![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dih1dimat | Structured version Visualization version GIF version |
Description: Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014.) |
Ref | Expression |
---|---|
dih1dimat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dih1dimat.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dih1dimat.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dih1dimat.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
Ref | Expression |
---|---|
dih1dimat | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ran 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dih1dimat.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dih1dimat.u | . 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | dih1dimat.i | . 2 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
4 | dih1dimat.a | . 2 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
5 | eqid 2736 | . 2 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | eqid 2736 | . 2 ⊢ (le‘𝐾) = (le‘𝐾) | |
7 | eqid 2736 | . 2 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
8 | eqid 2736 | . 2 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊) | |
9 | eqid 2736 | . 2 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
10 | eqid 2736 | . 2 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
11 | eqid 2736 | . 2 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊) | |
12 | eqid 2736 | . 2 ⊢ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) | |
13 | eqid 2736 | . 2 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
14 | eqid 2736 | . 2 ⊢ (invr‘(Scalar‘𝑈)) = (invr‘(Scalar‘𝑈)) | |
15 | eqid 2736 | . 2 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
16 | eqid 2736 | . 2 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
17 | eqid 2736 | . 2 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
18 | eqid 2736 | . 2 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈) | |
19 | eqid 2736 | . 2 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
20 | eqid 2736 | . 2 ⊢ (℩ℎ ∈ ((LTrn‘𝐾)‘𝑊)(ℎ‘((oc‘𝐾)‘𝑊)) = ((((invr‘(Scalar‘𝑈))‘𝑠)‘𝑓)‘((oc‘𝐾)‘𝑊))) = (℩ℎ ∈ ((LTrn‘𝐾)‘𝑊)(ℎ‘((oc‘𝐾)‘𝑊)) = ((((invr‘(Scalar‘𝑈))‘𝑠)‘𝑓)‘((oc‘𝐾)‘𝑊))) | |
21 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 | dih1dimatlem 39782 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ↦ cmpt 5188 I cid 5530 ran crn 5634 ↾ cres 5635 ‘cfv 6496 ℩crio 7311 Basecbs 17082 Scalarcsca 17135 ·𝑠 cvsca 17136 lecple 17139 occoc 17140 0gc0g 17320 invrcinvr 20098 LSubSpclss 20390 LSpanclspn 20430 LSAtomsclsa 37426 Atomscatm 37715 HLchlt 37802 LHypclh 38437 LTrncltrn 38554 trLctrl 38611 TEndoctendo 39205 DVecHcdvh 39531 DIsoHcdih 39681 |
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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-riotaBAD 37405 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-tpos 8156 df-undef 8203 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-map 8766 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-n0 12413 df-z 12499 df-uz 12763 df-fz 13424 df-struct 17018 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-mulr 17146 df-sca 17148 df-vsca 17149 df-0g 17322 df-proset 18183 df-poset 18201 df-plt 18218 df-lub 18234 df-glb 18235 df-join 18236 df-meet 18237 df-p0 18313 df-p1 18314 df-lat 18320 df-clat 18387 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-submnd 18601 df-grp 18750 df-minusg 18751 df-sbg 18752 df-subg 18923 df-cntz 19095 df-lsm 19416 df-cmn 19562 df-abl 19563 df-mgp 19895 df-ur 19912 df-ring 19964 df-oppr 20047 df-dvdsr 20068 df-unit 20069 df-invr 20099 df-dvr 20110 df-drng 20185 df-lmod 20322 df-lss 20391 df-lsp 20431 df-lvec 20562 df-lsatoms 37428 df-oposet 37628 df-ol 37630 df-oml 37631 df-covers 37718 df-ats 37719 df-atl 37750 df-cvlat 37774 df-hlat 37803 df-llines 37951 df-lplanes 37952 df-lvols 37953 df-lines 37954 df-psubsp 37956 df-pmap 37957 df-padd 38249 df-lhyp 38441 df-laut 38442 df-ldil 38557 df-ltrn 38558 df-trl 38612 df-tendo 39208 df-edring 39210 df-disoa 39482 df-dvech 39532 df-dib 39592 df-dic 39626 df-dih 39682 |
This theorem is referenced by: dihlsprn 39784 dihglblem6 39793 dochsat 39836 dihjat4 39886 dihjat6 39887 dvh4dimat 39891 dochsatshp 39904 dochpolN 39943 lcfrlem23 40018 mapdordlem2 40090 |
Copyright terms: Public domain | W3C validator |