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Theorem ldilfset 35996
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 3365 . 2 (𝐾𝐶𝐾 ∈ V)
2 fveq2 6375 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 ldilset.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2817 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6375 . . . . . 6 (𝑘 = 𝐾 → (LAut‘𝑘) = (LAut‘𝐾))
6 ldilset.i . . . . . 6 𝐼 = (LAut‘𝐾)
75, 6syl6eqr 2817 . . . . 5 (𝑘 = 𝐾 → (LAut‘𝑘) = 𝐼)
8 fveq2 6375 . . . . . . 7 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
9 ldilset.b . . . . . . 7 𝐵 = (Base‘𝐾)
108, 9syl6eqr 2817 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
11 fveq2 6375 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
12 ldilset.l . . . . . . . . 9 = (le‘𝐾)
1311, 12syl6eqr 2817 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = )
1413breqd 4820 . . . . . . 7 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑤𝑥 𝑤))
1514imbi1d 332 . . . . . 6 (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥) ↔ (𝑥 𝑤 → (𝑓𝑥) = 𝑥)))
1610, 15raleqbidv 3300 . . . . 5 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)))
177, 16rabeqbidv 3344 . . . 4 (𝑘 = 𝐾 → {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)} = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})
184, 17mpteq12dv 4892 . . 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 35992 . . 3 LDil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)}))
2018, 19, 3mptfvmpt 6683 . 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 1652  wcel 2155  wral 3055  {crab 3059  Vcvv 3350   class class class wbr 4809  cmpt 4888  cfv 6068  Basecbs 16132  lecple 16223  LHypclh 35872  LAutclaut 35873  LDilcldil 35988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ldil 35992
This theorem is referenced by:  ldilset  35997
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