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Theorem ltrnset 37134
Description: The set of lattice translations for a fiducial co-atom 𝑊. (Contributed by NM, 11-May-2012.)
Hypotheses
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)
Assertion
Ref Expression
ltrnset ((𝐾𝐵𝑊𝐻) → 𝑇 = {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))})
Distinct variable groups:   𝑞,𝑝,𝐴   𝐷,𝑓   𝑓,𝑝,𝑞,𝐾   𝑓,𝑊,𝑝,𝑞
Allowed substitution hints:   𝐴(𝑓)   𝐵(𝑓,𝑞,𝑝)   𝐷(𝑞,𝑝)   𝑇(𝑓,𝑞,𝑝)   𝐻(𝑓,𝑞,𝑝)   (𝑓,𝑞,𝑝)   (𝑓,𝑞,𝑝)   (𝑓,𝑞,𝑝)

Proof of Theorem ltrnset
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ltrnset.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
2 ltrnset.l . . . . 5 = (le‘𝐾)
3 ltrnset.j . . . . 5 = (join‘𝐾)
4 ltrnset.m . . . . 5 = (meet‘𝐾)
5 ltrnset.a . . . . 5 𝐴 = (Atoms‘𝐾)
6 ltrnset.h . . . . 5 𝐻 = (LHyp‘𝐾)
72, 3, 4, 5, 6ltrnfset 37133 . . . 4 (𝐾𝐵 → (LTrn‘𝐾) = (𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))}))
87fveq1d 6665 . . 3 (𝐾𝐵 → ((LTrn‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))})‘𝑊))
91, 8syl5eq 2865 . 2 (𝐾𝐵𝑇 = ((𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))})‘𝑊))
10 fveq2 6663 . . . . 5 (𝑤 = 𝑊 → ((LDil‘𝐾)‘𝑤) = ((LDil‘𝐾)‘𝑊))
11 ltrnset.d . . . . 5 𝐷 = ((LDil‘𝐾)‘𝑊)
1210, 11syl6eqr 2871 . . . 4 (𝑤 = 𝑊 → ((LDil‘𝐾)‘𝑤) = 𝐷)
13 breq2 5061 . . . . . . . 8 (𝑤 = 𝑊 → (𝑝 𝑤𝑝 𝑊))
1413notbid 319 . . . . . . 7 (𝑤 = 𝑊 → (¬ 𝑝 𝑤 ↔ ¬ 𝑝 𝑊))
15 breq2 5061 . . . . . . . 8 (𝑤 = 𝑊 → (𝑞 𝑤𝑞 𝑊))
1615notbid 319 . . . . . . 7 (𝑤 = 𝑊 → (¬ 𝑞 𝑤 ↔ ¬ 𝑞 𝑊))
1714, 16anbi12d 630 . . . . . 6 (𝑤 = 𝑊 → ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) ↔ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
18 oveq2 7153 . . . . . . 7 (𝑤 = 𝑊 → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑝 (𝑓𝑝)) 𝑊))
19 oveq2 7153 . . . . . . 7 (𝑤 = 𝑊 → ((𝑞 (𝑓𝑞)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑊))
2018, 19eqeq12d 2834 . . . . . 6 (𝑤 = 𝑊 → (((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤) ↔ ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊)))
2117, 20imbi12d 346 . . . . 5 (𝑤 = 𝑊 → (((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤)) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))))
22212ralbidv 3196 . . . 4 (𝑤 = 𝑊 → (∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤)) ↔ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))))
2312, 22rabeqbidv 3483 . . 3 (𝑤 = 𝑊 → {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))} = {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))})
24 eqid 2818 . . 3 (𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))}) = (𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))})
2511fvexi 6677 . . . 4 𝐷 ∈ V
2625rabex 5226 . . 3 {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))} ∈ V
2723, 24, 26fvmpt 6761 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))})‘𝑊) = {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))})
289, 27sylan9eq 2873 1 ((𝐾𝐵𝑊𝐻) → 𝑇 = {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  {crab 3139   class class class wbr 5057  cmpt 5137  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:  isltrn  37135
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