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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdrn | Structured version Visualization version GIF version |
Description: Range of the map defined by df-mapd 40201. (Contributed by NM, 12-Mar-2015.) |
Ref | Expression |
---|---|
mapdrn.h | β’ π» = (LHypβπΎ) |
mapdrn.o | β’ π = ((ocHβπΎ)βπ) |
mapdrn.m | β’ π = ((mapdβπΎ)βπ) |
mapdrn.u | β’ π = ((DVecHβπΎ)βπ) |
mapdrn.f | β’ πΉ = (LFnlβπ) |
mapdrn.l | β’ πΏ = (LKerβπ) |
mapdrn.d | β’ π· = (LDualβπ) |
mapdrn.t | β’ π = (LSubSpβπ·) |
mapdrn.c | β’ πΆ = {π β πΉ β£ (πβ(πβ(πΏβπ))) = (πΏβπ)} |
mapdrn.k | β’ (π β (πΎ β HL β§ π β π»)) |
Ref | Expression |
---|---|
mapdrn | β’ (π β ran π = (π β© π« πΆ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdrn.h | . . 3 β’ π» = (LHypβπΎ) | |
2 | mapdrn.o | . . 3 β’ π = ((ocHβπΎ)βπ) | |
3 | mapdrn.m | . . 3 β’ π = ((mapdβπΎ)βπ) | |
4 | mapdrn.u | . . 3 β’ π = ((DVecHβπΎ)βπ) | |
5 | eqid 2731 | . . 3 β’ (LSubSpβπ) = (LSubSpβπ) | |
6 | mapdrn.f | . . 3 β’ πΉ = (LFnlβπ) | |
7 | mapdrn.l | . . 3 β’ πΏ = (LKerβπ) | |
8 | mapdrn.d | . . 3 β’ π· = (LDualβπ) | |
9 | mapdrn.t | . . 3 β’ π = (LSubSpβπ·) | |
10 | mapdrn.c | . . 3 β’ πΆ = {π β πΉ β£ (πβ(πβ(πΏβπ))) = (πΏβπ)} | |
11 | mapdrn.k | . . 3 β’ (π β (πΎ β HL β§ π β π»)) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | mapd1o 40224 | . 2 β’ (π β π:(LSubSpβπ)β1-1-ontoβ(π β© π« πΆ)) |
13 | f1ofo 6818 | . 2 β’ (π:(LSubSpβπ)β1-1-ontoβ(π β© π« πΆ) β π:(LSubSpβπ)βontoβ(π β© π« πΆ)) | |
14 | forn 6786 | . 2 β’ (π:(LSubSpβπ)βontoβ(π β© π« πΆ) β ran π = (π β© π« πΆ)) | |
15 | 12, 13, 14 | 3syl 18 | 1 β’ (π β ran π = (π β© π« πΆ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 {crab 3425 β© cin 3934 π« cpw 4587 ran crn 5661 βontoβwfo 6521 β1-1-ontoβwf1o 6522 βcfv 6523 LSubSpclss 20471 LFnlclfn 37632 LKerclk 37660 LDualcld 37698 HLchlt 37925 LHypclh 38560 DVecHcdvh 39654 ocHcoch 39923 mapdcmpd 40200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5269 ax-sep 5283 ax-nul 5290 ax-pow 5347 ax-pr 5411 ax-un 7699 ax-cnex 11138 ax-resscn 11139 ax-1cn 11140 ax-icn 11141 ax-addcl 11142 ax-addrcl 11143 ax-mulcl 11144 ax-mulrcl 11145 ax-mulcom 11146 ax-addass 11147 ax-mulass 11148 ax-distr 11149 ax-i2m1 11150 ax-1ne0 11151 ax-1rid 11152 ax-rnegex 11153 ax-rrecex 11154 ax-cnre 11155 ax-pre-lttri 11156 ax-pre-lttrn 11157 ax-pre-ltadd 11158 ax-pre-mulgt0 11159 ax-riotaBAD 37528 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 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 3371 df-reu 3372 df-rab 3426 df-v 3468 df-sbc 3765 df-csb 3881 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-pss 3954 df-nul 4310 df-if 4514 df-pw 4589 df-sn 4614 df-pr 4616 df-tp 4618 df-op 4620 df-uni 4893 df-int 4935 df-iun 4983 df-iin 4984 df-br 5133 df-opab 5195 df-mpt 5216 df-tr 5250 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5615 df-we 5617 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6280 df-ord 6347 df-on 6348 df-lim 6349 df-suc 6350 df-iota 6475 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7340 df-ov 7387 df-oprab 7388 df-mpo 7389 df-of 7644 df-om 7830 df-1st 7948 df-2nd 7949 df-tpos 8184 df-undef 8231 df-frecs 8239 df-wrecs 8270 df-recs 8344 df-rdg 8383 df-1o 8439 df-er 8677 df-map 8796 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-pnf 11222 df-mnf 11223 df-xr 11224 df-ltxr 11225 df-le 11226 df-sub 11418 df-neg 11419 df-nn 12185 df-2 12247 df-3 12248 df-4 12249 df-5 12250 df-6 12251 df-n0 12445 df-z 12531 df-uz 12795 df-fz 13457 df-struct 17052 df-sets 17069 df-slot 17087 df-ndx 17099 df-base 17117 df-ress 17146 df-plusg 17182 df-mulr 17183 df-sca 17185 df-vsca 17186 df-0g 17359 df-mre 17502 df-mrc 17503 df-acs 17505 df-proset 18220 df-poset 18238 df-plt 18255 df-lub 18271 df-glb 18272 df-join 18273 df-meet 18274 df-p0 18350 df-p1 18351 df-lat 18357 df-clat 18424 df-mgm 18533 df-sgrp 18582 df-mnd 18593 df-submnd 18638 df-grp 18787 df-minusg 18788 df-sbg 18789 df-subg 18961 df-cntz 19133 df-oppg 19160 df-lsm 19454 df-cmn 19600 df-abl 19601 df-mgp 19933 df-ur 19950 df-ring 20002 df-oppr 20085 df-dvdsr 20106 df-unit 20107 df-invr 20137 df-dvr 20148 df-drng 20249 df-lmod 20402 df-lss 20472 df-lsp 20512 df-lvec 20643 df-lsatoms 37551 df-lshyp 37552 df-lcv 37594 df-lfl 37633 df-lkr 37661 df-ldual 37699 df-oposet 37751 df-ol 37753 df-oml 37754 df-covers 37841 df-ats 37842 df-atl 37873 df-cvlat 37897 df-hlat 37926 df-llines 38074 df-lplanes 38075 df-lvols 38076 df-lines 38077 df-psubsp 38079 df-pmap 38080 df-padd 38372 df-lhyp 38564 df-laut 38565 df-ldil 38680 df-ltrn 38681 df-trl 38735 df-tgrp 39319 df-tendo 39331 df-edring 39333 df-dveca 39579 df-disoa 39605 df-dvech 39655 df-dib 39715 df-dic 39749 df-dih 39805 df-doch 39924 df-djh 39971 df-mapd 40201 |
This theorem is referenced by: mapdunirnN 40226 mapdrn2 40227 |
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