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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 35539 | . . 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 4045 | . . 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 2664 | . 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 677 | 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 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: dihlsscpre 35541 dihvalcqpre 35542 |
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