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Theorem pmapglb 39753
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 3069 . . . . . . 7 (∃𝑥𝑆 𝑦 = 𝑥 ↔ ∃𝑥(𝑥𝑆𝑦 = 𝑥))
2 equcom 2015 . . . . . . . . 9 (𝑦 = 𝑥𝑥 = 𝑦)
32anbi1ci 626 . . . . . . . 8 ((𝑥𝑆𝑦 = 𝑥) ↔ (𝑥 = 𝑦𝑥𝑆))
43exbii 1845 . . . . . . 7 (∃𝑥(𝑥𝑆𝑦 = 𝑥) ↔ ∃𝑥(𝑥 = 𝑦𝑥𝑆))
5 eleq1w 2822 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑆𝑦𝑆))
65equsexvw 2002 . . . . . . 7 (∃𝑥(𝑥 = 𝑦𝑥𝑆) ↔ 𝑦𝑆)
71, 4, 63bitri 297 . . . . . 6 (∃𝑥𝑆 𝑦 = 𝑥𝑦𝑆)
87abbii 2807 . . . . 5 {𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥} = {𝑦𝑦𝑆}
9 abid2 2877 . . . . 5 {𝑦𝑦𝑆} = 𝑆
108, 9eqtr2i 2764 . . . 4 𝑆 = {𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥}
1110fveq2i 6910 . . 3 (𝐺𝑆) = (𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})
1211fveq2i 6910 . 2 (𝑀‘(𝐺𝑆)) = (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥}))
13 dfss3 3984 . . 3 (𝑆𝐵 ↔ ∀𝑥𝑆 𝑥𝐵)
14 pmapglb.b . . . 4 𝐵 = (Base‘𝐾)
15 pmapglb.g . . . 4 𝐺 = (glb‘𝐾)
16 pmapglb.m . . . 4 𝑀 = (pmap‘𝐾)
1714, 15, 16pmapglbx 39752 . . 3 ((𝐾 ∈ HL ∧ ∀𝑥𝑆 𝑥𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})) = 𝑥𝑆 (𝑀𝑥))
1813, 17syl3an2b 1403 . 2 ((𝐾 ∈ HL ∧ 𝑆𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})) = 𝑥𝑆 (𝑀𝑥))
1912, 18eqtrid 2787 1 ((𝐾 ∈ HL ∧ 𝑆𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺𝑆)) = 𝑥𝑆 (𝑀𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wne 2938  wral 3059  wrex 3068  wss 3963  c0 4339   ciin 4997  cfv 6563  Basecbs 17245  glbcglb 18368  HLchlt 39332  pmapcpmap 39480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-poset 18371  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-lat 18490  df-clat 18557  df-ats 39249  df-hlat 39333  df-pmap 39487
This theorem is referenced by:  pmapglb2N  39754  pmapmeet  39756
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