‘𝐾)‘𝑊))) |
24 | 1, 2, 3, 5, 22, 7, 8, 10 | hdmapcl 40506 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑌) ∈ (Base‘((LCDual‘𝐾)‘𝑊))) |
25 | 22, 14, 6 | grpsubeq0 18883 | . . . 4 ⊢ ((((LCDual‘𝐾)‘𝑊) ∈ Grp ∧ (𝑆‘𝑋) ∈ (Base‘((LCDual‘𝐾)‘𝑊)) ∧ (𝑆‘𝑌) ∈ (Base‘((LCDual‘𝐾)‘𝑊))) → (((𝑆‘𝑋)(-g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝑌)) = (0g‘((LCDual‘𝐾)‘𝑊)) ↔ (𝑆‘𝑋) = (𝑆‘𝑌))) |
26 | 21, 23, 24, 25 | syl3anc 1371 | . . 3 ⊢ (𝜑 → (((𝑆‘𝑋)(-g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝑌)) = (0g‘((LCDual‘𝐾)‘𝑊)) ↔ (𝑆‘𝑋) = (𝑆‘𝑌))) |
27 | 12, 18, 26 | 3bitr3rd 309 | . 2 ⊢ (𝜑 → ((𝑆‘𝑋) = (𝑆‘𝑌) ↔ (𝑋(-g‘𝑈)𝑌) = (0g‘𝑈))) |
28 | lmodgrp 20427 | . . . 4 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Grp) | |
29 | 15, 28 | syl 17 | . . 3 ⊢ (𝜑 → 𝑈 ∈ Grp) |
30 | 3, 13, 4 | grpsubeq0 18883 | . . 3 ⊢ ((𝑈 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑋(-g‘𝑈)𝑌) = (0g‘𝑈) ↔ 𝑋 = 𝑌)) |
31 | 29, 9, 10, 30 | syl3anc 1371 | . 2 ⊢ (𝜑 → ((𝑋(-g‘𝑈)𝑌) = (0g‘𝑈) ↔ 𝑋 = 𝑌)) |
32 | 27, 31 | bitrd 278 | 1 ⊢ (𝜑 → ((𝑆‘𝑋) = (𝑆‘𝑌) ↔ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 0gc0g 17367 Grpcgrp 18794 -gcsg 18796 LModclmod 20420 HLchlt 38025 LHypclh 38660 DVecHcdvh 39754 LCDualclcd 40262 HDMapchdma 40468 |
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 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-riotaBAD 37628 |
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 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-ot 4631 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-om 7839 df-1st 7957 df-2nd 7958 df-tpos 8193 df-undef 8240 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-n0 12455 df-z 12541 df-uz 12805 df-fz 13467 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-0g 17369 df-mre 17512 df-mrc 17513 df-acs 17515 df-proset 18230 df-poset 18248 df-plt 18265 df-lub 18281 df-glb 18282 df-join 18283 df-meet 18284 df-p0 18360 df-p1 18361 df-lat 18367 df-clat 18434 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-submnd 18648 df-grp 18797 df-minusg 18798 df-sbg 18799 df-subg 18975 df-cntz 19147 df-oppg 19174 df-lsm 19468 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-oppr 20102 df-dvdsr 20123 df-unit 20124 df-invr 20154 df-dvr 20165 df-drng 20267 df-lmod 20422 df-lss 20492 df-lsp 20532 df-lvec 20663 df-lsatoms 37651 df-lshyp 37652 df-lcv 37694 df-lfl 37733 df-lkr 37761 df-ldual 37799 df-oposet 37851 df-ol 37853 df-oml 37854 df-covers 37941 df-ats 37942 df-atl 37973 df-cvlat 37997 df-hlat 38026 df-llines 38174 df-lplanes 38175 df-lvols 38176 df-lines 38177 df-psubsp 38179 df-pmap 38180 df-padd 38472 df-lhyp 38664 df-laut 38665 df-ldil 38780 df-ltrn 38781 df-trl 38835 df-tgrp 39419 df-tendo 39431 df-edring 39433 df-dveca 39679 df-disoa 39705 df-dvech 39755 df-dib 39815 df-dic 39849 df-dih 39905 df-doch 40024 df-djh 40071 df-lcdual 40263 df-mapd 40301 df-hvmap 40433 df-hdmap1 40469 df-hdmap 40470 |
This theorem is referenced by: hdmapf1oN 40541 hgmap11 40578 |
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