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Theorem pmaplubN 39830
Description: The LUB of a projective map is the projective map's argument. (Contributed by NM, 13-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmaplub.b 𝐵 = (Base‘𝐾)
pmaplub.u 𝑈 = (lub‘𝐾)
pmaplub.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
pmaplubN ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑈‘(𝑀𝑋)) = 𝑋)

Proof of Theorem pmaplubN
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 pmaplub.b . . . 4 𝐵 = (Base‘𝐾)
2 eqid 2734 . . . 4 (le‘𝐾) = (le‘𝐾)
3 eqid 2734 . . . 4 (Atoms‘𝐾) = (Atoms‘𝐾)
4 pmaplub.m . . . 4 𝑀 = (pmap‘𝐾)
51, 2, 3, 4pmapval 39663 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑀𝑋) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑋})
65fveq2d 6923 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑈‘(𝑀𝑋)) = (𝑈‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑋}))
7 hlomcmat 39270 . . 3 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
8 pmaplub.u . . . 4 𝑈 = (lub‘𝐾)
91, 2, 8, 3atlatmstc 39224 . . 3 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋𝐵) → (𝑈‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑋}) = 𝑋)
107, 9sylan 579 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑈‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑋}) = 𝑋)
116, 10eqtrd 2774 1 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑈‘(𝑀𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1537  wcel 2103  {crab 3438   class class class wbr 5169  cfv 6572  Basecbs 17253  lecple 17313  lubclub 18374  CLatccla 18563  OMLcoml 39080  Atomscatm 39168  AtLatcal 39169  HLchlt 39255  pmapcpmap 39403
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
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 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-proset 18360  df-poset 18378  df-plt 18395  df-lub 18411  df-glb 18412  df-join 18413  df-meet 18414  df-p0 18490  df-lat 18497  df-clat 18564  df-oposet 39081  df-ol 39083  df-oml 39084  df-covers 39171  df-ats 39172  df-atl 39203  df-cvlat 39227  df-hlat 39256  df-pmap 39410
This theorem is referenced by: (None)
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