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Theorem pmapglb2xN 34076
 Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 34075, where we read 𝑆 as 𝑆(𝑖). Extension of Theorem 15.5.2 of [MaedaMaeda] p. 62 that allows 𝐼 = ∅. (Contributed by NM, 21-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmapglb2.b 𝐵 = (Base‘𝐾)
pmapglb2.g 𝐺 = (glb‘𝐾)
pmapglb2.a 𝐴 = (Atoms‘𝐾)
pmapglb2.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
pmapglb2xN ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆)))
Distinct variable groups:   𝐴,𝑖   𝑦,𝑖,𝐵   𝑖,𝐼,𝑦   𝑖,𝐾,𝑦   𝑦,𝑆
Allowed substitution hints:   𝐴(𝑦)   𝑆(𝑖)   𝐺(𝑦,𝑖)   𝑀(𝑦,𝑖)

Proof of Theorem pmapglb2xN
StepHypRef Expression
1 hlop 33667 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OP)
2 pmapglb2.g . . . . . . . 8 𝐺 = (glb‘𝐾)
3 eqid 2610 . . . . . . . 8 (1.‘𝐾) = (1.‘𝐾)
42, 3glb0N 33498 . . . . . . 7 (𝐾 ∈ OP → (𝐺‘∅) = (1.‘𝐾))
54fveq2d 6107 . . . . . 6 (𝐾 ∈ OP → (𝑀‘(𝐺‘∅)) = (𝑀‘(1.‘𝐾)))
6 pmapglb2.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
7 pmapglb2.m . . . . . . 7 𝑀 = (pmap‘𝐾)
83, 6, 7pmap1N 34071 . . . . . 6 (𝐾 ∈ OP → (𝑀‘(1.‘𝐾)) = 𝐴)
95, 8eqtrd 2644 . . . . 5 (𝐾 ∈ OP → (𝑀‘(𝐺‘∅)) = 𝐴)
101, 9syl 17 . . . 4 (𝐾 ∈ HL → (𝑀‘(𝐺‘∅)) = 𝐴)
11 rexeq 3116 . . . . . . . . 9 (𝐼 = ∅ → (∃𝑖𝐼 𝑦 = 𝑆 ↔ ∃𝑖 ∈ ∅ 𝑦 = 𝑆))
1211abbidv 2728 . . . . . . . 8 (𝐼 = ∅ → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} = {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆})
13 rex0 3894 . . . . . . . . 9 ¬ ∃𝑖 ∈ ∅ 𝑦 = 𝑆
1413abf 3930 . . . . . . . 8 {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆} = ∅
1512, 14syl6eq 2660 . . . . . . 7 (𝐼 = ∅ → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} = ∅)
1615fveq2d 6107 . . . . . 6 (𝐼 = ∅ → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) = (𝐺‘∅))
1716fveq2d 6107 . . . . 5 (𝐼 = ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝑀‘(𝐺‘∅)))
18 riin0 4530 . . . . 5 (𝐼 = ∅ → (𝐴 𝑖𝐼 (𝑀𝑆)) = 𝐴)
1917, 18eqeq12d 2625 . . . 4 (𝐼 = ∅ → ((𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆)) ↔ (𝑀‘(𝐺‘∅)) = 𝐴))
2010, 19syl5ibrcom 236 . . 3 (𝐾 ∈ HL → (𝐼 = ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆))))
2120adantr 480 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐼 = ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆))))
22 pmapglb2.b . . . . 5 𝐵 = (Base‘𝐾)
2322, 2, 7pmapglbx 34073 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 (𝑀𝑆))
24 nfv 1830 . . . . . . . . . 10 𝑖 𝐾 ∈ HL
25 nfra1 2925 . . . . . . . . . 10 𝑖𝑖𝐼 𝑆𝐵
2624, 25nfan 1816 . . . . . . . . 9 𝑖(𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵)
27 simpr 476 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → 𝑖𝐼)
28 simpll 786 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → 𝐾 ∈ HL)
29 rspa 2914 . . . . . . . . . . . . 13 ((∀𝑖𝐼 𝑆𝐵𝑖𝐼) → 𝑆𝐵)
3029adantll 746 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → 𝑆𝐵)
3122, 6, 7pmapssat 34063 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑀𝑆) ⊆ 𝐴)
3228, 30, 31syl2anc 691 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → (𝑀𝑆) ⊆ 𝐴)
3327, 32jca 553 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → (𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴))
3433ex 449 . . . . . . . . 9 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝑖𝐼 → (𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴)))
3526, 34eximd 2072 . . . . . . . 8 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (∃𝑖 𝑖𝐼 → ∃𝑖(𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴)))
36 n0 3890 . . . . . . . 8 (𝐼 ≠ ∅ ↔ ∃𝑖 𝑖𝐼)
37 df-rex 2902 . . . . . . . 8 (∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴 ↔ ∃𝑖(𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴))
3835, 36, 373imtr4g 284 . . . . . . 7 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐼 ≠ ∅ → ∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴))
39383impia 1253 . . . . . 6 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → ∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴)
40 iinss 4507 . . . . . 6 (∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴 𝑖𝐼 (𝑀𝑆) ⊆ 𝐴)
4139, 40syl 17 . . . . 5 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝑖𝐼 (𝑀𝑆) ⊆ 𝐴)
42 sseqin2 3779 . . . . 5 ( 𝑖𝐼 (𝑀𝑆) ⊆ 𝐴 ↔ (𝐴 𝑖𝐼 (𝑀𝑆)) = 𝑖𝐼 (𝑀𝑆))
4341, 42sylib 207 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝐴 𝑖𝐼 (𝑀𝑆)) = 𝑖𝐼 (𝑀𝑆))
4423, 43eqtr4d 2647 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆)))
45443expia 1259 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐼 ≠ ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆))))
4621, 45pm2.61dne 2868 1 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∩ ciin 4456  ‘cfv 5804  Basecbs 15695  glbcglb 16766  1.cp1 16861  OPcops 33477  Atomscatm 33568  HLchlt 33655  pmapcpmap 33801 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-ats 33572  df-hlat 33656  df-pmap 33808 This theorem is referenced by:  polval2N  34210
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