![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 40402. (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 2733 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
5 | mapdat.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | 1, 3, 5 | dvhlmod 39887 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
7 | eqid 2733 | . . . . . . 7 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
8 | 7, 4 | lsssn0 20535 | . . . . . 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 40431 | . . . 4 ⊢ (𝜑 → (◡𝑀‘(𝑀‘{(0g‘𝑈)})) = {(0g‘𝑈)}) |
11 | mapdat.c | . . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
12 | eqid 2733 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
13 | 1, 2, 3, 7, 11, 12, 5 | mapd0 40442 | . . . . 5 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)}) = {(0g‘𝐶)}) |
14 | 13 | fveq2d 6885 | . . . 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 2733 | . . . . . 6 ⊢ ( ⋖L ‘𝐶) = ( ⋖L ‘𝐶) | |
18 | 1, 11, 5 | lcdlvec 40368 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ LVec) |
19 | mapdcnvat.q | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐵) | |
20 | 12, 16, 17, 18, 19 | lsatcv0 37807 | . . . . 5 ⊢ (𝜑 → {(0g‘𝐶)} ( ⋖L ‘𝐶)𝑄) |
21 | 1, 11, 5 | lcdlmod 40369 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ LMod) |
22 | eqid 2733 | . . . . . . . . 9 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
23 | 12, 22 | lsssn0 20535 | . . . . . . . 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 40428 | . . . . . . 7 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶)) |
26 | 24, 25 | eleqtrrd 2837 | . . . . . 6 ⊢ (𝜑 → {(0g‘𝐶)} ∈ ran 𝑀) |
27 | 1, 2, 5, 26 | mapdcnvid2 40434 | . . . . 5 ⊢ (𝜑 → (𝑀‘(◡𝑀‘{(0g‘𝐶)})) = {(0g‘𝐶)}) |
28 | 22, 16, 21, 19 | lsatlssel 37773 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝐶)) |
29 | 28, 25 | eleqtrrd 2837 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ran 𝑀) |
30 | 1, 2, 5, 29 | mapdcnvid2 40434 | . . . . 5 ⊢ (𝜑 → (𝑀‘(◡𝑀‘𝑄)) = 𝑄) |
31 | 20, 27, 30 | 3brtr4d 5176 | . . . 4 ⊢ (𝜑 → (𝑀‘(◡𝑀‘{(0g‘𝐶)}))( ⋖L ‘𝐶)(𝑀‘(◡𝑀‘𝑄))) |
32 | eqid 2733 | . . . . 5 ⊢ ( ⋖L ‘𝑈) = ( ⋖L ‘𝑈) | |
33 | 1, 2, 3, 4, 5, 26 | mapdcnvcl 40429 | . . . . 5 ⊢ (𝜑 → (◡𝑀‘{(0g‘𝐶)}) ∈ (LSubSp‘𝑈)) |
34 | 1, 2, 3, 4, 5, 29 | mapdcnvcl 40429 | . . . . 5 ⊢ (𝜑 → (◡𝑀‘𝑄) ∈ (LSubSp‘𝑈)) |
35 | 1, 2, 3, 4, 32, 11, 17, 5, 33, 34 | mapdcv 40437 | . . . 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 5166 | . 2 ⊢ (𝜑 → {(0g‘𝑈)} ( ⋖L ‘𝑈)(◡𝑀‘𝑄)) |
38 | mapdat.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
39 | 1, 3, 5 | dvhlvec 39886 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LVec) |
40 | 7, 4, 38, 32, 39, 34 | lsat0cv 37809 | . 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 397 = wceq 1542 ∈ wcel 2107 {csn 4624 class class class wbr 5144 ◡ccnv 5671 ran crn 5673 ‘cfv 6535 0gc0g 17372 LModclmod 20448 LSubSpclss 20519 LSAtomsclsa 37750 ⋖L clcv 37794 HLchlt 38126 LHypclh 38761 DVecHcdvh 39855 LCDualclcd 40363 mapdcmpd 40401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-riotaBAD 37729 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-iin 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7657 df-om 7843 df-1st 7962 df-2nd 7963 df-tpos 8198 df-undef 8245 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-n0 12460 df-z 12546 df-uz 12810 df-fz 13472 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-sca 17200 df-vsca 17201 df-0g 17374 df-mre 17517 df-mrc 17518 df-acs 17520 df-proset 18235 df-poset 18253 df-plt 18270 df-lub 18286 df-glb 18287 df-join 18288 df-meet 18289 df-p0 18365 df-p1 18366 df-lat 18372 df-clat 18439 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-submnd 18659 df-grp 18809 df-minusg 18810 df-sbg 18811 df-subg 18988 df-cntz 19166 df-oppg 19194 df-lsm 19488 df-cmn 19634 df-abl 19635 df-mgp 19971 df-ur 19988 df-ring 20040 df-oppr 20128 df-dvdsr 20149 df-unit 20150 df-invr 20180 df-dvr 20193 df-drng 20295 df-lmod 20450 df-lss 20520 df-lsp 20560 df-lvec 20691 df-lsatoms 37752 df-lshyp 37753 df-lcv 37795 df-lfl 37834 df-lkr 37862 df-ldual 37900 df-oposet 37952 df-ol 37954 df-oml 37955 df-covers 38042 df-ats 38043 df-atl 38074 df-cvlat 38098 df-hlat 38127 df-llines 38275 df-lplanes 38276 df-lvols 38277 df-lines 38278 df-psubsp 38280 df-pmap 38281 df-padd 38573 df-lhyp 38765 df-laut 38766 df-ldil 38881 df-ltrn 38882 df-trl 38936 df-tgrp 39520 df-tendo 39532 df-edring 39534 df-dveca 39780 df-disoa 39806 df-dvech 39856 df-dib 39916 df-dic 39950 df-dih 40006 df-doch 40125 df-djh 40172 df-lcdual 40364 df-mapd 40402 |
This theorem is referenced by: hdmaprnlem3eN 40635 hdmaprnlem16N 40639 |
Copyright terms: Public domain | W3C validator |