Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 | ⊢ 𝐻 = (LHyp‘𝐾) |
dihlsprn.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dihlsprn.v | ⊢ 𝑉 = (Base‘𝑈) |
dihlsprn.n | ⊢ 𝑁 = (LSpan‘𝑈) |
dihlsprn.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dihlsprn | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 = (0g‘𝑈)) → 𝑋 = (0g‘𝑈)) | |
2 | 1 | sneqd 4579 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 = (0g‘𝑈)) → {𝑋} = {(0g‘𝑈)}) |
3 | 2 | fveq2d 6775 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 = (0g‘𝑈)) → (𝑁‘{𝑋}) = (𝑁‘{(0g‘𝑈)})) |
4 | dihlsprn.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | dihlsprn.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | simpll 764 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 = (0g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | 4, 5, 6 | dvhlmod 39120 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 = (0g‘𝑈)) → 𝑈 ∈ LMod) |
8 | eqid 2740 | . . . . . 6 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
9 | dihlsprn.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈) | |
10 | 8, 9 | lspsn0 20268 | . . . . 5 ⊢ (𝑈 ∈ LMod → (𝑁‘{(0g‘𝑈)}) = {(0g‘𝑈)}) |
11 | 7, 10 | syl 17 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 = (0g‘𝑈)) → (𝑁‘{(0g‘𝑈)}) = {(0g‘𝑈)}) |
12 | 3, 11 | eqtrd 2780 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 = (0g‘𝑈)) → (𝑁‘{𝑋}) = {(0g‘𝑈)}) |
13 | dihlsprn.i | . . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
14 | 4, 13, 5, 8 | dih0rn 39294 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → {(0g‘𝑈)} ∈ ran 𝐼) |
15 | 14 | ad2antrr 723 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 = (0g‘𝑈)) → {(0g‘𝑈)} ∈ ran 𝐼) |
16 | 12, 15 | eqeltrd 2841 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 = (0g‘𝑈)) → (𝑁‘{𝑋}) ∈ ran 𝐼) |
17 | simpll 764 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
18 | 4, 5, 17 | dvhlmod 39120 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0g‘𝑈)) → 𝑈 ∈ LMod) |
19 | simplr 766 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0g‘𝑈)) → 𝑋 ∈ 𝑉) | |
20 | simpr 485 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0g‘𝑈)) → 𝑋 ≠ (0g‘𝑈)) | |
21 | dihlsprn.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
22 | eqid 2740 | . . . . 5 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
23 | 21, 9, 8, 22 | lsatlspsn2 37002 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ (0g‘𝑈)) → (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) |
24 | 18, 19, 20, 23 | syl3anc 1370 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0g‘𝑈)) → (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) |
25 | 4, 5, 13, 22 | dih1dimat 39340 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) → (𝑁‘{𝑋}) ∈ ran 𝐼) |
26 | 17, 24, 25 | syl2anc 584 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0g‘𝑈)) → (𝑁‘{𝑋}) ∈ ran 𝐼) |
27 | 16, 26 | pm2.61dane 3034 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 {csn 4567 ran crn 5591 ‘cfv 6432 Basecbs 16910 0gc0g 17148 LModclmod 20121 LSpanclspn 20231 LSAtomsclsa 36984 HLchlt 37360 LHypclh 37994 DVecHcdvh 39088 DIsoHcdih 39238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-riotaBAD 36963 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-tpos 8033 df-undef 8080 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-0g 17150 df-proset 18011 df-poset 18029 df-plt 18046 df-lub 18062 df-glb 18063 df-join 18064 df-meet 18065 df-p0 18141 df-p1 18142 df-lat 18148 df-clat 18215 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-cntz 18921 df-lsm 19239 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-dvr 19923 df-drng 19991 df-lmod 20123 df-lss 20192 df-lsp 20232 df-lvec 20363 df-lsatoms 36986 df-oposet 37186 df-ol 37188 df-oml 37189 df-covers 37276 df-ats 37277 df-atl 37308 df-cvlat 37332 df-hlat 37361 df-llines 37508 df-lplanes 37509 df-lvols 37510 df-lines 37511 df-psubsp 37513 df-pmap 37514 df-padd 37806 df-lhyp 37998 df-laut 37999 df-ldil 38114 df-ltrn 38115 df-trl 38169 df-tendo 38765 df-edring 38767 df-disoa 39039 df-dvech 39089 df-dib 39149 df-dic 39183 df-dih 39239 |
This theorem is referenced by: dihlspsnssN 39342 dihlspsnat 39343 dihatexv2 39349 dochocsn 39391 dochsncom 39392 djhcvat42 39425 dihprrnlem1N 39434 dihprrnlem2 39435 dihprrn 39436 dihjat1lem 39438 dihsmsnrn 39445 dochsatshpb 39462 dochsnkr2cl 39484 lcfl7lem 39509 lclkrlem2a 39517 lclkrlem2c 39519 lcfrlem14 39566 hdmapoc 39941 |
Copyright terms: Public domain | W3C validator |