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| 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 41234 | . . 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 4534 | . . 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 2797 | . 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 466 | 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 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ifcif 4525 class class class wbr 5143 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 Basecbs 17247 lecple 17304 joincjn 18357 meetcmee 18358 LSSumclsm 19652 LSubSpclss 20929 Atomscatm 39264 LHypclh 39986 DVecHcdvh 41080 DIsoBcdib 41140 DIsoCcdic 41174 DIsoHcdih 41230 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-dih 41231 |
| This theorem is referenced by: dihlsscpre 41236 dihvalcqpre 41237 |
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