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Theorem isldil 40069
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 40068 . . 3 ((𝐾𝐶𝑊𝐻) → 𝐷 = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
76eleq2d 2830 . 2 ((𝐾𝐶𝑊𝐻) → (𝐹𝐷𝐹 ∈ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)}))
8 fveq1 6921 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
98eqeq1d 2742 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑥) = 𝑥 ↔ (𝐹𝑥) = 𝑥))
109imbi2d 340 . . . 4 (𝑓 = 𝐹 → ((𝑥 𝑊 → (𝑓𝑥) = 𝑥) ↔ (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
1110ralbidv 3184 . . 3 (𝑓 = 𝐹 → (∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
1211elrab 3708 . 2 (𝐹 ∈ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)} ↔ (𝐹𝐼 ∧ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
137, 12bitrdi 287 1 ((𝐾𝐶𝑊𝐻) → (𝐹𝐷 ↔ (𝐹𝐼 ∧ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2108  wral 3067  {crab 3443   class class class wbr 5166  cfv 6575  Basecbs 17260  lecple 17320  LHypclh 39943  LAutclaut 39944  LDilcldil 40059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-ldil 40063
This theorem is referenced by:  ldillaut  40070  ldilval  40072  idldil  40073  ldilcnv  40074  ldilco  40075  cdleme50ldil  40507
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