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Theorem diadmclN 38326
Description: A member of domain of the partial isomorphism A is a lattice element. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
diadmcl.b 𝐵 = (Base‘𝐾)
diadmcl.h 𝐻 = (LHyp‘𝐾)
diadmcl.i 𝐼 = ((DIsoA‘𝐾)‘𝑊)
Assertion
Ref Expression
diadmclN (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋𝐵)

Proof of Theorem diadmclN
StepHypRef Expression
1 diadmcl.b . . 3 𝐵 = (Base‘𝐾)
2 eqid 2801 . . 3 (le‘𝐾) = (le‘𝐾)
3 diadmcl.h . . 3 𝐻 = (LHyp‘𝐾)
4 diadmcl.i . . 3 𝐼 = ((DIsoA‘𝐾)‘𝑊)
51, 2, 3, 4diaeldm 38325 . 2 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋𝐵𝑋(le‘𝐾)𝑊)))
65simprbda 502 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112   class class class wbr 5033  dom cdm 5523  cfv 6328  Basecbs 16478  lecple 16567  LHypclh 37273  DIsoAcdia 38317
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-disoa 38318
This theorem is referenced by:  diameetN  38345  docaclN  38413  diaocN  38414  doca2N  38415  djajN  38426
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