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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihsslss | Structured version Visualization version GIF version |
Description: The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.) |
Ref | Expression |
---|---|
dihsslss.h | β’ π» = (LHypβπΎ) |
dihsslss.u | β’ π = ((DVecHβπΎ)βπ) |
dihsslss.i | β’ πΌ = ((DIsoHβπΎ)βπ) |
dihsslss.s | β’ π = (LSubSpβπ) |
Ref | Expression |
---|---|
dihsslss | β’ ((πΎ β HL β§ π β π») β ran πΌ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihsslss.h | . . . . 5 β’ π» = (LHypβπΎ) | |
2 | dihsslss.i | . . . . 5 β’ πΌ = ((DIsoHβπΎ)βπ) | |
3 | 1, 2 | dihcnvid2 40786 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π₯ β ran πΌ) β (πΌβ(β‘πΌβπ₯)) = π₯) |
4 | eqid 2728 | . . . . . 6 β’ (BaseβπΎ) = (BaseβπΎ) | |
5 | 4, 1, 2 | dihcnvcl 40784 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ π₯ β ran πΌ) β (β‘πΌβπ₯) β (BaseβπΎ)) |
6 | dihsslss.u | . . . . . 6 β’ π = ((DVecHβπΎ)βπ) | |
7 | dihsslss.s | . . . . . 6 β’ π = (LSubSpβπ) | |
8 | 4, 1, 2, 6, 7 | dihlss 40763 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ (β‘πΌβπ₯) β (BaseβπΎ)) β (πΌβ(β‘πΌβπ₯)) β π) |
9 | 5, 8 | syldan 589 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π₯ β ran πΌ) β (πΌβ(β‘πΌβπ₯)) β π) |
10 | 3, 9 | eqeltrrd 2830 | . . 3 β’ (((πΎ β HL β§ π β π») β§ π₯ β ran πΌ) β π₯ β π) |
11 | 10 | ex 411 | . 2 β’ ((πΎ β HL β§ π β π») β (π₯ β ran πΌ β π₯ β π)) |
12 | 11 | ssrdv 3988 | 1 β’ ((πΎ β HL β§ π β π») β ran πΌ β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wss 3949 β‘ccnv 5681 ran crn 5683 βcfv 6553 Basecbs 17189 LSubSpclss 20829 HLchlt 38862 LHypclh 39497 DVecHcdvh 40591 DIsoHcdih 40741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-riotaBAD 38465 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-tpos 8240 df-undef 8287 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-n0 12513 df-z 12599 df-uz 12863 df-fz 13527 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-sca 17258 df-vsca 17259 df-0g 17432 df-proset 18296 df-poset 18314 df-plt 18331 df-lub 18347 df-glb 18348 df-join 18349 df-meet 18350 df-p0 18426 df-p1 18427 df-lat 18433 df-clat 18500 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-submnd 18750 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19092 df-cntz 19282 df-lsm 19605 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-oppr 20287 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-dvr 20354 df-drng 20640 df-lmod 20759 df-lss 20830 df-lsp 20870 df-lvec 21002 df-oposet 38688 df-ol 38690 df-oml 38691 df-covers 38778 df-ats 38779 df-atl 38810 df-cvlat 38834 df-hlat 38863 df-llines 39011 df-lplanes 39012 df-lvols 39013 df-lines 39014 df-psubsp 39016 df-pmap 39017 df-padd 39309 df-lhyp 39501 df-laut 39502 df-ldil 39617 df-ltrn 39618 df-trl 39672 df-tendo 40268 df-edring 40270 df-disoa 40542 df-dvech 40592 df-dib 40652 df-dic 40686 df-dih 40742 |
This theorem is referenced by: dihrnlss 40790 dihlspsnssN 40845 dochspss 40891 dochpolN 41003 |
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