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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdat | Structured version Visualization version GIF version |
Description: Atoms are preserved by the map defined by df-mapd 37435. Property (g) in [Baer] p. 41. (Contributed by NM, 14-Mar-2015.) |
Ref | Expression |
---|---|
mapdat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdat.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdat.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdat.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
mapdat.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdat.b | ⊢ 𝐵 = (LSAtoms‘𝐶) |
mapdat.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdat.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
Ref | Expression |
---|---|
mapdat | ⊢ (𝜑 → (𝑀‘𝑄) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdat.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdat.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
3 | mapdat.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | eqid 2771 | . . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
5 | mapdat.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
6 | eqid 2771 | . . . 4 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
7 | mapdat.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | mapd0 37475 | . . 3 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)}) = {(0g‘𝐶)}) |
9 | mapdat.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
10 | eqid 2771 | . . . . 5 ⊢ ( ⋖L ‘𝑈) = ( ⋖L ‘𝑈) | |
11 | 1, 3, 7 | dvhlvec 36919 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec) |
12 | mapdat.q | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
13 | 4, 9, 10, 11, 12 | lsatcv0 34840 | . . . 4 ⊢ (𝜑 → {(0g‘𝑈)} ( ⋖L ‘𝑈)𝑄) |
14 | eqid 2771 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
15 | eqid 2771 | . . . . 5 ⊢ ( ⋖L ‘𝐶) = ( ⋖L ‘𝐶) | |
16 | 1, 3, 7 | dvhlmod 36920 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
17 | 4, 14 | lsssn0 19158 | . . . . . 6 ⊢ (𝑈 ∈ LMod → {(0g‘𝑈)} ∈ (LSubSp‘𝑈)) |
18 | 16, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → {(0g‘𝑈)} ∈ (LSubSp‘𝑈)) |
19 | 14, 9, 16, 12 | lsatlssel 34806 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑈)) |
20 | 1, 2, 3, 14, 10, 5, 15, 7, 18, 19 | mapdcv 37470 | . . . 4 ⊢ (𝜑 → ({(0g‘𝑈)} ( ⋖L ‘𝑈)𝑄 ↔ (𝑀‘{(0g‘𝑈)})( ⋖L ‘𝐶)(𝑀‘𝑄))) |
21 | 13, 20 | mpbid 222 | . . 3 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)})( ⋖L ‘𝐶)(𝑀‘𝑄)) |
22 | 8, 21 | eqbrtrrd 4810 | . 2 ⊢ (𝜑 → {(0g‘𝐶)} ( ⋖L ‘𝐶)(𝑀‘𝑄)) |
23 | eqid 2771 | . . 3 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
24 | mapdat.b | . . 3 ⊢ 𝐵 = (LSAtoms‘𝐶) | |
25 | 1, 5, 7 | lcdlvec 37401 | . . 3 ⊢ (𝜑 → 𝐶 ∈ LVec) |
26 | 1, 2, 3, 14, 5, 23, 7, 19 | mapdcl2 37466 | . . 3 ⊢ (𝜑 → (𝑀‘𝑄) ∈ (LSubSp‘𝐶)) |
27 | 6, 23, 24, 15, 25, 26 | lsat0cv 34842 | . 2 ⊢ (𝜑 → ((𝑀‘𝑄) ∈ 𝐵 ↔ {(0g‘𝐶)} ( ⋖L ‘𝐶)(𝑀‘𝑄))) |
28 | 22, 27 | mpbird 247 | 1 ⊢ (𝜑 → (𝑀‘𝑄) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 {csn 4316 class class class wbr 4786 ‘cfv 6031 0gc0g 16308 LModclmod 19073 LSubSpclss 19142 LSAtomsclsa 34783 ⋖L clcv 34827 HLchlt 35159 LHypclh 35792 DVecHcdvh 36888 LCDualclcd 37396 mapdcmpd 37434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-riotaBAD 34761 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-tpos 7504 df-undef 7551 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-n0 11495 df-z 11580 df-uz 11889 df-fz 12534 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-sca 16165 df-vsca 16166 df-0g 16310 df-mre 16454 df-mrc 16455 df-acs 16457 df-preset 17136 df-poset 17154 df-plt 17166 df-lub 17182 df-glb 17183 df-join 17184 df-meet 17185 df-p0 17247 df-p1 17248 df-lat 17254 df-clat 17316 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-grp 17633 df-minusg 17634 df-sbg 17635 df-subg 17799 df-cntz 17957 df-oppg 17983 df-lsm 18258 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-ring 18757 df-oppr 18831 df-dvdsr 18849 df-unit 18850 df-invr 18880 df-dvr 18891 df-drng 18959 df-lmod 19075 df-lss 19143 df-lsp 19185 df-lvec 19316 df-lsatoms 34785 df-lshyp 34786 df-lcv 34828 df-lfl 34867 df-lkr 34895 df-ldual 34933 df-oposet 34985 df-ol 34987 df-oml 34988 df-covers 35075 df-ats 35076 df-atl 35107 df-cvlat 35131 df-hlat 35160 df-llines 35306 df-lplanes 35307 df-lvols 35308 df-lines 35309 df-psubsp 35311 df-pmap 35312 df-padd 35604 df-lhyp 35796 df-laut 35797 df-ldil 35912 df-ltrn 35913 df-trl 35968 df-tgrp 36552 df-tendo 36564 df-edring 36566 df-dveca 36812 df-disoa 36839 df-dvech 36889 df-dib 36949 df-dic 36983 df-dih 37039 df-doch 37158 df-djh 37205 df-lcdual 37397 df-mapd 37435 |
This theorem is referenced by: mapdspex 37478 mapdpglem5N 37487 mapdpglem20 37501 mapdpglem30a 37505 mapdpglem30b 37506 |
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