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Theorem pmapglb 34074
 Description: The projective map of the GLB of a set of lattice elements 𝑆. Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)
Hypotheses
Ref Expression
pmapglb.b 𝐵 = (Base‘𝐾)
pmapglb.g 𝐺 = (glb‘𝐾)
pmapglb.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
pmapglb ((𝐾 ∈ HL ∧ 𝑆𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺𝑆)) = 𝑥𝑆 (𝑀𝑥))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐾   𝑥,𝑆
Allowed substitution hints:   𝐺(𝑥)   𝑀(𝑥)

Proof of Theorem pmapglb
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-rex 2902 . . . . . . 7 (∃𝑥𝑆 𝑦 = 𝑥 ↔ ∃𝑥(𝑥𝑆𝑦 = 𝑥))
2 equcom 1932 . . . . . . . . . . 11 (𝑦 = 𝑥𝑥 = 𝑦)
32anbi2i 726 . . . . . . . . . 10 ((𝑥𝑆𝑦 = 𝑥) ↔ (𝑥𝑆𝑥 = 𝑦))
4 ancom 465 . . . . . . . . . 10 ((𝑥𝑆𝑥 = 𝑦) ↔ (𝑥 = 𝑦𝑥𝑆))
53, 4bitri 263 . . . . . . . . 9 ((𝑥𝑆𝑦 = 𝑥) ↔ (𝑥 = 𝑦𝑥𝑆))
65exbii 1764 . . . . . . . 8 (∃𝑥(𝑥𝑆𝑦 = 𝑥) ↔ ∃𝑥(𝑥 = 𝑦𝑥𝑆))
7 vex 3176 . . . . . . . . 9 𝑦 ∈ V
8 eleq1 2676 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑆𝑦𝑆))
97, 8ceqsexv 3215 . . . . . . . 8 (∃𝑥(𝑥 = 𝑦𝑥𝑆) ↔ 𝑦𝑆)
106, 9bitri 263 . . . . . . 7 (∃𝑥(𝑥𝑆𝑦 = 𝑥) ↔ 𝑦𝑆)
111, 10bitri 263 . . . . . 6 (∃𝑥𝑆 𝑦 = 𝑥𝑦𝑆)
1211abbii 2726 . . . . 5 {𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥} = {𝑦𝑦𝑆}
13 abid2 2732 . . . . 5 {𝑦𝑦𝑆} = 𝑆
1412, 13eqtr2i 2633 . . . 4 𝑆 = {𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥}
1514fveq2i 6106 . . 3 (𝐺𝑆) = (𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})
1615fveq2i 6106 . 2 (𝑀‘(𝐺𝑆)) = (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥}))
17 dfss3 3558 . . 3 (𝑆𝐵 ↔ ∀𝑥𝑆 𝑥𝐵)
18 pmapglb.b . . . 4 𝐵 = (Base‘𝐾)
19 pmapglb.g . . . 4 𝐺 = (glb‘𝐾)
20 pmapglb.m . . . 4 𝑀 = (pmap‘𝐾)
2118, 19, 20pmapglbx 34073 . . 3 ((𝐾 ∈ HL ∧ ∀𝑥𝑆 𝑥𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})) = 𝑥𝑆 (𝑀𝑥))
2217, 21syl3an2b 1355 . 2 ((𝐾 ∈ HL ∧ 𝑆𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})) = 𝑥𝑆 (𝑀𝑥))
2316, 22syl5eq 2656 1 ((𝐾 ∈ HL ∧ 𝑆𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺𝑆)) = 𝑥𝑆 (𝑀𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874  ∩ ciin 4456  ‘cfv 5804  Basecbs 15695  glbcglb 16766  HLchlt 33655  pmapcpmap 33801 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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 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-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-poset 16769  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-lat 16869  df-clat 16931  df-ats 33572  df-hlat 33656  df-pmap 33808 This theorem is referenced by:  pmapglb2N  34075  pmapmeet  34077
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