![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 38536 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
3 | 2 | adantr 480 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ OP) |
4 | eqid 2731 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
5 | dih0.z | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
6 | 4, 5 | op0cl 38358 | . . . 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 39173 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
10 | eqid 2731 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
11 | 4, 10, 5 | op0le 38360 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑊) |
12 | 2, 9, 11 | syl2an 595 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 (le‘𝐾)𝑊) |
13 | dih0.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
14 | eqid 2731 | . . . 4 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊) | |
15 | 4, 10, 8, 13, 14 | dihvalb 40412 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( 0 ∈ (Base‘𝐾) ∧ 0 (le‘𝐾)𝑊)) → (𝐼‘ 0 ) = (((DIsoB‘𝐾)‘𝑊)‘ 0 )) |
16 | 1, 7, 12, 15 | syl12anc 834 | . 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 40339 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((DIsoB‘𝐾)‘𝑊)‘ 0 ) = {𝑂}) |
20 | 16, 19 | eqtrd 2771 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {𝑂}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 {csn 4628 class class class wbr 5148 ‘cfv 6543 Basecbs 17149 lecple 17209 0gc0g 17390 0.cp0 18381 OPcops 38346 HLchlt 38524 LHypclh 39159 DVecHcdvh 40253 DIsoBcdib 40313 DIsoHcdih 40403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-riotaBAD 38127 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8215 df-undef 8262 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-0g 17392 df-proset 18253 df-poset 18271 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-p1 18384 df-lat 18390 df-clat 18457 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-minusg 18860 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-drng 20503 df-lmod 20617 df-lvec 20859 df-oposet 38350 df-ol 38352 df-oml 38353 df-covers 38440 df-ats 38441 df-atl 38472 df-cvlat 38496 df-hlat 38525 df-llines 38673 df-lplanes 38674 df-lvols 38675 df-lines 38676 df-psubsp 38678 df-pmap 38679 df-padd 38971 df-lhyp 39163 df-laut 39164 df-ldil 39279 df-ltrn 39280 df-trl 39334 df-tendo 39930 df-edring 39932 df-disoa 40204 df-dvech 40254 df-dib 40314 df-dih 40404 |
This theorem is referenced by: dih0bN 40456 dih0vbN 40457 dih0cnv 40458 dih0rn 40459 dihmeetlem4preN 40481 dihmeetlem18N 40499 dihlspsnssN 40507 dihlspsnat 40508 dihatexv 40513 doch1 40534 dochnoncon 40566 |
Copyright terms: Public domain | W3C validator |