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Theorem isldil 39713
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 39712 . . 3 ((𝐾𝐶𝑊𝐻) → 𝐷 = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
76eleq2d 2811 . 2 ((𝐾𝐶𝑊𝐻) → (𝐹𝐷𝐹 ∈ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)}))
8 fveq1 6895 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
98eqeq1d 2727 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑥) = 𝑥 ↔ (𝐹𝑥) = 𝑥))
109imbi2d 339 . . . 4 (𝑓 = 𝐹 → ((𝑥 𝑊 → (𝑓𝑥) = 𝑥) ↔ (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
1110ralbidv 3167 . . 3 (𝑓 = 𝐹 → (∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
1211elrab 3679 . 2 (𝐹 ∈ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)} ↔ (𝐹𝐼 ∧ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
137, 12bitrdi 286 1 ((𝐾𝐶𝑊𝐻) → (𝐹𝐷 ↔ (𝐹𝐼 ∧ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  {crab 3418   class class class wbr 5149  cfv 6549  Basecbs 17183  lecple 17243  LHypclh 39587  LAutclaut 39588  LDilcldil 39703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ldil 39707
This theorem is referenced by:  ldillaut  39714  ldilval  39716  idldil  39717  ldilcnv  39718  ldilco  39719  cdleme50ldil  40151
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