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Theorem pmapglbx 39179
Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 39180, 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 38767 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ CLat)
21ad2antrr 725 . . . . . . 7 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ CLat)
3 pmapglb.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
4 eqid 2727 . . . . . . . . 9 (Atoms‘𝐾) = (Atoms‘𝐾)
53, 4atbase 38698 . . . . . . . 8 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
65adantl 481 . . . . . . 7 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝𝐵)
7 r19.29 3109 . . . . . . . . . . 11 ((∀𝑖𝐼 𝑆𝐵 ∧ ∃𝑖𝐼 𝑦 = 𝑆) → ∃𝑖𝐼 (𝑆𝐵𝑦 = 𝑆))
8 eleq1a 2823 . . . . . . . . . . . . 13 (𝑆𝐵 → (𝑦 = 𝑆𝑦𝐵))
98imp 406 . . . . . . . . . . . 12 ((𝑆𝐵𝑦 = 𝑆) → 𝑦𝐵)
109rexlimivw 3146 . . . . . . . . . . 11 (∃𝑖𝐼 (𝑆𝐵𝑦 = 𝑆) → 𝑦𝐵)
117, 10syl 17 . . . . . . . . . 10 ((∀𝑖𝐼 𝑆𝐵 ∧ ∃𝑖𝐼 𝑦 = 𝑆) → 𝑦𝐵)
1211ex 412 . . . . . . . . 9 (∀𝑖𝐼 𝑆𝐵 → (∃𝑖𝐼 𝑦 = 𝑆𝑦𝐵))
1312ad2antlr 726 . . . . . . . 8 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (∃𝑖𝐼 𝑦 = 𝑆𝑦𝐵))
1413abssdv 4061 . . . . . . 7 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ⊆ 𝐵)
15 eqid 2727 . . . . . . . 8 (le‘𝐾) = (le‘𝐾)
16 pmapglb.g . . . . . . . 8 𝐺 = (glb‘𝐾)
173, 15, 16clatleglb 18501 . . . . . . 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 1369 . . . . . 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 3473 . . . . . . . . . . . . 13 𝑧 ∈ V
20 eqeq1 2731 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑦 = 𝑆𝑧 = 𝑆))
2120rexbidv 3173 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (∃𝑖𝐼 𝑦 = 𝑆 ↔ ∃𝑖𝐼 𝑧 = 𝑆))
2219, 21elab 3665 . . . . . . . . . . . 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 3177 . . . . . . . . . . 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 1814 . . . . . . . . 9 (∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑧𝑖𝐼 (𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
27 df-ral 3057 . . . . . . . . 9 (∀𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧))
28 ralcom4 3278 . . . . . . . . 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 1910 . . . . . . . . . . 11 𝑧 𝑝(le‘𝐾)𝑆
31 breq2 5146 . . . . . . . . . . 11 (𝑧 = 𝑆 → (𝑝(le‘𝐾)𝑧𝑝(le‘𝐾)𝑆))
3230, 31ceqsalg 3503 . . . . . . . . . 10 (𝑆𝐵 → (∀𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
3332ralimi 3078 . . . . . . . . 9 (∀𝑖𝐼 𝑆𝐵 → ∀𝑖𝐼 (∀𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
34 ralbi 3098 . . . . . . . . 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 726 . . . . . 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 3434 . . . 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 1130 . . 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 1134 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝐾 ∈ HL)
4212abssdv 4061 . . . . . 6 (∀𝑖𝐼 𝑆𝐵 → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ⊆ 𝐵)
433, 16clatglbcl 18488 . . . . . 6 ((𝐾 ∈ CLat ∧ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ⊆ 𝐵) → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ∈ 𝐵)
441, 42, 43syl2an 595 . . . . 5 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ∈ 𝐵)
45443adant3 1130 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ∈ 𝐵)
46 pmapglb.m . . . . 5 𝑀 = (pmap‘𝐾)
473, 15, 4, 46pmapval 39167 . . . 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 583 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})})
49 iinrab 5066 . . . 4 (𝐼 ≠ ∅ → 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆})
50493ad2ant3 1133 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆})
5140, 48, 503eqtr4d 2777 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
52 nfv 1910 . . . 4 𝑖 𝐾 ∈ HL
53 nfra1 3276 . . . 4 𝑖𝑖𝐼 𝑆𝐵
54 nfv 1910 . . . 4 𝑖 𝐼 ≠ ∅
5552, 53, 54nf3an 1897 . . 3 𝑖(𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅)
56 simpl1 1189 . . . 4 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) ∧ 𝑖𝐼) → 𝐾 ∈ HL)
57 rspa 3240 . . . . 5 ((∀𝑖𝐼 𝑆𝐵𝑖𝐼) → 𝑆𝐵)
58573ad2antl2 1184 . . . 4 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) ∧ 𝑖𝐼) → 𝑆𝐵)
593, 15, 4, 46pmapval 39167 . . . 4 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑀𝑆) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
6056, 58, 59syl2anc 583 . . 3 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) ∧ 𝑖𝐼) → (𝑀𝑆) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
6155, 60iineq2d 5014 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝑖𝐼 (𝑀𝑆) = 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
6251, 61eqtr4d 2770 1 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 (𝑀𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085  wal 1532   = wceq 1534  wcel 2099  {cab 2704  wne 2935  wral 3056  wrex 3065  {crab 3427  wss 3944  c0 4318   ciin 4992   class class class wbr 5142  cfv 6542  Basecbs 17171  lecple 17231  glbcglb 18293  CLatccla 18481  Atomscatm 38672  HLchlt 38759  pmapcpmap 38907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-poset 18296  df-lub 18329  df-glb 18330  df-join 18331  df-meet 18332  df-lat 18415  df-clat 18482  df-ats 38676  df-hlat 38760  df-pmap 38914
This theorem is referenced by:  pmapglb  39180  pmapglb2xN  39182
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