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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrnldil | Structured version Visualization version GIF version |
Description: A lattice translation is a lattice dilation. (Contributed by NM, 20-May-2012.) |
Ref | Expression |
---|---|
ltrnldil.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ltrnldil.d | ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) |
ltrnldil.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
ltrnldil | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | eqid 2735 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
3 | eqid 2735 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
4 | eqid 2735 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
5 | ltrnldil.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | ltrnldil.d | . . 3 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) | |
7 | ltrnldil.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isltrn 40102 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ (𝐹 ∈ 𝐷 ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(𝐹‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(𝐹‘𝑞))(meet‘𝐾)𝑊))))) |
9 | 8 | simprbda 498 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 lecple 17305 joincjn 18369 meetcmee 18370 Atomscatm 39245 LHypclh 39967 LDilcldil 40083 LTrncltrn 40084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-ltrn 40088 |
This theorem is referenced by: ltrnlaut 40106 ltrnval1 40117 ltrncnv 40129 ltrnco 40702 |
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