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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 34466 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝑅 = (𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))) |
10 | 9 | fveq1d 6105 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑅‘𝐹) = ((𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))‘𝐹)) |
11 | fveq1 6102 | . . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑝) = (𝐹‘𝑝)) | |
12 | 11 | oveq2d 6565 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑝 ∨ (𝑓‘𝑝)) = (𝑝 ∨ (𝐹‘𝑝))) |
13 | 12 | oveq1d 6564 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊)) |
14 | 13 | eqeq2d 2620 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) ↔ 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊))) |
15 | 14 | imbi2d 329 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)) ↔ (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊)))) |
16 | 15 | ralbidv 2969 | . . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)) ↔ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊)))) |
17 | 16 | riotabidv 6513 | . . 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 2610 | . . 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 6515 | . . 3 ⊢ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊))) ∈ V | |
20 | 17, 18, 19 | fvmpt 6191 | . 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 2664 | 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 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 Basecbs 15695 lecple 15775 joincjn 16767 meetcmee 16768 Atomscatm 33568 LHypclh 34288 LTrncltrn 34405 trLctrl 34463 |
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-trl 34464 |
This theorem is referenced by: trlval2 34468 |
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