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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihlsprn | Structured version Visualization version GIF version |
Description: The span of a vector belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.) |
Ref | Expression |
---|---|
dihlsprn.h | β’ π» = (LHypβπΎ) |
dihlsprn.u | β’ π = ((DVecHβπΎ)βπ) |
dihlsprn.v | β’ π = (Baseβπ) |
dihlsprn.n | β’ π = (LSpanβπ) |
dihlsprn.i | β’ πΌ = ((DIsoHβπΎ)βπ) |
Ref | Expression |
---|---|
dihlsprn | β’ (((πΎ β HL β§ π β π») β§ π β π) β (πβ{π}) β ran πΌ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 486 | . . . . . 6 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π = (0gβπ)) β π = (0gβπ)) | |
2 | 1 | sneqd 4602 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π = (0gβπ)) β {π} = {(0gβπ)}) |
3 | 2 | fveq2d 6850 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π = (0gβπ)) β (πβ{π}) = (πβ{(0gβπ)})) |
4 | dihlsprn.h | . . . . . 6 β’ π» = (LHypβπΎ) | |
5 | dihlsprn.u | . . . . . 6 β’ π = ((DVecHβπΎ)βπ) | |
6 | simpll 766 | . . . . . 6 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π = (0gβπ)) β (πΎ β HL β§ π β π»)) | |
7 | 4, 5, 6 | dvhlmod 39623 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π = (0gβπ)) β π β LMod) |
8 | eqid 2733 | . . . . . 6 β’ (0gβπ) = (0gβπ) | |
9 | dihlsprn.n | . . . . . 6 β’ π = (LSpanβπ) | |
10 | 8, 9 | lspsn0 20513 | . . . . 5 β’ (π β LMod β (πβ{(0gβπ)}) = {(0gβπ)}) |
11 | 7, 10 | syl 17 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π = (0gβπ)) β (πβ{(0gβπ)}) = {(0gβπ)}) |
12 | 3, 11 | eqtrd 2773 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π = (0gβπ)) β (πβ{π}) = {(0gβπ)}) |
13 | dihlsprn.i | . . . . 5 β’ πΌ = ((DIsoHβπΎ)βπ) | |
14 | 4, 13, 5, 8 | dih0rn 39797 | . . . 4 β’ ((πΎ β HL β§ π β π») β {(0gβπ)} β ran πΌ) |
15 | 14 | ad2antrr 725 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π = (0gβπ)) β {(0gβπ)} β ran πΌ) |
16 | 12, 15 | eqeltrd 2834 | . 2 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π = (0gβπ)) β (πβ{π}) β ran πΌ) |
17 | simpll 766 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π β (0gβπ)) β (πΎ β HL β§ π β π»)) | |
18 | 4, 5, 17 | dvhlmod 39623 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π β (0gβπ)) β π β LMod) |
19 | simplr 768 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π β (0gβπ)) β π β π) | |
20 | simpr 486 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π β (0gβπ)) β π β (0gβπ)) | |
21 | dihlsprn.v | . . . . 5 β’ π = (Baseβπ) | |
22 | eqid 2733 | . . . . 5 β’ (LSAtomsβπ) = (LSAtomsβπ) | |
23 | 21, 9, 8, 22 | lsatlspsn2 37504 | . . . 4 β’ ((π β LMod β§ π β π β§ π β (0gβπ)) β (πβ{π}) β (LSAtomsβπ)) |
24 | 18, 19, 20, 23 | syl3anc 1372 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π β (0gβπ)) β (πβ{π}) β (LSAtomsβπ)) |
25 | 4, 5, 13, 22 | dih1dimat 39843 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (πβ{π}) β (LSAtomsβπ)) β (πβ{π}) β ran πΌ) |
26 | 17, 24, 25 | syl2anc 585 | . 2 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π β (0gβπ)) β (πβ{π}) β ran πΌ) |
27 | 16, 26 | pm2.61dane 3029 | 1 β’ (((πΎ β HL β§ π β π») β§ π β π) β (πβ{π}) β ran πΌ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 {csn 4590 ran crn 5638 βcfv 6500 Basecbs 17091 0gc0g 17329 LModclmod 20365 LSpanclspn 20476 LSAtomsclsa 37486 HLchlt 37862 LHypclh 38497 DVecHcdvh 39591 DIsoHcdih 39741 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-riotaBAD 37465 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-tpos 8161 df-undef 8208 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-0g 17331 df-proset 18192 df-poset 18210 df-plt 18227 df-lub 18243 df-glb 18244 df-join 18245 df-meet 18246 df-p0 18322 df-p1 18323 df-lat 18329 df-clat 18396 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-grp 18759 df-minusg 18760 df-sbg 18761 df-subg 18933 df-cntz 19105 df-lsm 19426 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-dvr 20120 df-drng 20221 df-lmod 20367 df-lss 20437 df-lsp 20477 df-lvec 20608 df-lsatoms 37488 df-oposet 37688 df-ol 37690 df-oml 37691 df-covers 37778 df-ats 37779 df-atl 37810 df-cvlat 37834 df-hlat 37863 df-llines 38011 df-lplanes 38012 df-lvols 38013 df-lines 38014 df-psubsp 38016 df-pmap 38017 df-padd 38309 df-lhyp 38501 df-laut 38502 df-ldil 38617 df-ltrn 38618 df-trl 38672 df-tendo 39268 df-edring 39270 df-disoa 39542 df-dvech 39592 df-dib 39652 df-dic 39686 df-dih 39742 |
This theorem is referenced by: dihlspsnssN 39845 dihlspsnat 39846 dihatexv2 39852 dochocsn 39894 dochsncom 39895 djhcvat42 39928 dihprrnlem1N 39937 dihprrnlem2 39938 dihprrn 39939 dihjat1lem 39941 dihsmsnrn 39948 dochsatshpb 39965 dochsnkr2cl 39987 lcfl7lem 40012 lclkrlem2a 40020 lclkrlem2c 40022 lcfrlem14 40069 hdmapoc 40444 |
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