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Theorem pmapglb 35578
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 3057 . . . . . . 7 (∃𝑥𝑆 𝑦 = 𝑥 ↔ ∃𝑥(𝑥𝑆𝑦 = 𝑥))
2 equcom 2101 . . . . . . . . . . 11 (𝑦 = 𝑥𝑥 = 𝑦)
32anbi2i 732 . . . . . . . . . 10 ((𝑥𝑆𝑦 = 𝑥) ↔ (𝑥𝑆𝑥 = 𝑦))
4 ancom 465 . . . . . . . . . 10 ((𝑥𝑆𝑥 = 𝑦) ↔ (𝑥 = 𝑦𝑥𝑆))
53, 4bitri 264 . . . . . . . . 9 ((𝑥𝑆𝑦 = 𝑥) ↔ (𝑥 = 𝑦𝑥𝑆))
65exbii 1923 . . . . . . . 8 (∃𝑥(𝑥𝑆𝑦 = 𝑥) ↔ ∃𝑥(𝑥 = 𝑦𝑥𝑆))
7 vex 3344 . . . . . . . . 9 𝑦 ∈ V
8 eleq1w 2823 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑆𝑦𝑆))
97, 8ceqsexv 3383 . . . . . . . 8 (∃𝑥(𝑥 = 𝑦𝑥𝑆) ↔ 𝑦𝑆)
106, 9bitri 264 . . . . . . 7 (∃𝑥(𝑥𝑆𝑦 = 𝑥) ↔ 𝑦𝑆)
111, 10bitri 264 . . . . . 6 (∃𝑥𝑆 𝑦 = 𝑥𝑦𝑆)
1211abbii 2878 . . . . 5 {𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥} = {𝑦𝑦𝑆}
13 abid2 2884 . . . . 5 {𝑦𝑦𝑆} = 𝑆
1412, 13eqtr2i 2784 . . . 4 𝑆 = {𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥}
1514fveq2i 6357 . . 3 (𝐺𝑆) = (𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})
1615fveq2i 6357 . 2 (𝑀‘(𝐺𝑆)) = (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥}))
17 dfss3 3734 . . 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 35577 . . 3 ((𝐾 ∈ HL ∧ ∀𝑥𝑆 𝑥𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})) = 𝑥𝑆 (𝑀𝑥))
2217, 21syl3an2b 1511 . 2 ((𝐾 ∈ HL ∧ 𝑆𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})) = 𝑥𝑆 (𝑀𝑥))
2316, 22syl5eq 2807 1 ((𝐾 ∈ HL ∧ 𝑆𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺𝑆)) = 𝑥𝑆 (𝑀𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2140  {cab 2747  wne 2933  wral 3051  wrex 3052  wss 3716  c0 4059   ciin 4674  cfv 6050  Basecbs 16080  glbcglb 17165  HLchlt 35159  pmapcpmap 35305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-iin 4676  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-poset 17168  df-lub 17196  df-glb 17197  df-join 17198  df-meet 17199  df-lat 17268  df-clat 17330  df-ats 35076  df-hlat 35160  df-pmap 35312
This theorem is referenced by:  pmapglb2N  35579  pmapmeet  35581
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