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Theorem pmapglbx 35373
Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 35374, 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 34963 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ CLat)
21ad2antrr 762 . . . . . . 7 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ CLat)
3 pmapglb.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
4 eqid 2651 . . . . . . . . 9 (Atoms‘𝐾) = (Atoms‘𝐾)
53, 4atbase 34894 . . . . . . . 8 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
65adantl 481 . . . . . . 7 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝𝐵)
7 r19.29 3101 . . . . . . . . . . 11 ((∀𝑖𝐼 𝑆𝐵 ∧ ∃𝑖𝐼 𝑦 = 𝑆) → ∃𝑖𝐼 (𝑆𝐵𝑦 = 𝑆))
8 eleq1a 2725 . . . . . . . . . . . . 13 (𝑆𝐵 → (𝑦 = 𝑆𝑦𝐵))
98imp 444 . . . . . . . . . . . 12 ((𝑆𝐵𝑦 = 𝑆) → 𝑦𝐵)
109rexlimivw 3058 . . . . . . . . . . 11 (∃𝑖𝐼 (𝑆𝐵𝑦 = 𝑆) → 𝑦𝐵)
117, 10syl 17 . . . . . . . . . 10 ((∀𝑖𝐼 𝑆𝐵 ∧ ∃𝑖𝐼 𝑦 = 𝑆) → 𝑦𝐵)
1211ex 449 . . . . . . . . 9 (∀𝑖𝐼 𝑆𝐵 → (∃𝑖𝐼 𝑦 = 𝑆𝑦𝐵))
1312ad2antlr 763 . . . . . . . 8 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (∃𝑖𝐼 𝑦 = 𝑆𝑦𝐵))
1413abssdv 3709 . . . . . . 7 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ⊆ 𝐵)
15 eqid 2651 . . . . . . . 8 (le‘𝐾) = (le‘𝐾)
16 pmapglb.g . . . . . . . 8 𝐺 = (glb‘𝐾)
173, 15, 16clatleglb 17173 . . . . . . 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 1366 . . . . . 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 3234 . . . . . . . . . . . . 13 𝑧 ∈ V
20 eqeq1 2655 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑦 = 𝑆𝑧 = 𝑆))
2120rexbidv 3081 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (∃𝑖𝐼 𝑦 = 𝑆 ↔ ∃𝑖𝐼 𝑧 = 𝑆))
2219, 21elab 3382 . . . . . . . . . . . 12 (𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ↔ ∃𝑖𝐼 𝑧 = 𝑆)
2322imbi1i 338 . . . . . . . . . . 11 ((𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ (∃𝑖𝐼 𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
24 r19.23v 3052 . . . . . . . . . . 11 (∀𝑖𝐼 (𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ (∃𝑖𝐼 𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
2523, 24bitr4i 267 . . . . . . . . . 10 ((𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑖𝐼 (𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
2625albii 1787 . . . . . . . . 9 (∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑧𝑖𝐼 (𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
27 df-ral 2946 . . . . . . . . 9 (∀𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧))
28 ralcom4 3255 . . . . . . . . 9 (∀𝑖𝐼𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ ∀𝑧𝑖𝐼 (𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
2926, 27, 283bitr4i 292 . . . . . . . 8 (∀𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑖𝐼𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧))
30 nfv 1883 . . . . . . . . . . 11 𝑧 𝑝(le‘𝐾)𝑆
31 breq2 4689 . . . . . . . . . . 11 (𝑧 = 𝑆 → (𝑝(le‘𝐾)𝑧𝑝(le‘𝐾)𝑆))
3230, 31ceqsalg 3261 . . . . . . . . . 10 (𝑆𝐵 → (∀𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
3332ralimi 2981 . . . . . . . . 9 (∀𝑖𝐼 𝑆𝐵 → ∀𝑖𝐼 (∀𝑧(𝑧 = 𝑆𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
34 ralbi 3097 . . . . . . . . 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 272 . . . . . . 7 (∀𝑖𝐼 𝑆𝐵 → (∀𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆))
3736ad2antlr 763 . . . . . 6 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (∀𝑧 ∈ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆))
3818, 37bitrd 268 . . . . 5 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ↔ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆))
3938rabbidva 3219 . . . 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 1101 . . 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 1081 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝐾 ∈ HL)
4212abssdv 3709 . . . . . 6 (∀𝑖𝐼 𝑆𝐵 → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ⊆ 𝐵)
433, 16clatglbcl 17161 . . . . . 6 ((𝐾 ∈ CLat ∧ {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} ⊆ 𝐵) → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ∈ 𝐵)
441, 42, 43syl2an 493 . . . . 5 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ∈ 𝐵)
45443adant3 1101 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) ∈ 𝐵)
46 pmapglb.m . . . . 5 𝑀 = (pmap‘𝐾)
473, 15, 4, 46pmapval 35361 . . . 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 694 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})})
49 iinrab 4614 . . . 4 (𝐼 ≠ ∅ → 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆})
50493ad2ant3 1104 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖𝐼 𝑝(le‘𝐾)𝑆})
5140, 48, 503eqtr4d 2695 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
52 nfv 1883 . . . 4 𝑖 𝐾 ∈ HL
53 nfra1 2970 . . . 4 𝑖𝑖𝐼 𝑆𝐵
54 nfv 1883 . . . 4 𝑖 𝐼 ≠ ∅
5552, 53, 54nf3an 1871 . . 3 𝑖(𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅)
56 simpl1 1084 . . . 4 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) ∧ 𝑖𝐼) → 𝐾 ∈ HL)
57 rspa 2959 . . . . 5 ((∀𝑖𝐼 𝑆𝐵𝑖𝐼) → 𝑆𝐵)
58573ad2antl2 1244 . . . 4 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) ∧ 𝑖𝐼) → 𝑆𝐵)
593, 15, 4, 46pmapval 35361 . . . 4 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑀𝑆) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
6056, 58, 59syl2anc 694 . . 3 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) ∧ 𝑖𝐼) → (𝑀𝑆) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
6155, 60iineq2d 4573 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝑖𝐼 (𝑀𝑆) = 𝑖𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
6251, 61eqtr4d 2688 1 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 (𝑀𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054  wal 1521   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  {crab 2945  wss 3607  c0 3948   ciin 4553   class class class wbr 4685  cfv 5926  Basecbs 15904  lecple 15995  glbcglb 16990  CLatccla 17154  Atomscatm 34868  HLchlt 34955  pmapcpmap 35101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-poset 16993  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-lat 17093  df-clat 17155  df-ats 34872  df-hlat 34956  df-pmap 35108
This theorem is referenced by:  pmapglb  35374  pmapglb2xN  35376
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