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Theorem ldilfset 39435
Description: The mapping from fiducial co-atom 𝑤 to its set of lattice dilations. (Contributed by NM, 11-May-2012.)
Hypotheses
Ref Expression
ldilset.b 𝐵 = (Base‘𝐾)
ldilset.l = (le‘𝐾)
ldilset.h 𝐻 = (LHyp‘𝐾)
ldilset.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
ldilfset (𝐾𝐶 → (LDil‘𝐾) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}))
Distinct variable groups:   𝑥,𝐵   𝑤,𝐻   𝑓,𝐼   𝑤,𝑓,𝑥,𝐾
Allowed substitution hints:   𝐵(𝑤,𝑓)   𝐶(𝑥,𝑤,𝑓)   𝐻(𝑥,𝑓)   𝐼(𝑥,𝑤)   (𝑥,𝑤,𝑓)

Proof of Theorem ldilfset
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3485 . 2 (𝐾𝐶𝐾 ∈ V)
2 fveq2 6881 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 ldilset.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2782 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6881 . . . . . 6 (𝑘 = 𝐾 → (LAut‘𝑘) = (LAut‘𝐾))
6 ldilset.i . . . . . 6 𝐼 = (LAut‘𝐾)
75, 6eqtr4di 2782 . . . . 5 (𝑘 = 𝐾 → (LAut‘𝑘) = 𝐼)
8 fveq2 6881 . . . . . . 7 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
9 ldilset.b . . . . . . 7 𝐵 = (Base‘𝐾)
108, 9eqtr4di 2782 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
11 fveq2 6881 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
12 ldilset.l . . . . . . . . 9 = (le‘𝐾)
1311, 12eqtr4di 2782 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = )
1413breqd 5149 . . . . . . 7 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑤𝑥 𝑤))
1514imbi1d 341 . . . . . 6 (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥) ↔ (𝑥 𝑤 → (𝑓𝑥) = 𝑥)))
1610, 15raleqbidv 3334 . . . . 5 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)))
177, 16rabeqbidv 3441 . . . 4 (𝑘 = 𝐾 → {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)} = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})
184, 17mpteq12dv 5229 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)}) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}))
19 df-ldil 39431 . . 3 LDil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)}))
2018, 19, 3mptfvmpt 7221 . 2 (𝐾 ∈ V → (LDil‘𝐾) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}))
211, 20syl 17 1 (𝐾𝐶 → (LDil‘𝐾) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wral 3053  {crab 3424  Vcvv 3466   class class class wbr 5138  cmpt 5221  cfv 6533  Basecbs 17142  lecple 17202  LHypclh 39311  LAutclaut 39312  LDilcldil 39427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ldil 39431
This theorem is referenced by:  ldilset  39436
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