Nearby theorems
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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 483 | . . . . . 6 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π = (0gβπ)) β π = (0gβπ)) | |
| 2 | 1 | sneqd 4644 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π = (0gβπ)) β {π} = {(0gβπ)}) |
| 3 | 2 | fveq2d 6906 | . . . 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 765 | . . . . . 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 40615 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π = (0gβπ)) β π β LMod) |
| 8 | eqid 2728 | . . . . . 6 β’ (0gβπ) = (0gβπ) | |
| 9 | dihlsprn.n | . . . . . 6 β’ π = (LSpanβπ) | |
| 10 | 8, 9 | lspsn0 20899 | . . . . 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 2768 | . . 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 40789 | . . . 4 β’ ((πΎ β HL β§ π β π») β {(0gβπ)} β ran πΌ) |
| 15 | 14 | ad2antrr 724 | . . 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 2829 | . 2 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π = (0gβπ)) β (πβ{π}) β ran πΌ) |
| 17 | simpll 765 | . . 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 40615 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π β (0gβπ)) β π β LMod) |
| 19 | simplr 767 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ π β π) β§ π β (0gβπ)) β π β π) | |
| 20 | simpr 483 | . . . 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 2728 | . . . . 5 β’ (LSAtomsβπ) = (LSAtomsβπ) | |
| 23 | 21, 9, 8, 22 | lsatlspsn2 38496 | . . . 4 β’ ((π β LMod β§ π β π β§ π β (0gβπ)) β (πβ{π}) β (LSAtomsβπ)) |
| 24 | 18, 19, 20, 23 | syl3anc 1368 | . . 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 40835 | . . 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 582 | . 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 3026 | 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 394 = wceq 1533 β wcel 2098 β wne 2937 {csn 4632 ran crn 5683 βcfv 6553 Basecbs 17187 0gc0g 17428 LModclmod 20750 LSpanclspn 20862 LSAtomsclsa 38478 HLchlt 38854 LHypclh 39489 DVecHcdvh 40583 DIsoHcdih 40733 |
| 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 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-riotaBAD 38457 |
| 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 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-undef 8285 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-0g 17430 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-p1 18425 df-lat 18431 df-clat 18498 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-cntz 19275 df-lsm 19598 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-dvr 20347 df-drng 20633 df-lmod 20752 df-lss 20823 df-lsp 20863 df-lvec 20995 df-lsatoms 38480 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-llines 39003 df-lplanes 39004 df-lvols 39005 df-lines 39006 df-psubsp 39008 df-pmap 39009 df-padd 39301 df-lhyp 39493 df-laut 39494 df-ldil 39609 df-ltrn 39610 df-trl 39664 df-tendo 40260 df-edring 40262 df-disoa 40534 df-dvech 40584 df-dib 40644 df-dic 40678 df-dih 40734 |
| This theorem is referenced by: dihlspsnssN 40837 dihlspsnat 40838 dihatexv2 40844 dochocsn 40886 dochsncom 40887 djhcvat42 40920 dihprrnlem1N 40929 dihprrnlem2 40930 dihprrn 40931 dihjat1lem 40933 dihsmsnrn 40940 dochsatshpb 40957 dochsnkr2cl 40979 lcfl7lem 41004 lclkrlem2a 41012 lclkrlem2c 41014 lcfrlem14 41061 hdmapoc 41436 |
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