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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 2821 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | eqid 2821 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
3 | eqid 2821 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
4 | eqid 2821 | . . 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 37249 | . 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 501 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 lecple 16566 joincjn 17548 meetcmee 17549 Atomscatm 36393 LHypclh 37114 LDilcldil 37230 LTrncltrn 37231 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-ltrn 37235 |
This theorem is referenced by: ltrnlaut 37253 ltrnval1 37264 ltrncnv 37276 ltrnco 37849 |
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