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Theorem pmapglb 35550
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 3102 . . . . . . 7 (∃𝑥𝑆 𝑦 = 𝑥 ↔ ∃𝑥(𝑥𝑆𝑦 = 𝑥))
2 equcom 2114 . . . . . . . . . . 11 (𝑦 = 𝑥𝑥 = 𝑦)
32anbi2i 611 . . . . . . . . . 10 ((𝑥𝑆𝑦 = 𝑥) ↔ (𝑥𝑆𝑥 = 𝑦))
4 ancom 450 . . . . . . . . . 10 ((𝑥𝑆𝑥 = 𝑦) ↔ (𝑥 = 𝑦𝑥𝑆))
53, 4bitri 266 . . . . . . . . 9 ((𝑥𝑆𝑦 = 𝑥) ↔ (𝑥 = 𝑦𝑥𝑆))
65exbii 1933 . . . . . . . 8 (∃𝑥(𝑥𝑆𝑦 = 𝑥) ↔ ∃𝑥(𝑥 = 𝑦𝑥𝑆))
7 vex 3394 . . . . . . . . 9 𝑦 ∈ V
8 eleq1w 2868 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑆𝑦𝑆))
97, 8ceqsexv 3436 . . . . . . . 8 (∃𝑥(𝑥 = 𝑦𝑥𝑆) ↔ 𝑦𝑆)
106, 9bitri 266 . . . . . . 7 (∃𝑥(𝑥𝑆𝑦 = 𝑥) ↔ 𝑦𝑆)
111, 10bitri 266 . . . . . 6 (∃𝑥𝑆 𝑦 = 𝑥𝑦𝑆)
1211abbii 2923 . . . . 5 {𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥} = {𝑦𝑦𝑆}
13 abid2 2929 . . . . 5 {𝑦𝑦𝑆} = 𝑆
1412, 13eqtr2i 2829 . . . 4 𝑆 = {𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥}
1514fveq2i 6411 . . 3 (𝐺𝑆) = (𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})
1615fveq2i 6411 . 2 (𝑀‘(𝐺𝑆)) = (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥}))
17 dfss3 3787 . . 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 35549 . . 3 ((𝐾 ∈ HL ∧ ∀𝑥𝑆 𝑥𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})) = 𝑥𝑆 (𝑀𝑥))
2217, 21syl3an2b 1516 . 2 ((𝐾 ∈ HL ∧ 𝑆𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥𝑆 𝑦 = 𝑥})) = 𝑥𝑆 (𝑀𝑥))
2316, 22syl5eq 2852 1 ((𝐾 ∈ HL ∧ 𝑆𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺𝑆)) = 𝑥𝑆 (𝑀𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2156  {cab 2792  wne 2978  wral 3096  wrex 3097  wss 3769  c0 4116   ciin 4713  cfv 6101  Basecbs 16068  glbcglb 17148  HLchlt 35130  pmapcpmap 35277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-oprab 6878  df-poset 17151  df-lub 17179  df-glb 17180  df-join 17181  df-meet 17182  df-lat 17251  df-clat 17313  df-ats 35047  df-hlat 35131  df-pmap 35284
This theorem is referenced by:  pmapglb2N  35551  pmapmeet  35553
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