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Theorem ldilset 34413
 Description: The set of lattice dilations for a fiducial co-atom 𝑊. (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)
ldilset.d 𝐷 = ((LDil‘𝐾)‘𝑊)
Assertion
Ref Expression
ldilset ((𝐾𝐶𝑊𝐻) → 𝐷 = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
Distinct variable groups:   𝑥,𝐵   𝑓,𝐼   𝑥,𝑓,𝐾   𝑓,𝑊,𝑥
Allowed substitution hints:   𝐵(𝑓)   𝐶(𝑥,𝑓)   𝐷(𝑥,𝑓)   𝐻(𝑥,𝑓)   𝐼(𝑥)   (𝑥,𝑓)

Proof of Theorem ldilset
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ldilset.d . 2 𝐷 = ((LDil‘𝐾)‘𝑊)
2 ldilset.b . . . . 5 𝐵 = (Base‘𝐾)
3 ldilset.l . . . . 5 = (le‘𝐾)
4 ldilset.h . . . . 5 𝐻 = (LHyp‘𝐾)
5 ldilset.i . . . . 5 𝐼 = (LAut‘𝐾)
62, 3, 4, 5ldilfset 34412 . . . 4 (𝐾𝐶 → (LDil‘𝐾) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}))
76fveq1d 6105 . . 3 (𝐾𝐶 → ((LDil‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})‘𝑊))
8 breq2 4587 . . . . . . 7 (𝑤 = 𝑊 → (𝑥 𝑤𝑥 𝑊))
98imbi1d 330 . . . . . 6 (𝑤 = 𝑊 → ((𝑥 𝑤 → (𝑓𝑥) = 𝑥) ↔ (𝑥 𝑊 → (𝑓𝑥) = 𝑥)))
109ralbidv 2969 . . . . 5 (𝑤 = 𝑊 → (∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)))
1110rabbidv 3164 . . . 4 (𝑤 = 𝑊 → {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)} = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
12 eqid 2610 . . . 4 (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})
13 fvex 6113 . . . . . 6 (LAut‘𝐾) ∈ V
145, 13eqeltri 2684 . . . . 5 𝐼 ∈ V
1514rabex 4740 . . . 4 {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)} ∈ V
1611, 12, 15fvmpt 6191 . . 3 (𝑊𝐻 → ((𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})‘𝑊) = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
177, 16sylan9eq 2664 . 2 ((𝐾𝐶𝑊𝐻) → ((LDil‘𝐾)‘𝑊) = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
181, 17syl5eq 2656 1 ((𝐾𝐶𝑊𝐻) → 𝐷 = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  Basecbs 15695  lecple 15775  LHypclh 34288  LAutclaut 34289  LDilcldil 34404 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ldil 34408 This theorem is referenced by:  isldil  34414
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