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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 36838 | . . 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 4128 | . . 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 2705 | . 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 680 | 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 1523 ∈ wcel 2030 ∀wral 2941 ifcif 4119 class class class wbr 4685 ‘cfv 5926 ℩crio 6650 (class class class)co 6690 Basecbs 15904 lecple 15995 joincjn 16991 meetcmee 16992 LSSumclsm 18095 LSubSpclss 18980 Atomscatm 34868 LHypclh 35588 DVecHcdvh 36684 DIsoBcdib 36744 DIsoCcdic 36778 DIsoHcdih 36834 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-dih 36835 |
This theorem is referenced by: dihlsscpre 36840 dihvalcqpre 36841 |
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