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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihvalcqat | Structured version Visualization version GIF version |
Description: Value of isomorphism H for a lattice 𝐾 at an atom not under 𝑊. (Contributed by NM, 27-Mar-2014.) |
Ref | Expression |
---|---|
dihvalcqat.l | ⊢ ≤ = (le‘𝐾) |
dihvalcqat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dihvalcqat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihvalcqat.j | ⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊) |
dihvalcqat.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dihvalcqat | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = (𝐽‘𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | eqid 2818 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | dihvalcqat.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 2, 3 | atbase 36305 | . . . 4 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
5 | 4 | ad2antrl 724 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝑄 ∈ (Base‘𝐾)) |
6 | simprr 769 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ¬ 𝑄 ≤ 𝑊) | |
7 | simpr 485 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) | |
8 | dihvalcqat.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
9 | eqid 2818 | . . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
10 | eqid 2818 | . . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
11 | dihvalcqat.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
12 | 8, 9, 10, 3, 11 | lhpmat 37046 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄(meet‘𝐾)𝑊) = (0.‘𝐾)) |
13 | 12 | oveq2d 7161 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄(join‘𝐾)(𝑄(meet‘𝐾)𝑊)) = (𝑄(join‘𝐾)(0.‘𝐾))) |
14 | hlol 36377 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) | |
15 | 14 | ad2antrr 722 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐾 ∈ OL) |
16 | eqid 2818 | . . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾) | |
17 | 2, 16, 10 | olj01 36241 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑄(join‘𝐾)(0.‘𝐾)) = 𝑄) |
18 | 15, 5, 17 | syl2anc 584 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄(join‘𝐾)(0.‘𝐾)) = 𝑄) |
19 | 13, 18 | eqtrd 2853 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄(join‘𝐾)(𝑄(meet‘𝐾)𝑊)) = 𝑄) |
20 | dihvalcqat.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
21 | eqid 2818 | . . . 4 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊) | |
22 | dihvalcqat.j | . . . 4 ⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊) | |
23 | eqid 2818 | . . . 4 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
24 | eqid 2818 | . . . 4 ⊢ (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊)) | |
25 | 2, 8, 16, 9, 3, 11, 20, 21, 22, 23, 24 | dihvalcq 38252 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ (Base‘𝐾) ∧ ¬ 𝑄 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄(join‘𝐾)(𝑄(meet‘𝐾)𝑊)) = 𝑄)) → (𝐼‘𝑄) = ((𝐽‘𝑄)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑄(meet‘𝐾)𝑊)))) |
26 | 1, 5, 6, 7, 19, 25 | syl122anc 1371 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = ((𝐽‘𝑄)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑄(meet‘𝐾)𝑊)))) |
27 | 12 | fveq2d 6667 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((DIsoB‘𝐾)‘𝑊)‘(𝑄(meet‘𝐾)𝑊)) = (((DIsoB‘𝐾)‘𝑊)‘(0.‘𝐾))) |
28 | eqid 2818 | . . . . . . 7 ⊢ (0g‘((DVecH‘𝐾)‘𝑊)) = (0g‘((DVecH‘𝐾)‘𝑊)) | |
29 | 10, 11, 21, 23, 28 | dib0 38180 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((DIsoB‘𝐾)‘𝑊)‘(0.‘𝐾)) = {(0g‘((DVecH‘𝐾)‘𝑊))}) |
30 | 29 | adantr 481 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((DIsoB‘𝐾)‘𝑊)‘(0.‘𝐾)) = {(0g‘((DVecH‘𝐾)‘𝑊))}) |
31 | 27, 30 | eqtrd 2853 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((DIsoB‘𝐾)‘𝑊)‘(𝑄(meet‘𝐾)𝑊)) = {(0g‘((DVecH‘𝐾)‘𝑊))}) |
32 | 31 | oveq2d 7161 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐽‘𝑄)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑄(meet‘𝐾)𝑊))) = ((𝐽‘𝑄)(LSSum‘((DVecH‘𝐾)‘𝑊)){(0g‘((DVecH‘𝐾)‘𝑊))})) |
33 | 11, 23, 1 | dvhlmod 38126 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((DVecH‘𝐾)‘𝑊) ∈ LMod) |
34 | eqid 2818 | . . . . . 6 ⊢ (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊)) | |
35 | 8, 3, 11, 23, 22, 34 | diclss 38209 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐽‘𝑄) ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊))) |
36 | 34 | lsssubg 19658 | . . . . 5 ⊢ ((((DVecH‘𝐾)‘𝑊) ∈ LMod ∧ (𝐽‘𝑄) ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊))) → (𝐽‘𝑄) ∈ (SubGrp‘((DVecH‘𝐾)‘𝑊))) |
37 | 33, 35, 36 | syl2anc 584 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐽‘𝑄) ∈ (SubGrp‘((DVecH‘𝐾)‘𝑊))) |
38 | 28, 24 | lsm01 18726 | . . . 4 ⊢ ((𝐽‘𝑄) ∈ (SubGrp‘((DVecH‘𝐾)‘𝑊)) → ((𝐽‘𝑄)(LSSum‘((DVecH‘𝐾)‘𝑊)){(0g‘((DVecH‘𝐾)‘𝑊))}) = (𝐽‘𝑄)) |
39 | 37, 38 | syl 17 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐽‘𝑄)(LSSum‘((DVecH‘𝐾)‘𝑊)){(0g‘((DVecH‘𝐾)‘𝑊))}) = (𝐽‘𝑄)) |
40 | 32, 39 | eqtrd 2853 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐽‘𝑄)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑄(meet‘𝐾)𝑊))) = (𝐽‘𝑄)) |
41 | 26, 40 | eqtrd 2853 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = (𝐽‘𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {csn 4557 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 lecple 16560 0gc0g 16701 joincjn 17542 meetcmee 17543 0.cp0 17635 SubGrpcsubg 18211 LSSumclsm 18688 LModclmod 19563 LSubSpclss 19632 OLcol 36190 Atomscatm 36279 HLchlt 36366 LHypclh 37000 DVecHcdvh 38094 DIsoBcdib 38154 DIsoCcdic 38188 DIsoHcdih 38244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-riotaBAD 35969 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-undef 7928 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-0g 16703 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 df-lsm 18690 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 df-lvols 36516 df-lines 36517 df-psubsp 36519 df-pmap 36520 df-padd 36812 df-lhyp 37004 df-laut 37005 df-ldil 37120 df-ltrn 37121 df-trl 37175 df-tendo 37771 df-edring 37773 df-disoa 38045 df-dvech 38095 df-dib 38155 df-dic 38189 df-dih 38245 |
This theorem is referenced by: dih1dimc 38258 dihopelvalcqat 38262 dihvalcq2 38263 dih1dimatlem 38345 dihpN 38352 |
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