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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 2610 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
4 | eqid 2610 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
5 | eqid 2610 | . . . 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 2610 | . . . 4 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊) | |
10 | eqid 2610 | . . . 4 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
11 | eqid 2610 | . . . 4 ⊢ (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊)) | |
12 | eqid 2610 | . . . 4 ⊢ (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊)) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | dihval 35539 | . . 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 4042 | . . 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 2664 | . 2 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 ≤ 𝑊) → (𝐼‘𝑋) = (𝐷‘𝑋)) |
16 | 15 | anasss 677 | 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 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ifcif 4036 class class class wbr 4583 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 Basecbs 15695 lecple 15775 joincjn 16767 meetcmee 16768 LSSumclsm 17872 LSubSpclss 18753 Atomscatm 33568 LHypclh 34288 DVecHcdvh 35385 DIsoBcdib 35445 DIsoCcdic 35479 DIsoHcdih 35535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-dih 35536 |
This theorem is referenced by: dihopelvalbN 35545 dih1dimb 35547 dih2dimb 35551 dih2dimbALTN 35552 dihvalcq2 35554 dihlss 35557 dihord6apre 35563 dihord3 35564 dihord5b 35566 dihord5apre 35569 dih0 35587 dihwN 35596 dihglblem3N 35602 dihmeetlem2N 35606 dih1dimatlem 35636 dihjatcclem4 35728 |
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