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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihvalb | Structured version Visualization version GIF version |
Description: Value of isomorphism H for a lattice 𝐾 when 𝑋 ≤ 𝑊. (Contributed by NM, 4-Mar-2014.) |
Ref | Expression |
---|---|
dihvalb.b | ⊢ 𝐵 = (Base‘𝐾) |
dihvalb.l | ⊢ ≤ = (le‘𝐾) |
dihvalb.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihvalb.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihvalb.d | ⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dihvalb | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (𝐷‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihvalb.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dihvalb.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | eqid 2725 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
4 | eqid 2725 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
5 | eqid 2725 | . . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
6 | dihvalb.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | dihvalb.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
8 | dihvalb.d | . . . 4 ⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊) | |
9 | eqid 2725 | . . . 4 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊) | |
10 | eqid 2725 | . . . 4 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
11 | eqid 2725 | . . . 4 ⊢ (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊)) | |
12 | eqid 2725 | . . . 4 ⊢ (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊)) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | dihval 40855 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) = if(𝑋 ≤ 𝑊, (𝐷‘𝑋), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊))∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) → 𝑢 = ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(𝐷‘(𝑋(meet‘𝐾)𝑊))))))) |
14 | iftrue 4536 | . . 3 ⊢ (𝑋 ≤ 𝑊 → if(𝑋 ≤ 𝑊, (𝐷‘𝑋), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊))∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) → 𝑢 = ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(𝐷‘(𝑋(meet‘𝐾)𝑊)))))) = (𝐷‘𝑋)) | |
15 | 13, 14 | sylan9eq 2785 | . 2 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 ≤ 𝑊) → (𝐼‘𝑋) = (𝐷‘𝑋)) |
16 | 15 | anasss 465 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (𝐷‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ifcif 4530 class class class wbr 5149 ‘cfv 6549 ℩crio 7374 (class class class)co 7419 Basecbs 17199 lecple 17259 joincjn 18322 meetcmee 18323 LSSumclsm 19618 LSubSpclss 20844 Atomscatm 38885 LHypclh 39607 DVecHcdvh 40701 DIsoBcdib 40761 DIsoCcdic 40795 DIsoHcdih 40851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-dih 40852 |
This theorem is referenced by: dihopelvalbN 40861 dih1dimb 40863 dih2dimb 40867 dih2dimbALTN 40868 dihvalcq2 40870 dihlss 40873 dihord6apre 40879 dihord3 40880 dihord5b 40882 dihord5apre 40885 dih0 40903 dihwN 40912 dihglblem3N 40918 dihmeetlem2N 40922 dih1dimatlem 40952 dihjatcclem4 41044 |
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