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Theorem ldilset 35713
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 35712 . . . 4 (𝐾𝐶 → (LDil‘𝐾) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}))
76fveq1d 6231 . . 3 (𝐾𝐶 → ((LDil‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})‘𝑊))
8 breq2 4689 . . . . . . 7 (𝑤 = 𝑊 → (𝑥 𝑤𝑥 𝑊))
98imbi1d 330 . . . . . 6 (𝑤 = 𝑊 → ((𝑥 𝑤 → (𝑓𝑥) = 𝑥) ↔ (𝑥 𝑊 → (𝑓𝑥) = 𝑥)))
109ralbidv 3015 . . . . 5 (𝑤 = 𝑊 → (∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)))
1110rabbidv 3220 . . . 4 (𝑤 = 𝑊 → {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)} = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
12 eqid 2651 . . . 4 (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})
13 fvex 6239 . . . . . 6 (LAut‘𝐾) ∈ V
145, 13eqeltri 2726 . . . . 5 𝐼 ∈ V
1514rabex 4845 . . . 4 {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)} ∈ V
1611, 12, 15fvmpt 6321 . . 3 (𝑊𝐻 → ((𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})‘𝑊) = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
177, 16sylan9eq 2705 . 2 ((𝐾𝐶𝑊𝐻) → ((LDil‘𝐾)‘𝑊) = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
181, 17syl5eq 2697 1 ((𝐾𝐶𝑊𝐻) → 𝐷 = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231   class class class wbr 4685  cmpt 4762  cfv 5926  Basecbs 15904  lecple 15995  LHypclh 35588  LAutclaut 35589  LDilcldil 35704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ldil 35708
This theorem is referenced by:  isldil  35714
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