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Theorem pmapglbx 39771
Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 39772, 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 39359 . . . . . . . 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 2737 . . . . . . . . 9 (Atoms‘𝐾) = (Atoms‘𝐾)
53, 4atbase 39290 . . . . . . . 8 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
65adantl 481 . . . . . . 7 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝𝐵)
7 r19.29 3114 . . . . . . . . . . 11 ((∀𝑖𝐼 𝑆𝐵 ∧ ∃𝑖𝐼 𝑦 = 𝑆) → ∃𝑖𝐼 (𝑆𝐵𝑦 = 𝑆))
8 eleq1a 2836 . . . . . . . . . . . . 13 (𝑆𝐵 → (𝑦 = 𝑆𝑦𝐵))
98imp 406 . . . . . . . . . . . 12 ((𝑆𝐵𝑦 = 𝑆) → 𝑦𝐵)
109rexlimivw 3151 . . . . . . . . . . 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 4068 . . . . . . 7 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ⊆ 𝐵)
15 eqid 2737 . . . . . . . 8 (le‘𝐾) = (le‘𝐾)
16 pmapglb.g . . . . . . . 8 𝐺 = (glb‘𝐾)
173, 15, 16clatleglb 18563 . . . . . . 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 1373 . . . . . 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 3484 . . . . . . . . . . . . 13 𝑧 ∈ V
20 eqeq1 2741 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑦 = 𝑆𝑧 = 𝑆))
2120rexbidv 3179 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (∃𝑖𝐼 𝑦 = 𝑆 ↔ ∃𝑖𝐼 𝑧 = 𝑆))
2219, 21elab 3679 . . . . . . . . . . . 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 3183 . . . . . . . . . . 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 1819 . . . . . . . . 9 (∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑧𝑖𝐼 (𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
27 df-ral 3062 . . . . . . . . 9 (∀𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧))
28 ralcom4 3286 . . . . . . . . 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 1914 . . . . . . . . . . 11 𝑧 𝑝(le‘𝐾)𝑆
31 breq2 5147 . . . . . . . . . . 11 (𝑧 = 𝑆 → (𝑝(le‘𝐾)𝑧𝑝(le‘𝐾)𝑆))
3230, 31ceqsalg 3517 . . . . . . . . . 10 (𝑆𝐵 → (∀𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
3332ralimi 3083 . . . . . . . . 9 (∀𝑖𝐼 𝑆𝐵 → ∀𝑖𝐼 (∀𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
34 ralbi 3103 . . . . . . . . 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 3443 . . . 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 1133 . . 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 1137 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝐾 ∈ HL)
4212abssdv 4068 . . . . . 6 (∀𝑖𝐼 𝑆𝐵 → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ⊆ 𝐵)
433, 16clatglbcl 18550 . . . . . 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 1133 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ∈ 𝐵)
46 pmapglb.m . . . . 5 𝑀 = (pmap‘𝐾)
473, 15, 4, 46pmapval 39759 . . . 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 5069 . . . 4 (𝐼 ≠ ∅ → 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆})
50493ad2ant3 1136 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆})
5140, 48, 503eqtr4d 2787 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
52 nfv 1914 . . . 4 𝑖 𝐾 ∈ HL
53 nfra1 3284 . . . 4 𝑖𝑖𝐼 𝑆𝐵
54 nfv 1914 . . . 4 𝑖 𝐼 ≠ ∅
5552, 53, 54nf3an 1901 . . 3 𝑖(𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅)
56 simpl1 1192 . . . 4 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) ∧ 𝑖𝐼) → 𝐾 ∈ HL)
57 rspa 3248 . . . . 5 ((∀𝑖𝐼 𝑆𝐵𝑖𝐼) → 𝑆𝐵)
58573ad2antl2 1187 . . . 4 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) ∧ 𝑖𝐼) → 𝑆𝐵)
593, 15, 4, 46pmapval 39759 . . . 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 5015 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝑖𝐼 (𝑀𝑆) = 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
6251, 61eqtr4d 2780 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 1087  wal 1538   = wceq 1540  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  {crab 3436  wss 3951  c0 4333   ciin 4992   class class class wbr 5143  cfv 6561  Basecbs 17247  lecple 17304  glbcglb 18356  CLatccla 18543  Atomscatm 39264  HLchlt 39351  pmapcpmap 39499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-poset 18359  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-lat 18477  df-clat 18544  df-ats 39268  df-hlat 39352  df-pmap 39506
This theorem is referenced by:  pmapglb  39772  pmapglb2xN  39774
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