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Theorem pmapglb 33857
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 2901 . . . . . . 7 (∃𝑥𝑆 𝑦 = 𝑥 ↔ ∃𝑥(𝑥𝑆𝑦 = 𝑥))
2 equcom 1931 . . . . . . . . . . 11 (𝑦 = 𝑥𝑥 = 𝑦)
32anbi2i 725 . . . . . . . . . 10 ((𝑥𝑆𝑦 = 𝑥) ↔ (𝑥𝑆𝑥 = 𝑦))
4 ancom 464 . . . . . . . . . 10 ((𝑥𝑆𝑥 = 𝑦) ↔ (𝑥 = 𝑦𝑥𝑆))
53, 4bitri 262 . . . . . . . . 9 ((𝑥𝑆𝑦 = 𝑥) ↔ (𝑥 = 𝑦𝑥𝑆))
65exbii 1763 . . . . . . . 8 (∃𝑥(𝑥𝑆𝑦 = 𝑥) ↔ ∃𝑥(𝑥 = 𝑦𝑥𝑆))
7 vex 3175 . . . . . . . . 9 𝑦 ∈ V
8 eleq1 2675 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑆𝑦𝑆))
97, 8ceqsexv 3214 . . . . . . . 8 (∃𝑥(𝑥 = 𝑦𝑥𝑆) ↔ 𝑦𝑆)
106, 9bitri 262 . . . . . . 7 (∃𝑥(𝑥𝑆𝑦 = 𝑥) ↔ 𝑦𝑆)
111, 10bitri 262 . . . . . 6 (∃𝑥𝑆 𝑦 = 𝑥𝑦𝑆)
1211abbii 2725 . . . . 5 {𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥} = {𝑦𝑦𝑆}
13 abid2 2731 . . . . 5 {𝑦𝑦𝑆} = 𝑆
1412, 13eqtr2i 2632 . . . 4 𝑆 = {𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥}
1514fveq2i 6090 . . 3 (𝐺𝑆) = (𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})
1615fveq2i 6090 . 2 (𝑀‘(𝐺𝑆)) = (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥}))
17 dfss3 3557 . . 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 33856 . . 3 ((𝐾 ∈ HL ∧ ∀𝑥𝑆 𝑥𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})) = 𝑥𝑆 (𝑀𝑥))
2217, 21syl3an2b 1354 . 2 ((𝐾 ∈ HL ∧ 𝑆𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})) = 𝑥𝑆 (𝑀𝑥))
2316, 22syl5eq 2655 1 ((𝐾 ∈ HL ∧ 𝑆𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺𝑆)) = 𝑥𝑆 (𝑀𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1976  {cab 2595  wne 2779  wral 2895  wrex 2896  wss 3539  c0 3873   ciin 4450  cfv 5789  Basecbs 15643  glbcglb 16714  HLchlt 33438  pmapcpmap 33584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-poset 16717  df-lub 16745  df-glb 16746  df-join 16747  df-meet 16748  df-lat 16817  df-clat 16879  df-ats 33355  df-hlat 33439  df-pmap 33591
This theorem is referenced by:  pmapglb2N  33858  pmapmeet  33860
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