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Theorem isldil 37355
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 37354 . . 3 ((𝐾𝐶𝑊𝐻) → 𝐷 = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
76eleq2d 2901 . 2 ((𝐾𝐶𝑊𝐻) → (𝐹𝐷𝐹 ∈ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)}))
8 fveq1 6660 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
98eqeq1d 2826 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑥) = 𝑥 ↔ (𝐹𝑥) = 𝑥))
109imbi2d 344 . . . 4 (𝑓 = 𝐹 → ((𝑥 𝑊 → (𝑓𝑥) = 𝑥) ↔ (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
1110ralbidv 3192 . . 3 (𝑓 = 𝐹 → (∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
1211elrab 3666 . 2 (𝐹 ∈ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)} ↔ (𝐹𝐼 ∧ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
137, 12syl6bb 290 1 ((𝐾𝐶𝑊𝐻) → (𝐹𝐷 ↔ (𝐹𝐼 ∧ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  {crab 3137   class class class wbr 5052  cfv 6343  Basecbs 16483  lecple 16572  LHypclh 37229  LAutclaut 37230  LDilcldil 37345
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ldil 37349
This theorem is referenced by:  ldillaut  37356  ldilval  37358  idldil  37359  ldilcnv  37360  ldilco  37361  cdleme50ldil  37793
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