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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dih0 | Structured version Visualization version GIF version |
Description: The value of isomorphism H at the lattice zero is the singleton of the zero vector i.e. the zero subspace. (Contributed by NM, 9-Mar-2014.) |
Ref | Expression |
---|---|
dih0.z | ⊢ 0 = (0.‘𝐾) |
dih0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dih0.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dih0.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dih0.o | ⊢ 𝑂 = (0g‘𝑈) |
Ref | Expression |
---|---|
dih0 | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {𝑂}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | hlop 35891 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
3 | 2 | adantr 473 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ OP) |
4 | eqid 2772 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
5 | dih0.z | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
6 | 4, 5 | op0cl 35713 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) |
7 | 3, 6 | syl 17 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ (Base‘𝐾)) |
8 | dih0.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | 4, 8 | lhpbase 36527 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
10 | eqid 2772 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
11 | 4, 10, 5 | op0le 35715 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑊) |
12 | 2, 9, 11 | syl2an 586 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 (le‘𝐾)𝑊) |
13 | dih0.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
14 | eqid 2772 | . . . 4 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊) | |
15 | 4, 10, 8, 13, 14 | dihvalb 37766 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( 0 ∈ (Base‘𝐾) ∧ 0 (le‘𝐾)𝑊)) → (𝐼‘ 0 ) = (((DIsoB‘𝐾)‘𝑊)‘ 0 )) |
16 | 1, 7, 12, 15 | syl12anc 824 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = (((DIsoB‘𝐾)‘𝑊)‘ 0 )) |
17 | dih0.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
18 | dih0.o | . . 3 ⊢ 𝑂 = (0g‘𝑈) | |
19 | 5, 8, 14, 17, 18 | dib0 37693 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((DIsoB‘𝐾)‘𝑊)‘ 0 ) = {𝑂}) |
20 | 16, 19 | eqtrd 2808 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {𝑂}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 {csn 4435 class class class wbr 4923 ‘cfv 6182 Basecbs 16329 lecple 16418 0gc0g 16559 0.cp0 17495 OPcops 35701 HLchlt 35879 LHypclh 36513 DVecHcdvh 37607 DIsoBcdib 37667 DIsoHcdih 37757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-riotaBAD 35482 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-tpos 7688 df-undef 7735 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-n0 11701 df-z 11787 df-uz 12052 df-fz 12702 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-sca 16427 df-vsca 16428 df-0g 16561 df-proset 17386 df-poset 17404 df-plt 17416 df-lub 17432 df-glb 17433 df-join 17434 df-meet 17435 df-p0 17497 df-p1 17498 df-lat 17504 df-clat 17566 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-grp 17884 df-minusg 17885 df-mgp 18953 df-ur 18965 df-ring 19012 df-oppr 19086 df-dvdsr 19104 df-unit 19105 df-invr 19135 df-dvr 19146 df-drng 19217 df-lmod 19348 df-lvec 19587 df-oposet 35705 df-ol 35707 df-oml 35708 df-covers 35795 df-ats 35796 df-atl 35827 df-cvlat 35851 df-hlat 35880 df-llines 36027 df-lplanes 36028 df-lvols 36029 df-lines 36030 df-psubsp 36032 df-pmap 36033 df-padd 36325 df-lhyp 36517 df-laut 36518 df-ldil 36633 df-ltrn 36634 df-trl 36688 df-tendo 37284 df-edring 37286 df-disoa 37558 df-dvech 37608 df-dib 37668 df-dih 37758 |
This theorem is referenced by: dih0bN 37810 dih0vbN 37811 dih0cnv 37812 dih0rn 37813 dihmeetlem4preN 37835 dihmeetlem18N 37853 dihlspsnssN 37861 dihlspsnat 37862 dihatexv 37867 doch1 37888 dochnoncon 37920 |
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