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Theorem ldilfset 35915
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 3352 . 2 (𝐾𝐶𝐾 ∈ V)
2 fveq2 6353 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 ldilset.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2812 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6353 . . . . . 6 (𝑘 = 𝐾 → (LAut‘𝑘) = (LAut‘𝐾))
6 ldilset.i . . . . . 6 𝐼 = (LAut‘𝐾)
75, 6syl6eqr 2812 . . . . 5 (𝑘 = 𝐾 → (LAut‘𝑘) = 𝐼)
8 fveq2 6353 . . . . . . 7 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
9 ldilset.b . . . . . . 7 𝐵 = (Base‘𝐾)
108, 9syl6eqr 2812 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
11 fveq2 6353 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
12 ldilset.l . . . . . . . . 9 = (le‘𝐾)
1311, 12syl6eqr 2812 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = )
1413breqd 4815 . . . . . . 7 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑤𝑥 𝑤))
1514imbi1d 330 . . . . . 6 (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥) ↔ (𝑥 𝑤 → (𝑓𝑥) = 𝑥)))
1610, 15raleqbidv 3291 . . . . 5 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)))
177, 16rabeqbidv 3335 . . . 4 (𝑘 = 𝐾 → {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)} = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})
184, 17mpteq12dv 4885 . . 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 35911 . . 3 LDil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)}))
20 fvex 6363 . . . . 5 (LHyp‘𝐾) ∈ V
213, 20eqeltri 2835 . . . 4 𝐻 ∈ V
2221mptex 6651 . . 3 (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}) ∈ V
2318, 19, 22fvmpt 6445 . 2 (𝐾 ∈ V → (LDil‘𝐾) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}))
241, 23syl 17 1 (𝐾𝐶 → (LDil‘𝐾) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340   class class class wbr 4804  cmpt 4881  cfv 6049  Basecbs 16079  lecple 16170  LHypclh 35791  LAutclaut 35792  LDilcldil 35907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ldil 35911
This theorem is referenced by:  ldilset  35916
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