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Theorem pmapglbx 35782
Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 35783, 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 35371 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ CLat)
21ad2antrr 718 . . . . . . 7 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ CLat)
3 pmapglb.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
4 eqid 2797 . . . . . . . . 9 (Atoms‘𝐾) = (Atoms‘𝐾)
53, 4atbase 35302 . . . . . . . 8 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
65adantl 474 . . . . . . 7 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝𝐵)
7 r19.29 3251 . . . . . . . . . . 11 ((∀𝑖𝐼 𝑆𝐵 ∧ ∃𝑖𝐼 𝑦 = 𝑆) → ∃𝑖𝐼 (𝑆𝐵𝑦 = 𝑆))
8 eleq1a 2871 . . . . . . . . . . . . 13 (𝑆𝐵 → (𝑦 = 𝑆𝑦𝐵))
98imp 396 . . . . . . . . . . . 12 ((𝑆𝐵𝑦 = 𝑆) → 𝑦𝐵)
109rexlimivw 3208 . . . . . . . . . . 11 (∃𝑖𝐼 (𝑆𝐵𝑦 = 𝑆) → 𝑦𝐵)
117, 10syl 17 . . . . . . . . . 10 ((∀𝑖𝐼 𝑆𝐵 ∧ ∃𝑖𝐼 𝑦 = 𝑆) → 𝑦𝐵)
1211ex 402 . . . . . . . . 9 (∀𝑖𝐼 𝑆𝐵 → (∃𝑖𝐼 𝑦 = 𝑆𝑦𝐵))
1312ad2antlr 719 . . . . . . . 8 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (∃𝑖𝐼 𝑦 = 𝑆𝑦𝐵))
1413abssdv 3870 . . . . . . 7 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ⊆ 𝐵)
15 eqid 2797 . . . . . . . 8 (le‘𝐾) = (le‘𝐾)
16 pmapglb.g . . . . . . . 8 𝐺 = (glb‘𝐾)
173, 15, 16clatleglb 17438 . . . . . . 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 1491 . . . . . 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 3386 . . . . . . . . . . . . 13 𝑧 ∈ V
20 eqeq1 2801 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑦 = 𝑆𝑧 = 𝑆))
2120rexbidv 3231 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (∃𝑖𝐼 𝑦 = 𝑆 ↔ ∃𝑖𝐼 𝑧 = 𝑆))
2219, 21elab 3540 . . . . . . . . . . . 12 (𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ↔ ∃𝑖𝐼 𝑧 = 𝑆)
2322imbi1i 341 . . . . . . . . . . 11 ((𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ (∃𝑖𝐼 𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
24 r19.23v 3202 . . . . . . . . . . 11 (∀𝑖𝐼 (𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ (∃𝑖𝐼 𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
2523, 24bitr4i 270 . . . . . . . . . 10 ((𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑖𝐼 (𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
2625albii 1915 . . . . . . . . 9 (∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑧𝑖𝐼 (𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
27 df-ral 3092 . . . . . . . . 9 (∀𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧))
28 ralcom4 3410 . . . . . . . . 9 (∀𝑖𝐼𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ ∀𝑧𝑖𝐼 (𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
2926, 27, 283bitr4i 295 . . . . . . . 8 (∀𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑖𝐼𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
30 nfv 2010 . . . . . . . . . . 11 𝑧 𝑝(le‘𝐾)𝑆
31 breq2 4845 . . . . . . . . . . 11 (𝑧 = 𝑆 → (𝑝(le‘𝐾)𝑧𝑝(le‘𝐾)𝑆))
3230, 31ceqsalg 3416 . . . . . . . . . 10 (𝑆𝐵 → (∀𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
3332ralimi 3131 . . . . . . . . 9 (∀𝑖𝐼 𝑆𝐵 → ∀𝑖𝐼 (∀𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
34 ralbi 3247 . . . . . . . . 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, 35syl5bb 275 . . . . . . 7 (∀𝑖𝐼 𝑆𝐵 → (∀𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆))
3736ad2antlr 719 . . . . . 6 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (∀𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆))
3818, 37bitrd 271 . . . . 5 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ↔ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆))
3938rabbidva 3370 . . . 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 1163 . . 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 1167 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝐾 ∈ HL)
4212abssdv 3870 . . . . . 6 (∀𝑖𝐼 𝑆𝐵 → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ⊆ 𝐵)
433, 16clatglbcl 17426 . . . . . 6 ((𝐾 ∈ CLat ∧ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ⊆ 𝐵) → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ∈ 𝐵)
441, 42, 43syl2an 590 . . . . 5 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ∈ 𝐵)
45443adant3 1163 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ∈ 𝐵)
46 pmapglb.m . . . . 5 𝑀 = (pmap‘𝐾)
473, 15, 4, 46pmapval 35770 . . . 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 580 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})})
49 iinrab 4770 . . . 4 (𝐼 ≠ ∅ → 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆})
50493ad2ant3 1166 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆})
5140, 48, 503eqtr4d 2841 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
52 nfv 2010 . . . 4 𝑖 𝐾 ∈ HL
53 nfra1 3120 . . . 4 𝑖𝑖𝐼 𝑆𝐵
54 nfv 2010 . . . 4 𝑖 𝐼 ≠ ∅
5552, 53, 54nf3an 2001 . . 3 𝑖(𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅)
56 simpl1 1243 . . . 4 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) ∧ 𝑖𝐼) → 𝐾 ∈ HL)
57 rspa 3109 . . . . 5 ((∀𝑖𝐼 𝑆𝐵𝑖𝐼) → 𝑆𝐵)
58573ad2antl2 1238 . . . 4 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) ∧ 𝑖𝐼) → 𝑆𝐵)
593, 15, 4, 46pmapval 35770 . . . 4 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑀𝑆) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
6056, 58, 59syl2anc 580 . . 3 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) ∧ 𝑖𝐼) → (𝑀𝑆) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
6155, 60iineq2d 4729 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝑖𝐼 (𝑀𝑆) = 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
6251, 61eqtr4d 2834 1 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 (𝑀𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108  wal 1651   = wceq 1653  wcel 2157  {cab 2783  wne 2969  wral 3087  wrex 3088  {crab 3091  wss 3767  c0 4113   ciin 4709   class class class wbr 4841  cfv 6099  Basecbs 16181  lecple 16271  glbcglb 17255  CLatccla 17419  Atomscatm 35276  HLchlt 35363  pmapcpmap 35510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-iin 4711  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-poset 17258  df-lub 17286  df-glb 17287  df-join 17288  df-meet 17289  df-lat 17358  df-clat 17420  df-ats 35280  df-hlat 35364  df-pmap 35517
This theorem is referenced by:  pmapglb  35783  pmapglb2xN  35785
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