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Theorem pmapglb 35374
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 2947 . . . . . . 7 (∃𝑥𝑆 𝑦 = 𝑥 ↔ ∃𝑥(𝑥𝑆𝑦 = 𝑥))
2 equcom 1991 . . . . . . . . . . 11 (𝑦 = 𝑥𝑥 = 𝑦)
32anbi2i 730 . . . . . . . . . 10 ((𝑥𝑆𝑦 = 𝑥) ↔ (𝑥𝑆𝑥 = 𝑦))
4 ancom 465 . . . . . . . . . 10 ((𝑥𝑆𝑥 = 𝑦) ↔ (𝑥 = 𝑦𝑥𝑆))
53, 4bitri 264 . . . . . . . . 9 ((𝑥𝑆𝑦 = 𝑥) ↔ (𝑥 = 𝑦𝑥𝑆))
65exbii 1814 . . . . . . . 8 (∃𝑥(𝑥𝑆𝑦 = 𝑥) ↔ ∃𝑥(𝑥 = 𝑦𝑥𝑆))
7 vex 3234 . . . . . . . . 9 𝑦 ∈ V
8 eleq1 2718 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑆𝑦𝑆))
97, 8ceqsexv 3273 . . . . . . . 8 (∃𝑥(𝑥 = 𝑦𝑥𝑆) ↔ 𝑦𝑆)
106, 9bitri 264 . . . . . . 7 (∃𝑥(𝑥𝑆𝑦 = 𝑥) ↔ 𝑦𝑆)
111, 10bitri 264 . . . . . 6 (∃𝑥𝑆 𝑦 = 𝑥𝑦𝑆)
1211abbii 2768 . . . . 5 {𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥} = {𝑦𝑦𝑆}
13 abid2 2774 . . . . 5 {𝑦𝑦𝑆} = 𝑆
1412, 13eqtr2i 2674 . . . 4 𝑆 = {𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥}
1514fveq2i 6232 . . 3 (𝐺𝑆) = (𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})
1615fveq2i 6232 . 2 (𝑀‘(𝐺𝑆)) = (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥}))
17 dfss3 3625 . . 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 35373 . . 3 ((𝐾 ∈ HL ∧ ∀𝑥𝑆 𝑥𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})) = 𝑥𝑆 (𝑀𝑥))
2217, 21syl3an2b 1403 . 2 ((𝐾 ∈ HL ∧ 𝑆𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})) = 𝑥𝑆 (𝑀𝑥))
2316, 22syl5eq 2697 1 ((𝐾 ∈ HL ∧ 𝑆𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺𝑆)) = 𝑥𝑆 (𝑀𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  wss 3607  c0 3948   ciin 4553  cfv 5926  Basecbs 15904  glbcglb 16990  HLchlt 34955  pmapcpmap 35101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-poset 16993  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-lat 17093  df-clat 17155  df-ats 34872  df-hlat 34956  df-pmap 35108
This theorem is referenced by:  pmapglb2N  35375  pmapmeet  35377
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