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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 39262. (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 2738 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
5 | mapdat.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | 1, 3, 5 | dvhlmod 38747 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
7 | eqid 2738 | . . . . . . 7 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
8 | 7, 4 | lsssn0 19838 | . . . . . 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 39291 | . . . 4 ⊢ (𝜑 → (◡𝑀‘(𝑀‘{(0g‘𝑈)})) = {(0g‘𝑈)}) |
11 | mapdat.c | . . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
12 | eqid 2738 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
13 | 1, 2, 3, 7, 11, 12, 5 | mapd0 39302 | . . . . 5 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)}) = {(0g‘𝐶)}) |
14 | 13 | fveq2d 6678 | . . . 4 ⊢ (𝜑 → (◡𝑀‘(𝑀‘{(0g‘𝑈)})) = (◡𝑀‘{(0g‘𝐶)})) |
15 | 10, 14 | eqtr3d 2775 | . . 3 ⊢ (𝜑 → {(0g‘𝑈)} = (◡𝑀‘{(0g‘𝐶)})) |
16 | mapdat.b | . . . . . 6 ⊢ 𝐵 = (LSAtoms‘𝐶) | |
17 | eqid 2738 | . . . . . 6 ⊢ ( ⋖L ‘𝐶) = ( ⋖L ‘𝐶) | |
18 | 1, 11, 5 | lcdlvec 39228 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ LVec) |
19 | mapdcnvat.q | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐵) | |
20 | 12, 16, 17, 18, 19 | lsatcv0 36668 | . . . . 5 ⊢ (𝜑 → {(0g‘𝐶)} ( ⋖L ‘𝐶)𝑄) |
21 | 1, 11, 5 | lcdlmod 39229 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ LMod) |
22 | eqid 2738 | . . . . . . . . 9 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
23 | 12, 22 | lsssn0 19838 | . . . . . . . 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 39288 | . . . . . . 7 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶)) |
26 | 24, 25 | eleqtrrd 2836 | . . . . . 6 ⊢ (𝜑 → {(0g‘𝐶)} ∈ ran 𝑀) |
27 | 1, 2, 5, 26 | mapdcnvid2 39294 | . . . . 5 ⊢ (𝜑 → (𝑀‘(◡𝑀‘{(0g‘𝐶)})) = {(0g‘𝐶)}) |
28 | 22, 16, 21, 19 | lsatlssel 36634 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝐶)) |
29 | 28, 25 | eleqtrrd 2836 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ran 𝑀) |
30 | 1, 2, 5, 29 | mapdcnvid2 39294 | . . . . 5 ⊢ (𝜑 → (𝑀‘(◡𝑀‘𝑄)) = 𝑄) |
31 | 20, 27, 30 | 3brtr4d 5062 | . . . 4 ⊢ (𝜑 → (𝑀‘(◡𝑀‘{(0g‘𝐶)}))( ⋖L ‘𝐶)(𝑀‘(◡𝑀‘𝑄))) |
32 | eqid 2738 | . . . . 5 ⊢ ( ⋖L ‘𝑈) = ( ⋖L ‘𝑈) | |
33 | 1, 2, 3, 4, 5, 26 | mapdcnvcl 39289 | . . . . 5 ⊢ (𝜑 → (◡𝑀‘{(0g‘𝐶)}) ∈ (LSubSp‘𝑈)) |
34 | 1, 2, 3, 4, 5, 29 | mapdcnvcl 39289 | . . . . 5 ⊢ (𝜑 → (◡𝑀‘𝑄) ∈ (LSubSp‘𝑈)) |
35 | 1, 2, 3, 4, 32, 11, 17, 5, 33, 34 | mapdcv 39297 | . . . 4 ⊢ (𝜑 → ((◡𝑀‘{(0g‘𝐶)})( ⋖L ‘𝑈)(◡𝑀‘𝑄) ↔ (𝑀‘(◡𝑀‘{(0g‘𝐶)}))( ⋖L ‘𝐶)(𝑀‘(◡𝑀‘𝑄)))) |
36 | 31, 35 | mpbird 260 | . . 3 ⊢ (𝜑 → (◡𝑀‘{(0g‘𝐶)})( ⋖L ‘𝑈)(◡𝑀‘𝑄)) |
37 | 15, 36 | eqbrtrd 5052 | . 2 ⊢ (𝜑 → {(0g‘𝑈)} ( ⋖L ‘𝑈)(◡𝑀‘𝑄)) |
38 | mapdat.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
39 | 1, 3, 5 | dvhlvec 38746 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LVec) |
40 | 7, 4, 38, 32, 39, 34 | lsat0cv 36670 | . 2 ⊢ (𝜑 → ((◡𝑀‘𝑄) ∈ 𝐴 ↔ {(0g‘𝑈)} ( ⋖L ‘𝑈)(◡𝑀‘𝑄))) |
41 | 37, 40 | mpbird 260 | 1 ⊢ (𝜑 → (◡𝑀‘𝑄) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {csn 4516 class class class wbr 5030 ◡ccnv 5524 ran crn 5526 ‘cfv 6339 0gc0g 16816 LModclmod 19753 LSubSpclss 19822 LSAtomsclsa 36611 ⋖L clcv 36655 HLchlt 36987 LHypclh 37621 DVecHcdvh 38715 LCDualclcd 39223 mapdcmpd 39261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-riotaBAD 36590 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-tpos 7921 df-undef 7968 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-0g 16818 df-mre 16960 df-mrc 16961 df-acs 16963 df-proset 17654 df-poset 17672 df-plt 17684 df-lub 17700 df-glb 17701 df-join 17702 df-meet 17703 df-p0 17765 df-p1 17766 df-lat 17772 df-clat 17834 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-grp 18222 df-minusg 18223 df-sbg 18224 df-subg 18394 df-cntz 18565 df-oppg 18592 df-lsm 18879 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-ring 19418 df-oppr 19495 df-dvdsr 19513 df-unit 19514 df-invr 19544 df-dvr 19555 df-drng 19623 df-lmod 19755 df-lss 19823 df-lsp 19863 df-lvec 19994 df-lsatoms 36613 df-lshyp 36614 df-lcv 36656 df-lfl 36695 df-lkr 36723 df-ldual 36761 df-oposet 36813 df-ol 36815 df-oml 36816 df-covers 36903 df-ats 36904 df-atl 36935 df-cvlat 36959 df-hlat 36988 df-llines 37135 df-lplanes 37136 df-lvols 37137 df-lines 37138 df-psubsp 37140 df-pmap 37141 df-padd 37433 df-lhyp 37625 df-laut 37626 df-ldil 37741 df-ltrn 37742 df-trl 37796 df-tgrp 38380 df-tendo 38392 df-edring 38394 df-dveca 38640 df-disoa 38666 df-dvech 38716 df-dib 38776 df-dic 38810 df-dih 38866 df-doch 38985 df-djh 39032 df-lcdual 39224 df-mapd 39262 |
This theorem is referenced by: hdmaprnlem3eN 39495 hdmaprnlem16N 39499 |
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