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Theorem ldilval 34211
Description: Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
ldilval.b 𝐵 = (Base‘𝐾)
ldilval.l = (le‘𝐾)
ldilval.h 𝐻 = (LHyp‘𝐾)
ldilval.d 𝐷 = ((LDil‘𝐾)‘𝑊)
Assertion
Ref Expression
ldilval (((𝐾𝑉𝑊𝐻) ∧ 𝐹𝐷 ∧ (𝑋𝐵𝑋 𝑊)) → (𝐹𝑋) = 𝑋)

Proof of Theorem ldilval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ldilval.b . . . . 5 𝐵 = (Base‘𝐾)
2 ldilval.l . . . . 5 = (le‘𝐾)
3 ldilval.h . . . . 5 𝐻 = (LHyp‘𝐾)
4 eqid 2610 . . . . 5 (LAut‘𝐾) = (LAut‘𝐾)
5 ldilval.d . . . . 5 𝐷 = ((LDil‘𝐾)‘𝑊)
61, 2, 3, 4, 5isldil 34208 . . . 4 ((𝐾𝑉𝑊𝐻) → (𝐹𝐷 ↔ (𝐹 ∈ (LAut‘𝐾) ∧ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥))))
7 simpr 476 . . . 4 ((𝐹 ∈ (LAut‘𝐾) ∧ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥)) → ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥))
86, 7syl6bi 242 . . 3 ((𝐾𝑉𝑊𝐻) → (𝐹𝐷 → ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
9 breq1 4581 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
10 fveq2 6088 . . . . . . 7 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
11 id 22 . . . . . . 7 (𝑥 = 𝑋𝑥 = 𝑋)
1210, 11eqeq12d 2625 . . . . . 6 (𝑥 = 𝑋 → ((𝐹𝑥) = 𝑥 ↔ (𝐹𝑋) = 𝑋))
139, 12imbi12d 333 . . . . 5 (𝑥 = 𝑋 → ((𝑥 𝑊 → (𝐹𝑥) = 𝑥) ↔ (𝑋 𝑊 → (𝐹𝑋) = 𝑋)))
1413rspccv 3279 . . . 4 (∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥) → (𝑋𝐵 → (𝑋 𝑊 → (𝐹𝑋) = 𝑋)))
1514impd 446 . . 3 (∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥) → ((𝑋𝐵𝑋 𝑊) → (𝐹𝑋) = 𝑋))
168, 15syl6 34 . 2 ((𝐾𝑉𝑊𝐻) → (𝐹𝐷 → ((𝑋𝐵𝑋 𝑊) → (𝐹𝑋) = 𝑋)))
17163imp 1249 1 (((𝐾𝑉𝑊𝐻) ∧ 𝐹𝐷 ∧ (𝑋𝐵𝑋 𝑊)) → (𝐹𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896   class class class wbr 4578  cfv 5790  Basecbs 15644  lecple 15724  LHypclh 34082  LAutclaut 34083  LDilcldil 34198
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 4694  ax-sep 4704  ax-nul 4712  ax-pr 4828
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 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ldil 34202
This theorem is referenced by:  ldilcnv  34213  ldilco  34214  ltrnval1  34232
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