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

Theorem isldil 34414
Description: The predicate "is a lattice dilation". Similar to definition of dilation in [Crawley] p. 111. (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
isldil ((𝐾𝐶𝑊𝐻) → (𝐹𝐷 ↔ (𝐹𝐼 ∧ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥))))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐾   𝑥,𝑊   𝑥,𝐹
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)   𝐻(𝑥)   𝐼(𝑥)   (𝑥)

Proof of Theorem isldil
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 ldilset.b . . . 4 𝐵 = (Base‘𝐾)
2 ldilset.l . . . 4 = (le‘𝐾)
3 ldilset.h . . . 4 𝐻 = (LHyp‘𝐾)
4 ldilset.i . . . 4 𝐼 = (LAut‘𝐾)
5 ldilset.d . . . 4 𝐷 = ((LDil‘𝐾)‘𝑊)
61, 2, 3, 4, 5ldilset 34413 . . 3 ((𝐾𝐶𝑊𝐻) → 𝐷 = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
76eleq2d 2673 . 2 ((𝐾𝐶𝑊𝐻) → (𝐹𝐷𝐹 ∈ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)}))
8 fveq1 6102 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
98eqeq1d 2612 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑥) = 𝑥 ↔ (𝐹𝑥) = 𝑥))
109imbi2d 329 . . . 4 (𝑓 = 𝐹 → ((𝑥 𝑊 → (𝑓𝑥) = 𝑥) ↔ (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
1110ralbidv 2969 . . 3 (𝑓 = 𝐹 → (∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
1211elrab 3331 . 2 (𝐹 ∈ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)} ↔ (𝐹𝐼 ∧ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
137, 12syl6bb 275 1 ((𝐾𝐶𝑊𝐻) → (𝐹𝐷 ↔ (𝐹𝐼 ∧ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  {crab 2900   class class class wbr 4583  cfv 5804  Basecbs 15695  lecple 15775  LHypclh 34288  LAutclaut 34289  LDilcldil 34404
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-ldil 34408
This theorem is referenced by:  ldillaut  34415  ldilval  34417  idldil  34418  ldilcnv  34419  ldilco  34420  cdleme50ldil  34854
  Copyright terms: Public domain W3C validator