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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 2818 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | eqid 2818 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
3 | eqid 2818 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
4 | eqid 2818 | . . 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 37135 | . 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 499 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 lecple 16560 joincjn 17542 meetcmee 17543 Atomscatm 36279 LHypclh 37000 LDilcldil 37116 LTrncltrn 37117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-ltrn 37121 |
This theorem is referenced by: ltrnlaut 37139 ltrnval1 37150 ltrncnv 37162 ltrnco 37735 |
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