Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > dihvalc | 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 |
---|---|
dihval.b | ⊢ 𝐵 = (Base‘𝐾) |
dihval.l | ⊢ ≤ = (le‘𝐾) |
dihval.j | ⊢ ∨ = (join‘𝐾) |
dihval.m | ⊢ ∧ = (meet‘𝐾) |
dihval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dihval.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihval.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihval.d | ⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊) |
dihval.c | ⊢ 𝐶 = ((DIsoC‘𝐾)‘𝑊) |
dihval.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dihval.s | ⊢ 𝑆 = (LSubSp‘𝑈) |
dihval.p | ⊢ ⊕ = (LSSum‘𝑈) |
Ref | Expression |
---|---|
dihvalc | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dihval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | dihval.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
4 | dihval.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
5 | dihval.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | dihval.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | dihval.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
8 | dihval.d | . . . 4 ⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊) | |
9 | dihval.c | . . . 4 ⊢ 𝐶 = ((DIsoC‘𝐾)‘𝑊) | |
10 | dihval.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
11 | dihval.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑈) | |
12 | dihval.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑈) | |
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 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))))) |
14 | iffalse 4434 | . . 3 ⊢ (¬ 𝑋 ≤ 𝑊 → if(𝑋 ≤ 𝑊, (𝐷‘𝑋), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))))) = (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))))) | |
15 | 13, 14 | sylan9eq 2791 | . 2 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑋 ≤ 𝑊) → (𝐼‘𝑋) = (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))))) |
16 | 15 | anasss 470 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))))) |
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: dihlsscpre 38934 dihvalcqpre 38935 |
Copyright terms: Public domain | W3C validator |