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Theorem ldilval 35922
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 2771 . . . . 5 (LAut‘𝐾) = (LAut‘𝐾)
5 ldilval.d . . . . 5 𝐷 = ((LDil‘𝐾)‘𝑊)
61, 2, 3, 4, 5isldil 35919 . . . 4 ((𝐾𝑉𝑊𝐻) → (𝐹𝐷 ↔ (𝐹 ∈ (LAut‘𝐾) ∧ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥))))
7 simpr 471 . . . 4 ((𝐹 ∈ (LAut‘𝐾) ∧ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥)) → ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥))
86, 7syl6bi 243 . . 3 ((𝐾𝑉𝑊𝐻) → (𝐹𝐷 → ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
9 breq1 4790 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
10 fveq2 6333 . . . . . . 7 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
11 id 22 . . . . . . 7 (𝑥 = 𝑋𝑥 = 𝑋)
1210, 11eqeq12d 2786 . . . . . 6 (𝑥 = 𝑋 → ((𝐹𝑥) = 𝑥 ↔ (𝐹𝑋) = 𝑋))
139, 12imbi12d 333 . . . . 5 (𝑥 = 𝑋 → ((𝑥 𝑊 → (𝐹𝑥) = 𝑥) ↔ (𝑋 𝑊 → (𝐹𝑋) = 𝑋)))
1413rspccv 3457 . . . 4 (∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥) → (𝑋𝐵 → (𝑋 𝑊 → (𝐹𝑋) = 𝑋)))
1514impd 396 . . 3 (∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥) → ((𝑋𝐵𝑋 𝑊) → (𝐹𝑋) = 𝑋))
168, 15syl6 35 . 2 ((𝐾𝑉𝑊𝐻) → (𝐹𝐷 → ((𝑋𝐵𝑋 𝑊) → (𝐹𝑋) = 𝑋)))
17163imp 1101 1 (((𝐾𝑉𝑊𝐻) ∧ 𝐹𝐷 ∧ (𝑋𝐵𝑋 𝑊)) → (𝐹𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061   class class class wbr 4787  cfv 6030  Basecbs 16064  lecple 16156  LHypclh 35793  LAutclaut 35794  LDilcldil 35909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ldil 35913
This theorem is referenced by:  ldilcnv  35924  ldilco  35925  ltrnval1  35943
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