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Theorem pmapglbx 39752
Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 39753, where we read 𝑆 as 𝑆(𝑖). 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
pmapglbx ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 (𝑀𝑆))
Distinct variable groups:   𝑦,𝑖,𝐵   𝑖,𝐼,𝑦   𝑖,𝐾,𝑦   𝑦,𝑆
Allowed substitution hints:   𝑆(𝑖)   𝐺(𝑦,𝑖)   𝑀(𝑦,𝑖)

Proof of Theorem pmapglbx
Dummy variables 𝑝 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hlclat 39340 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ CLat)
21ad2antrr 726 . . . . . . 7 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ CLat)
3 pmapglb.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
4 eqid 2735 . . . . . . . . 9 (Atoms‘𝐾) = (Atoms‘𝐾)
53, 4atbase 39271 . . . . . . . 8 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
65adantl 481 . . . . . . 7 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝𝐵)
7 r19.29 3112 . . . . . . . . . . 11 ((∀𝑖𝐼 𝑆𝐵 ∧ ∃𝑖𝐼 𝑦 = 𝑆) → ∃𝑖𝐼 (𝑆𝐵𝑦 = 𝑆))
8 eleq1a 2834 . . . . . . . . . . . . 13 (𝑆𝐵 → (𝑦 = 𝑆𝑦𝐵))
98imp 406 . . . . . . . . . . . 12 ((𝑆𝐵𝑦 = 𝑆) → 𝑦𝐵)
109rexlimivw 3149 . . . . . . . . . . 11 (∃𝑖𝐼 (𝑆𝐵𝑦 = 𝑆) → 𝑦𝐵)
117, 10syl 17 . . . . . . . . . 10 ((∀𝑖𝐼 𝑆𝐵 ∧ ∃𝑖𝐼 𝑦 = 𝑆) → 𝑦𝐵)
1211ex 412 . . . . . . . . 9 (∀𝑖𝐼 𝑆𝐵 → (∃𝑖𝐼 𝑦 = 𝑆𝑦𝐵))
1312ad2antlr 727 . . . . . . . 8 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (∃𝑖𝐼 𝑦 = 𝑆𝑦𝐵))
1413abssdv 4078 . . . . . . 7 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ⊆ 𝐵)
15 eqid 2735 . . . . . . . 8 (le‘𝐾) = (le‘𝐾)
16 pmapglb.g . . . . . . . 8 𝐺 = (glb‘𝐾)
173, 15, 16clatleglb 18576 . . . . . . 7 ((𝐾 ∈ CLat ∧ 𝑝𝐵 ∧ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ⊆ 𝐵) → (𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧))
182, 6, 14, 17syl3anc 1370 . . . . . 6 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧))
19 vex 3482 . . . . . . . . . . . . 13 𝑧 ∈ V
20 eqeq1 2739 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑦 = 𝑆𝑧 = 𝑆))
2120rexbidv 3177 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (∃𝑖𝐼 𝑦 = 𝑆 ↔ ∃𝑖𝐼 𝑧 = 𝑆))
2219, 21elab 3681 . . . . . . . . . . . 12 (𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ↔ ∃𝑖𝐼 𝑧 = 𝑆)
2322imbi1i 349 . . . . . . . . . . 11 ((𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ (∃𝑖𝐼 𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
24 r19.23v 3181 . . . . . . . . . . 11 (∀𝑖𝐼 (𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ (∃𝑖𝐼 𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
2523, 24bitr4i 278 . . . . . . . . . 10 ((𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑖𝐼 (𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
2625albii 1816 . . . . . . . . 9 (∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑧𝑖𝐼 (𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
27 df-ral 3060 . . . . . . . . 9 (∀𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧))
28 ralcom4 3284 . . . . . . . . 9 (∀𝑖𝐼𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ ∀𝑧𝑖𝐼 (𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
2926, 27, 283bitr4i 303 . . . . . . . 8 (∀𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑖𝐼𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
30 nfv 1912 . . . . . . . . . . 11 𝑧 𝑝(le‘𝐾)𝑆
31 breq2 5152 . . . . . . . . . . 11 (𝑧 = 𝑆 → (𝑝(le‘𝐾)𝑧𝑝(le‘𝐾)𝑆))
3230, 31ceqsalg 3515 . . . . . . . . . 10 (𝑆𝐵 → (∀𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
3332ralimi 3081 . . . . . . . . 9 (∀𝑖𝐼 𝑆𝐵 → ∀𝑖𝐼 (∀𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
34 ralbi 3101 . . . . . . . . 9 (∀𝑖𝐼 (∀𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆) → (∀𝑖𝐼𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆))
3533, 34syl 17 . . . . . . . 8 (∀𝑖𝐼 𝑆𝐵 → (∀𝑖𝐼𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆))
3629, 35bitrid 283 . . . . . . 7 (∀𝑖𝐼 𝑆𝐵 → (∀𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆))
3736ad2antlr 727 . . . . . 6 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (∀𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆))
3818, 37bitrd 279 . . . . 5 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ↔ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆))
3938rabbidva 3440 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆})
40393adant3 1131 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆})
41 simp1 1135 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝐾 ∈ HL)
4212abssdv 4078 . . . . . 6 (∀𝑖𝐼 𝑆𝐵 → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ⊆ 𝐵)
433, 16clatglbcl 18563 . . . . . 6 ((𝐾 ∈ CLat ∧ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ⊆ 𝐵) → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ∈ 𝐵)
441, 42, 43syl2an 596 . . . . 5 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ∈ 𝐵)
45443adant3 1131 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ∈ 𝐵)
46 pmapglb.m . . . . 5 𝑀 = (pmap‘𝐾)
473, 15, 4, 46pmapval 39740 . . . 4 ((𝐾 ∈ HL ∧ (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ∈ 𝐵) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})})
4841, 45, 47syl2anc 584 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})})
49 iinrab 5074 . . . 4 (𝐼 ≠ ∅ → 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆})
50493ad2ant3 1134 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆})
5140, 48, 503eqtr4d 2785 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
52 nfv 1912 . . . 4 𝑖 𝐾 ∈ HL
53 nfra1 3282 . . . 4 𝑖𝑖𝐼 𝑆𝐵
54 nfv 1912 . . . 4 𝑖 𝐼 ≠ ∅
5552, 53, 54nf3an 1899 . . 3 𝑖(𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅)
56 simpl1 1190 . . . 4 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) ∧ 𝑖𝐼) → 𝐾 ∈ HL)
57 rspa 3246 . . . . 5 ((∀𝑖𝐼 𝑆𝐵𝑖𝐼) → 𝑆𝐵)
58573ad2antl2 1185 . . . 4 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) ∧ 𝑖𝐼) → 𝑆𝐵)
593, 15, 4, 46pmapval 39740 . . . 4 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑀𝑆) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
6056, 58, 59syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) ∧ 𝑖𝐼) → (𝑀𝑆) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
6155, 60iineq2d 5020 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝑖𝐼 (𝑀𝑆) = 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
6251, 61eqtr4d 2778 1 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 (𝑀𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1535   = wceq 1537  wcel 2106  {cab 2712  wne 2938  wral 3059  wrex 3068  {crab 3433  wss 3963  c0 4339   ciin 4997   class class class wbr 5148  cfv 6563  Basecbs 17245  lecple 17305  glbcglb 18368  CLatccla 18556  Atomscatm 39245  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:  pmapglb  39753  pmapglb2xN  39755
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