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Theorem ldilset 39296
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 39295 . . . 4 (𝐾𝐶 → (LDil‘𝐾) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}))
76fveq1d 6893 . . 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 3439 . . . 4 (𝑤 = 𝑊 → {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)} = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
12 eqid 2731 . . . 4 (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})
135fvexi 6905 . . . . 5 𝐼 ∈ V
1413rabex 5332 . . . 4 {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)} ∈ V
1511, 12, 14fvmpt 6998 . . 3 (𝑊𝐻 → ((𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})‘𝑊) = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
167, 15sylan9eq 2791 . 2 ((𝐾𝐶𝑊𝐻) → ((LDil‘𝐾)‘𝑊) = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
171, 16eqtrid 2783 1 ((𝐾𝐶𝑊𝐻) → 𝐷 = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  wral 3060  {crab 3431   class class class wbr 5148  cmpt 5231  cfv 6543  Basecbs 17151  lecple 17211  LHypclh 39171  LAutclaut 39172  LDilcldil 39287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ldil 39291
This theorem is referenced by:  isldil  39297
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