Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isltrn | Structured version Visualization version GIF version |
Description: The predicate "is a lattice translation". Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.) |
Ref | Expression |
---|---|
ltrnset.l | ⊢ ≤ = (le‘𝐾) |
ltrnset.j | ⊢ ∨ = (join‘𝐾) |
ltrnset.m | ⊢ ∧ = (meet‘𝐾) |
ltrnset.a | ⊢ 𝐴 = (Atoms‘𝐾) |
ltrnset.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ltrnset.d | ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) |
ltrnset.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
isltrn | ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ (𝐹 ∈ 𝐷 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrnset.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
2 | ltrnset.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
3 | ltrnset.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
4 | ltrnset.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | ltrnset.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | ltrnset.d | . . . 4 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) | |
7 | ltrnset.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ltrnset 34422 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑊 ∈ 𝐻) → 𝑇 = {𝑓 ∈ 𝐷 ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))}) |
9 | 8 | eleq2d 2673 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ 𝐹 ∈ {𝑓 ∈ 𝐷 ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))})) |
10 | fveq1 6102 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑝) = (𝐹‘𝑝)) | |
11 | 10 | oveq2d 6565 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑝 ∨ (𝑓‘𝑝)) = (𝑝 ∨ (𝐹‘𝑝))) |
12 | 11 | oveq1d 6564 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊)) |
13 | fveq1 6102 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑞) = (𝐹‘𝑞)) | |
14 | 13 | oveq2d 6565 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑞 ∨ (𝑓‘𝑞)) = (𝑞 ∨ (𝐹‘𝑞))) |
15 | 14 | oveq1d 6564 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) |
16 | 12, 15 | eqeq12d 2625 | . . . . 5 ⊢ (𝑓 = 𝐹 → (((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊) ↔ ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))) |
17 | 16 | imbi2d 329 | . . . 4 ⊢ (𝑓 = 𝐹 → (((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊)) ↔ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))) |
18 | 17 | 2ralbidv 2972 | . . 3 ⊢ (𝑓 = 𝐹 → (∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊)) ↔ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))) |
19 | 18 | elrab 3331 | . 2 ⊢ (𝐹 ∈ {𝑓 ∈ 𝐷 ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))} ↔ (𝐹 ∈ 𝐷 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))) |
20 | 9, 19 | syl6bb 275 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ (𝐹 ∈ 𝐷 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 lecple 15775 joincjn 16767 meetcmee 16768 Atomscatm 33568 LHypclh 34288 LDilcldil 34404 LTrncltrn 34405 |
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-ov 6552 df-ltrn 34409 |
This theorem is referenced by: isltrn2N 34424 ltrnu 34425 ltrnldil 34426 ltrncnv 34450 idltrn 34454 cdleme50ltrn 34863 ltrnco 35025 |
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