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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 2760 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
4 | eqid 2760 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
5 | eqid 2760 | . . . 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 2760 | . . . 4 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊) | |
10 | eqid 2760 | . . . 4 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
11 | eqid 2760 | . . . 4 ⊢ (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊)) | |
12 | eqid 2760 | . . . 4 ⊢ (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊)) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | dihval 37023 | . . 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 4236 | . . 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 2814 | . 2 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 ≤ 𝑊) → (𝐼‘𝑋) = (𝐷‘𝑋)) |
16 | 15 | anasss 682 | 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 1632 ∈ wcel 2139 ∀wral 3050 ifcif 4230 class class class wbr 4804 ‘cfv 6049 ℩crio 6773 (class class class)co 6813 Basecbs 16059 lecple 16150 joincjn 17145 meetcmee 17146 LSSumclsm 18249 LSubSpclss 19134 Atomscatm 35053 LHypclh 35773 DVecHcdvh 36869 DIsoBcdib 36929 DIsoCcdic 36963 DIsoHcdih 37019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-dih 37020 |
This theorem is referenced by: dihopelvalbN 37029 dih1dimb 37031 dih2dimb 37035 dih2dimbALTN 37036 dihvalcq2 37038 dihlss 37041 dihord6apre 37047 dihord3 37048 dihord5b 37050 dihord5apre 37053 dih0 37071 dihwN 37080 dihglblem3N 37086 dihmeetlem2N 37090 dih1dimatlem 37120 dihjatcclem4 37212 |
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