Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmapmeet Structured version   Visualization version   GIF version

Theorem pmapmeet 36776
Description: The projective map of a meet. (Contributed by NM, 25-Jan-2012.)
Hypotheses
Ref Expression
pmapmeet.b 𝐵 = (Base‘𝐾)
pmapmeet.m = (meet‘𝐾)
pmapmeet.a 𝐴 = (Atoms‘𝐾)
pmapmeet.p 𝑃 = (pmap‘𝐾)
Assertion
Ref Expression
pmapmeet ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑃‘(𝑋 𝑌)) = ((𝑃𝑋) ∩ (𝑃𝑌)))

Proof of Theorem pmapmeet
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2826 . . . 4 (glb‘𝐾) = (glb‘𝐾)
2 pmapmeet.m . . . 4 = (meet‘𝐾)
3 simp1 1130 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ HL)
4 simp2 1131 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
5 simp3 1132 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
61, 2, 3, 4, 5meetval 17619 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌}))
76fveq2d 6671 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑃‘(𝑋 𝑌)) = (𝑃‘((glb‘𝐾)‘{𝑋, 𝑌})))
8 prssi 4753 . . . 4 ((𝑋𝐵𝑌𝐵) → {𝑋, 𝑌} ⊆ 𝐵)
983adant1 1124 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → {𝑋, 𝑌} ⊆ 𝐵)
10 prnzg 4712 . . . 4 (𝑋𝐵 → {𝑋, 𝑌} ≠ ∅)
11103ad2ant2 1128 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → {𝑋, 𝑌} ≠ ∅)
12 pmapmeet.b . . . 4 𝐵 = (Base‘𝐾)
13 pmapmeet.p . . . 4 𝑃 = (pmap‘𝐾)
1412, 1, 13pmapglb 36773 . . 3 ((𝐾 ∈ HL ∧ {𝑋, 𝑌} ⊆ 𝐵 ∧ {𝑋, 𝑌} ≠ ∅) → (𝑃‘((glb‘𝐾)‘{𝑋, 𝑌})) = 𝑥 ∈ {𝑋, 𝑌} (𝑃𝑥))
153, 9, 11, 14syl3anc 1365 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑃‘((glb‘𝐾)‘{𝑋, 𝑌})) = 𝑥 ∈ {𝑋, 𝑌} (𝑃𝑥))
16 fveq2 6667 . . . 4 (𝑥 = 𝑋 → (𝑃𝑥) = (𝑃𝑋))
17 fveq2 6667 . . . 4 (𝑥 = 𝑌 → (𝑃𝑥) = (𝑃𝑌))
1816, 17iinxprg 5008 . . 3 ((𝑋𝐵𝑌𝐵) → 𝑥 ∈ {𝑋, 𝑌} (𝑃𝑥) = ((𝑃𝑋) ∩ (𝑃𝑌)))
19183adant1 1124 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑥 ∈ {𝑋, 𝑌} (𝑃𝑥) = ((𝑃𝑋) ∩ (𝑃𝑌)))
207, 15, 193eqtrd 2865 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑃‘(𝑋 𝑌)) = ((𝑃𝑋) ∩ (𝑃𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1081   = wceq 1530  wcel 2107  wne 3021  cin 3939  wss 3940  c0 4295  {cpr 4566   ciin 4918  cfv 6352  (class class class)co 7148  Basecbs 16473  glbcglb 17543  meetcmee 17545  Atomscatm 36266  HLchlt 36353  pmapcpmap 36500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-poset 17546  df-lub 17574  df-glb 17575  df-join 17576  df-meet 17577  df-lat 17646  df-clat 17708  df-ats 36270  df-hlat 36354  df-pmap 36507
This theorem is referenced by:  hlmod1i  36859  poldmj1N  36931  pmapj2N  36932  pnonsingN  36936  psubclinN  36951  poml4N  36956  pl42lem1N  36982  pl42lem2N  36983
  Copyright terms: Public domain W3C validator