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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdcnvatN | Structured version Visualization version GIF version |
Description: Atoms are preserved by the map defined by df-mapd 41531. (Contributed by NM, 29-May-2015.) (New usage is discouraged.) |
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)) |
mapdcnvat.q | ⊢ (𝜑 → 𝑄 ∈ 𝐵) |
Ref | Expression |
---|---|
mapdcnvatN | ⊢ (𝜑 → (◡𝑀‘𝑄) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdat.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdat.m | . . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
3 | mapdat.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | eqid 2734 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
5 | mapdat.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | 1, 3, 5 | dvhlmod 41016 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
7 | eqid 2734 | . . . . . . 7 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
8 | 7, 4 | lsssn0 20964 | . . . . . 6 ⊢ (𝑈 ∈ LMod → {(0g‘𝑈)} ∈ (LSubSp‘𝑈)) |
9 | 6, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → {(0g‘𝑈)} ∈ (LSubSp‘𝑈)) |
10 | 1, 2, 3, 4, 5, 9 | mapdcnvid1N 41560 | . . . 4 ⊢ (𝜑 → (◡𝑀‘(𝑀‘{(0g‘𝑈)})) = {(0g‘𝑈)}) |
11 | mapdat.c | . . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
12 | eqid 2734 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
13 | 1, 2, 3, 7, 11, 12, 5 | mapd0 41571 | . . . . 5 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)}) = {(0g‘𝐶)}) |
14 | 13 | fveq2d 6923 | . . . 4 ⊢ (𝜑 → (◡𝑀‘(𝑀‘{(0g‘𝑈)})) = (◡𝑀‘{(0g‘𝐶)})) |
15 | 10, 14 | eqtr3d 2776 | . . 3 ⊢ (𝜑 → {(0g‘𝑈)} = (◡𝑀‘{(0g‘𝐶)})) |
16 | mapdat.b | . . . . . 6 ⊢ 𝐵 = (LSAtoms‘𝐶) | |
17 | eqid 2734 | . . . . . 6 ⊢ ( ⋖L ‘𝐶) = ( ⋖L ‘𝐶) | |
18 | 1, 11, 5 | lcdlvec 41497 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ LVec) |
19 | mapdcnvat.q | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐵) | |
20 | 12, 16, 17, 18, 19 | lsatcv0 38936 | . . . . 5 ⊢ (𝜑 → {(0g‘𝐶)} ( ⋖L ‘𝐶)𝑄) |
21 | 1, 11, 5 | lcdlmod 41498 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ LMod) |
22 | eqid 2734 | . . . . . . . . 9 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
23 | 12, 22 | lsssn0 20964 | . . . . . . . 8 ⊢ (𝐶 ∈ LMod → {(0g‘𝐶)} ∈ (LSubSp‘𝐶)) |
24 | 21, 23 | syl 17 | . . . . . . 7 ⊢ (𝜑 → {(0g‘𝐶)} ∈ (LSubSp‘𝐶)) |
25 | 1, 2, 11, 22, 5 | mapdrn2 41557 | . . . . . . 7 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶)) |
26 | 24, 25 | eleqtrrd 2841 | . . . . . 6 ⊢ (𝜑 → {(0g‘𝐶)} ∈ ran 𝑀) |
27 | 1, 2, 5, 26 | mapdcnvid2 41563 | . . . . 5 ⊢ (𝜑 → (𝑀‘(◡𝑀‘{(0g‘𝐶)})) = {(0g‘𝐶)}) |
28 | 22, 16, 21, 19 | lsatlssel 38902 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝐶)) |
29 | 28, 25 | eleqtrrd 2841 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ran 𝑀) |
30 | 1, 2, 5, 29 | mapdcnvid2 41563 | . . . . 5 ⊢ (𝜑 → (𝑀‘(◡𝑀‘𝑄)) = 𝑄) |
31 | 20, 27, 30 | 3brtr4d 5201 | . . . 4 ⊢ (𝜑 → (𝑀‘(◡𝑀‘{(0g‘𝐶)}))( ⋖L ‘𝐶)(𝑀‘(◡𝑀‘𝑄))) |
32 | eqid 2734 | . . . . 5 ⊢ ( ⋖L ‘𝑈) = ( ⋖L ‘𝑈) | |
33 | 1, 2, 3, 4, 5, 26 | mapdcnvcl 41558 | . . . . 5 ⊢ (𝜑 → (◡𝑀‘{(0g‘𝐶)}) ∈ (LSubSp‘𝑈)) |
34 | 1, 2, 3, 4, 5, 29 | mapdcnvcl 41558 | . . . . 5 ⊢ (𝜑 → (◡𝑀‘𝑄) ∈ (LSubSp‘𝑈)) |
35 | 1, 2, 3, 4, 32, 11, 17, 5, 33, 34 | mapdcv 41566 | . . . 4 ⊢ (𝜑 → ((◡𝑀‘{(0g‘𝐶)})( ⋖L ‘𝑈)(◡𝑀‘𝑄) ↔ (𝑀‘(◡𝑀‘{(0g‘𝐶)}))( ⋖L ‘𝐶)(𝑀‘(◡𝑀‘𝑄)))) |
36 | 31, 35 | mpbird 257 | . . 3 ⊢ (𝜑 → (◡𝑀‘{(0g‘𝐶)})( ⋖L ‘𝑈)(◡𝑀‘𝑄)) |
37 | 15, 36 | eqbrtrd 5191 | . 2 ⊢ (𝜑 → {(0g‘𝑈)} ( ⋖L ‘𝑈)(◡𝑀‘𝑄)) |
38 | mapdat.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
39 | 1, 3, 5 | dvhlvec 41015 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LVec) |
40 | 7, 4, 38, 32, 39, 34 | lsat0cv 38938 | . 2 ⊢ (𝜑 → ((◡𝑀‘𝑄) ∈ 𝐴 ↔ {(0g‘𝑈)} ( ⋖L ‘𝑈)(◡𝑀‘𝑄))) |
41 | 37, 40 | mpbird 257 | 1 ⊢ (𝜑 → (◡𝑀‘𝑄) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2103 {csn 4648 class class class wbr 5169 ◡ccnv 5698 ran crn 5700 ‘cfv 6572 0gc0g 17494 LModclmod 20875 LSubSpclss 20947 LSAtomsclsa 38879 ⋖L clcv 38923 HLchlt 39255 LHypclh 39890 DVecHcdvh 40984 LCDualclcd 41492 mapdcmpd 41530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-riotaBAD 38858 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-iin 5022 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-om 7900 df-1st 8026 df-2nd 8027 df-tpos 8263 df-undef 8310 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-er 8759 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-n0 12550 df-z 12636 df-uz 12900 df-fz 13564 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-sca 17322 df-vsca 17323 df-0g 17496 df-mre 17639 df-mrc 17640 df-acs 17642 df-proset 18360 df-poset 18378 df-plt 18395 df-lub 18411 df-glb 18412 df-join 18413 df-meet 18414 df-p0 18490 df-p1 18491 df-lat 18497 df-clat 18564 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-submnd 18814 df-grp 18971 df-minusg 18972 df-sbg 18973 df-subg 19158 df-cntz 19352 df-oppg 19381 df-lsm 19673 df-cmn 19819 df-abl 19820 df-mgp 20157 df-rng 20175 df-ur 20204 df-ring 20257 df-oppr 20355 df-dvdsr 20378 df-unit 20379 df-invr 20409 df-dvr 20422 df-nzr 20534 df-rlreg 20711 df-domn 20712 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38881 df-lshyp 38882 df-lcv 38924 df-lfl 38963 df-lkr 38991 df-ldual 39029 df-oposet 39081 df-ol 39083 df-oml 39084 df-covers 39171 df-ats 39172 df-atl 39203 df-cvlat 39227 df-hlat 39256 df-llines 39404 df-lplanes 39405 df-lvols 39406 df-lines 39407 df-psubsp 39409 df-pmap 39410 df-padd 39702 df-lhyp 39894 df-laut 39895 df-ldil 40010 df-ltrn 40011 df-trl 40065 df-tgrp 40649 df-tendo 40661 df-edring 40663 df-dveca 40909 df-disoa 40935 df-dvech 40985 df-dib 41045 df-dic 41079 df-dih 41135 df-doch 41254 df-djh 41301 df-lcdual 41493 df-mapd 41531 |
This theorem is referenced by: hdmaprnlem3eN 41764 hdmaprnlem16N 41768 |
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