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

Theorem dilfsetN 34457
 Description: The mapping from fiducial atom to set of dilations. (Contributed by NM, 30-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
dilset.a 𝐴 = (Atoms‘𝐾)
dilset.s 𝑆 = (PSubSp‘𝐾)
dilset.w 𝑊 = (WAtoms‘𝐾)
dilset.m 𝑀 = (PAut‘𝐾)
dilset.l 𝐿 = (Dil‘𝐾)
Assertion
Ref Expression
dilfsetN (𝐾𝐵𝐿 = (𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)}))
Distinct variable groups:   𝐴,𝑑   𝑓,𝑑,𝑥,𝐾   𝑓,𝑀   𝑥,𝑆
Allowed substitution hints:   𝐴(𝑥,𝑓)   𝐵(𝑥,𝑓,𝑑)   𝑆(𝑓,𝑑)   𝐿(𝑥,𝑓,𝑑)   𝑀(𝑥,𝑑)   𝑊(𝑥,𝑓,𝑑)

Proof of Theorem dilfsetN
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3185 . 2 (𝐾𝐵𝐾 ∈ V)
2 dilset.l . . 3 𝐿 = (Dil‘𝐾)
3 fveq2 6103 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 dilset.a . . . . . 6 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2662 . . . . 5 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6 fveq2 6103 . . . . . . 7 (𝑘 = 𝐾 → (PAut‘𝑘) = (PAut‘𝐾))
7 dilset.m . . . . . . 7 𝑀 = (PAut‘𝐾)
86, 7syl6eqr 2662 . . . . . 6 (𝑘 = 𝐾 → (PAut‘𝑘) = 𝑀)
9 fveq2 6103 . . . . . . . 8 (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
10 dilset.s . . . . . . . 8 𝑆 = (PSubSp‘𝐾)
119, 10syl6eqr 2662 . . . . . . 7 (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
12 fveq2 6103 . . . . . . . . . . 11 (𝑘 = 𝐾 → (WAtoms‘𝑘) = (WAtoms‘𝐾))
13 dilset.w . . . . . . . . . . 11 𝑊 = (WAtoms‘𝐾)
1412, 13syl6eqr 2662 . . . . . . . . . 10 (𝑘 = 𝐾 → (WAtoms‘𝑘) = 𝑊)
1514fveq1d 6105 . . . . . . . . 9 (𝑘 = 𝐾 → ((WAtoms‘𝑘)‘𝑑) = (𝑊𝑑))
1615sseq2d 3596 . . . . . . . 8 (𝑘 = 𝐾 → (𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) ↔ 𝑥 ⊆ (𝑊𝑑)))
1716imbi1d 330 . . . . . . 7 (𝑘 = 𝐾 → ((𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥) ↔ (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)))
1811, 17raleqbidv 3129 . . . . . 6 (𝑘 = 𝐾 → (∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)))
198, 18rabeqbidv 3168 . . . . 5 (𝑘 = 𝐾 → {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥)} = {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)})
205, 19mpteq12dv 4663 . . . 4 (𝑘 = 𝐾 → (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥)}) = (𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)}))
21 df-dilN 34410 . . . 4 Dil = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥)}))
22 fvex 6113 . . . . . 6 (Atoms‘𝐾) ∈ V
234, 22eqeltri 2684 . . . . 5 𝐴 ∈ V
2423mptex 6390 . . . 4 (𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)}) ∈ V
2520, 21, 24fvmpt 6191 . . 3 (𝐾 ∈ V → (Dil‘𝐾) = (𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)}))
262, 25syl5eq 2656 . 2 (𝐾 ∈ V → 𝐿 = (𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)}))
271, 26syl 17 1 (𝐾𝐵𝐿 = (𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540   ↦ cmpt 4643  ‘cfv 5804  Atomscatm 33568  PSubSpcpsubsp 33800  WAtomscwpointsN 34290  PAutcpautN 34291  DilcdilN 34406 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-dilN 34410 This theorem is referenced by:  dilsetN  34458
 Copyright terms: Public domain W3C validator