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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 38370 | . . 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 4478 | . . 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 2878 | . 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 469 | 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 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ifcif 4469 class class class wbr 5068 ‘cfv 6357 ℩crio 7115 (class class class)co 7158 Basecbs 16485 lecple 16574 joincjn 17556 meetcmee 17557 LSSumclsm 18761 LSubSpclss 19705 Atomscatm 36401 LHypclh 37122 DVecHcdvh 38216 DIsoBcdib 38276 DIsoCcdic 38310 DIsoHcdih 38366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-dih 38367 |
This theorem is referenced by: dihlsscpre 38372 dihvalcqpre 38373 |
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