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Mathbox for Norm Megill |
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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 40800. (Contributed by NM, 29-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mapdat.h | β’ π» = (LHypβπΎ) |
mapdat.m | β’ π = ((mapdβπΎ)βπ) |
mapdat.u | β’ π = ((DVecHβπΎ)βπ) |
mapdat.a | β’ π΄ = (LSAtomsβπ) |
mapdat.c | β’ πΆ = ((LCDualβπΎ)βπ) |
mapdat.b | β’ π΅ = (LSAtomsβπΆ) |
mapdat.k | β’ (π β (πΎ β HL β§ π β π»)) |
mapdcnvat.q | β’ (π β π β π΅) |
Ref | Expression |
---|---|
mapdcnvatN | β’ (π β (β‘πβπ) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdat.h | . . . . 5 β’ π» = (LHypβπΎ) | |
2 | mapdat.m | . . . . 5 β’ π = ((mapdβπΎ)βπ) | |
3 | mapdat.u | . . . . 5 β’ π = ((DVecHβπΎ)βπ) | |
4 | eqid 2731 | . . . . 5 β’ (LSubSpβπ) = (LSubSpβπ) | |
5 | mapdat.k | . . . . 5 β’ (π β (πΎ β HL β§ π β π»)) | |
6 | 1, 3, 5 | dvhlmod 40285 | . . . . . 6 β’ (π β π β LMod) |
7 | eqid 2731 | . . . . . . 7 β’ (0gβπ) = (0gβπ) | |
8 | 7, 4 | lsssn0 20703 | . . . . . 6 β’ (π β LMod β {(0gβπ)} β (LSubSpβπ)) |
9 | 6, 8 | syl 17 | . . . . 5 β’ (π β {(0gβπ)} β (LSubSpβπ)) |
10 | 1, 2, 3, 4, 5, 9 | mapdcnvid1N 40829 | . . . 4 β’ (π β (β‘πβ(πβ{(0gβπ)})) = {(0gβπ)}) |
11 | mapdat.c | . . . . . 6 β’ πΆ = ((LCDualβπΎ)βπ) | |
12 | eqid 2731 | . . . . . 6 β’ (0gβπΆ) = (0gβπΆ) | |
13 | 1, 2, 3, 7, 11, 12, 5 | mapd0 40840 | . . . . 5 β’ (π β (πβ{(0gβπ)}) = {(0gβπΆ)}) |
14 | 13 | fveq2d 6895 | . . . 4 β’ (π β (β‘πβ(πβ{(0gβπ)})) = (β‘πβ{(0gβπΆ)})) |
15 | 10, 14 | eqtr3d 2773 | . . 3 β’ (π β {(0gβπ)} = (β‘πβ{(0gβπΆ)})) |
16 | mapdat.b | . . . . . 6 β’ π΅ = (LSAtomsβπΆ) | |
17 | eqid 2731 | . . . . . 6 β’ ( βL βπΆ) = ( βL βπΆ) | |
18 | 1, 11, 5 | lcdlvec 40766 | . . . . . 6 β’ (π β πΆ β LVec) |
19 | mapdcnvat.q | . . . . . 6 β’ (π β π β π΅) | |
20 | 12, 16, 17, 18, 19 | lsatcv0 38205 | . . . . 5 β’ (π β {(0gβπΆ)} ( βL βπΆ)π) |
21 | 1, 11, 5 | lcdlmod 40767 | . . . . . . . 8 β’ (π β πΆ β LMod) |
22 | eqid 2731 | . . . . . . . . 9 β’ (LSubSpβπΆ) = (LSubSpβπΆ) | |
23 | 12, 22 | lsssn0 20703 | . . . . . . . 8 β’ (πΆ β LMod β {(0gβπΆ)} β (LSubSpβπΆ)) |
24 | 21, 23 | syl 17 | . . . . . . 7 β’ (π β {(0gβπΆ)} β (LSubSpβπΆ)) |
25 | 1, 2, 11, 22, 5 | mapdrn2 40826 | . . . . . . 7 β’ (π β ran π = (LSubSpβπΆ)) |
26 | 24, 25 | eleqtrrd 2835 | . . . . . 6 β’ (π β {(0gβπΆ)} β ran π) |
27 | 1, 2, 5, 26 | mapdcnvid2 40832 | . . . . 5 β’ (π β (πβ(β‘πβ{(0gβπΆ)})) = {(0gβπΆ)}) |
28 | 22, 16, 21, 19 | lsatlssel 38171 | . . . . . . 7 β’ (π β π β (LSubSpβπΆ)) |
29 | 28, 25 | eleqtrrd 2835 | . . . . . 6 β’ (π β π β ran π) |
30 | 1, 2, 5, 29 | mapdcnvid2 40832 | . . . . 5 β’ (π β (πβ(β‘πβπ)) = π) |
31 | 20, 27, 30 | 3brtr4d 5180 | . . . 4 β’ (π β (πβ(β‘πβ{(0gβπΆ)}))( βL βπΆ)(πβ(β‘πβπ))) |
32 | eqid 2731 | . . . . 5 β’ ( βL βπ) = ( βL βπ) | |
33 | 1, 2, 3, 4, 5, 26 | mapdcnvcl 40827 | . . . . 5 β’ (π β (β‘πβ{(0gβπΆ)}) β (LSubSpβπ)) |
34 | 1, 2, 3, 4, 5, 29 | mapdcnvcl 40827 | . . . . 5 β’ (π β (β‘πβπ) β (LSubSpβπ)) |
35 | 1, 2, 3, 4, 32, 11, 17, 5, 33, 34 | mapdcv 40835 | . . . 4 β’ (π β ((β‘πβ{(0gβπΆ)})( βL βπ)(β‘πβπ) β (πβ(β‘πβ{(0gβπΆ)}))( βL βπΆ)(πβ(β‘πβπ)))) |
36 | 31, 35 | mpbird 257 | . . 3 β’ (π β (β‘πβ{(0gβπΆ)})( βL βπ)(β‘πβπ)) |
37 | 15, 36 | eqbrtrd 5170 | . 2 β’ (π β {(0gβπ)} ( βL βπ)(β‘πβπ)) |
38 | mapdat.a | . . 3 β’ π΄ = (LSAtomsβπ) | |
39 | 1, 3, 5 | dvhlvec 40284 | . . 3 β’ (π β π β LVec) |
40 | 7, 4, 38, 32, 39, 34 | lsat0cv 38207 | . 2 β’ (π β ((β‘πβπ) β π΄ β {(0gβπ)} ( βL βπ)(β‘πβπ))) |
41 | 37, 40 | mpbird 257 | 1 β’ (π β (β‘πβπ) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 {csn 4628 class class class wbr 5148 β‘ccnv 5675 ran crn 5677 βcfv 6543 0gc0g 17390 LModclmod 20615 LSubSpclss 20687 LSAtomsclsa 38148 βL clcv 38192 HLchlt 38524 LHypclh 39159 DVecHcdvh 40253 LCDualclcd 40761 mapdcmpd 40799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-riotaBAD 38127 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8215 df-undef 8262 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-0g 17392 df-mre 17535 df-mrc 17536 df-acs 17538 df-proset 18253 df-poset 18271 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-p1 18384 df-lat 18390 df-clat 18457 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 df-cntz 19223 df-oppg 19252 df-lsm 19546 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-drng 20503 df-lmod 20617 df-lss 20688 df-lsp 20728 df-lvec 20859 df-lsatoms 38150 df-lshyp 38151 df-lcv 38193 df-lfl 38232 df-lkr 38260 df-ldual 38298 df-oposet 38350 df-ol 38352 df-oml 38353 df-covers 38440 df-ats 38441 df-atl 38472 df-cvlat 38496 df-hlat 38525 df-llines 38673 df-lplanes 38674 df-lvols 38675 df-lines 38676 df-psubsp 38678 df-pmap 38679 df-padd 38971 df-lhyp 39163 df-laut 39164 df-ldil 39279 df-ltrn 39280 df-trl 39334 df-tgrp 39918 df-tendo 39930 df-edring 39932 df-dveca 40178 df-disoa 40204 df-dvech 40254 df-dib 40314 df-dic 40348 df-dih 40404 df-doch 40523 df-djh 40570 df-lcdual 40762 df-mapd 40800 |
This theorem is referenced by: hdmaprnlem3eN 41033 hdmaprnlem16N 41037 |
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