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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 2736 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
4 | eqid 2736 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
5 | eqid 2736 | . . . 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 2736 | . . . 4 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊) | |
10 | eqid 2736 | . . . 4 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
11 | eqid 2736 | . . . 4 ⊢ (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊)) | |
12 | eqid 2736 | . . . 4 ⊢ (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊)) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | dihval 38932 | . . 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 4431 | . . 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 2791 | . 2 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 ≤ 𝑊) → (𝐼‘𝑋) = (𝐷‘𝑋)) |
16 | 15 | anasss 470 | 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 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ifcif 4425 class class class wbr 5039 ‘cfv 6358 ℩crio 7147 (class class class)co 7191 Basecbs 16666 lecple 16756 joincjn 17772 meetcmee 17773 LSSumclsm 18977 LSubSpclss 19922 Atomscatm 36963 LHypclh 37684 DVecHcdvh 38778 DIsoBcdib 38838 DIsoCcdic 38872 DIsoHcdih 38928 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-dih 38929 |
This theorem is referenced by: dihopelvalbN 38938 dih1dimb 38940 dih2dimb 38944 dih2dimbALTN 38945 dihvalcq2 38947 dihlss 38950 dihord6apre 38956 dihord3 38957 dihord5b 38959 dihord5apre 38962 dih0 38980 dihwN 38989 dihglblem3N 38995 dihmeetlem2N 38999 dih1dimatlem 39029 dihjatcclem4 39121 |
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