Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ldilfset Structured version   Visualization version   GIF version

Theorem ldilfset 37710
 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 3428 . 2 (𝐾𝐶𝐾 ∈ V)
2 fveq2 6662 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 ldilset.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2811 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6662 . . . . . 6 (𝑘 = 𝐾 → (LAut‘𝑘) = (LAut‘𝐾))
6 ldilset.i . . . . . 6 𝐼 = (LAut‘𝐾)
75, 6eqtr4di 2811 . . . . 5 (𝑘 = 𝐾 → (LAut‘𝑘) = 𝐼)
8 fveq2 6662 . . . . . . 7 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
9 ldilset.b . . . . . . 7 𝐵 = (Base‘𝐾)
108, 9eqtr4di 2811 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
11 fveq2 6662 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
12 ldilset.l . . . . . . . . 9 = (le‘𝐾)
1311, 12eqtr4di 2811 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = )
1413breqd 5046 . . . . . . 7 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑤𝑥 𝑤))
1514imbi1d 345 . . . . . 6 (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥) ↔ (𝑥 𝑤 → (𝑓𝑥) = 𝑥)))
1610, 15raleqbidv 3319 . . . . 5 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)))
177, 16rabeqbidv 3398 . . . 4 (𝑘 = 𝐾 → {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)} = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})
184, 17mpteq12dv 5120 . . 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 37706 . . 3 LDil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)}))
2018, 19, 3mptfvmpt 6987 . 2 (𝐾 ∈ V → (LDil‘𝐾) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}))
211, 20syl 17 1 (𝐾𝐶 → (LDil‘𝐾) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074  Vcvv 3409   class class class wbr 5035   ↦ cmpt 5115  ‘cfv 6339  Basecbs 16546  lecple 16635  LHypclh 37586  LAutclaut 37587  LDilcldil 37702 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ldil 37706 This theorem is referenced by:  ldilset  37711
 Copyright terms: Public domain W3C validator