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Theorem ldilset 40092
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 40091 . . . 4 (𝐾𝐶 → (LDil‘𝐾) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}))
76fveq1d 6909 . . 3 (𝐾𝐶 → ((LDil‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})‘𝑊))
8 breq2 5152 . . . . . . 7 (𝑤 = 𝑊 → (𝑥 𝑤𝑥 𝑊))
98imbi1d 341 . . . . . 6 (𝑤 = 𝑊 → ((𝑥 𝑤 → (𝑓𝑥) = 𝑥) ↔ (𝑥 𝑊 → (𝑓𝑥) = 𝑥)))
109ralbidv 3176 . . . . 5 (𝑤 = 𝑊 → (∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)))
1110rabbidv 3441 . . . 4 (𝑤 = 𝑊 → {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)} = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
12 eqid 2735 . . . 4 (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})
135fvexi 6921 . . . . 5 𝐼 ∈ V
1413rabex 5345 . . . 4 {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)} ∈ V
1511, 12, 14fvmpt 7016 . . 3 (𝑊𝐻 → ((𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})‘𝑊) = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
167, 15sylan9eq 2795 . 2 ((𝐾𝐶𝑊𝐻) → ((LDil‘𝐾)‘𝑊) = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
171, 16eqtrid 2787 1 ((𝐾𝐶𝑊𝐻) → 𝐷 = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433   class class class wbr 5148  cmpt 5231  cfv 6563  Basecbs 17245  lecple 17305  LHypclh 39967  LAutclaut 39968  LDilcldil 40083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ldil 40087
This theorem is referenced by:  isldil  40093
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