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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > trlval | Structured version Visualization version GIF version |
Description: The value of the trace of a lattice translation. (Contributed by NM, 20-May-2012.) |
Ref | Expression |
---|---|
trlset.b | ⊢ 𝐵 = (Base‘𝐾) |
trlset.l | ⊢ ≤ = (le‘𝐾) |
trlset.j | ⊢ ∨ = (join‘𝐾) |
trlset.m | ⊢ ∧ = (meet‘𝐾) |
trlset.a | ⊢ 𝐴 = (Atoms‘𝐾) |
trlset.h | ⊢ 𝐻 = (LHyp‘𝐾) |
trlset.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
trlset.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
trlval | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) = (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlset.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | trlset.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | trlset.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
4 | trlset.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
5 | trlset.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | trlset.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | trlset.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | trlset.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | trlset 39629 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝑅 = (𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))) |
10 | 9 | fveq1d 6894 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑅‘𝐹) = ((𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))‘𝐹)) |
11 | fveq1 6891 | . . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑝) = (𝐹‘𝑝)) | |
12 | 11 | oveq2d 7431 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑝 ∨ (𝑓‘𝑝)) = (𝑝 ∨ (𝐹‘𝑝))) |
13 | 12 | oveq1d 7430 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊)) |
14 | 13 | eqeq2d 2739 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) ↔ 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊))) |
15 | 14 | imbi2d 340 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)) ↔ (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊)))) |
16 | 15 | ralbidv 3173 | . . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)) ↔ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊)))) |
17 | 16 | riotabidv 7373 | . . 3 ⊢ (𝑓 = 𝐹 → (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))) = (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊)))) |
18 | eqid 2728 | . . 3 ⊢ (𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))) = (𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))) | |
19 | riotaex 7375 | . . 3 ⊢ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊))) ∈ V | |
20 | 17, 18, 19 | fvmpt 7000 | . 2 ⊢ (𝐹 ∈ 𝑇 → ((𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))‘𝐹) = (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊)))) |
21 | 10, 20 | sylan9eq 2788 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) = (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 class class class wbr 5143 ↦ cmpt 5226 ‘cfv 6543 ℩crio 7370 (class class class)co 7415 Basecbs 17174 lecple 17234 joincjn 18297 meetcmee 18298 Atomscatm 38730 LHypclh 39452 LTrncltrn 39569 trLctrl 39626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-trl 39627 |
This theorem is referenced by: trlval2 39631 |
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