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Theorem pmapglb2xN 35376
Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 35375, 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 34967 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OP)
2 pmapglb2.g . . . . . . . 8 𝐺 = (glb‘𝐾)
3 eqid 2651 . . . . . . . 8 (1.‘𝐾) = (1.‘𝐾)
42, 3glb0N 34798 . . . . . . 7 (𝐾 ∈ OP → (𝐺‘∅) = (1.‘𝐾))
54fveq2d 6233 . . . . . 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 35371 . . . . . 6 (𝐾 ∈ OP → (𝑀‘(1.‘𝐾)) = 𝐴)
95, 8eqtrd 2685 . . . . 5 (𝐾 ∈ OP → (𝑀‘(𝐺‘∅)) = 𝐴)
101, 9syl 17 . . . 4 (𝐾 ∈ HL → (𝑀‘(𝐺‘∅)) = 𝐴)
11 rexeq 3169 . . . . . . . . 9 (𝐼 = ∅ → (∃𝑖𝐼 𝑦 = 𝑆 ↔ ∃𝑖 ∈ ∅ 𝑦 = 𝑆))
1211abbidv 2770 . . . . . . . 8 (𝐼 = ∅ → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} = {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆})
13 rex0 3971 . . . . . . . . 9 ¬ ∃𝑖 ∈ ∅ 𝑦 = 𝑆
1413abf 4011 . . . . . . . 8 {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆} = ∅
1512, 14syl6eq 2701 . . . . . . 7 (𝐼 = ∅ → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} = ∅)
1615fveq2d 6233 . . . . . 6 (𝐼 = ∅ → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) = (𝐺‘∅))
1716fveq2d 6233 . . . . 5 (𝐼 = ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝑀‘(𝐺‘∅)))
18 riin0 4626 . . . . 5 (𝐼 = ∅ → (𝐴 𝑖𝐼 (𝑀𝑆)) = 𝐴)
1917, 18eqeq12d 2666 . . . 4 (𝐼 = ∅ → ((𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆)) ↔ (𝑀‘(𝐺‘∅)) = 𝐴))
2010, 19syl5ibrcom 237 . . 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 35373 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 (𝑀𝑆))
24 nfv 1883 . . . . . . . . . 10 𝑖 𝐾 ∈ HL
25 nfra1 2970 . . . . . . . . . 10 𝑖𝑖𝐼 𝑆𝐵
2624, 25nfan 1868 . . . . . . . . 9 𝑖(𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵)
27 simpr 476 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → 𝑖𝐼)
28 simpll 805 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → 𝐾 ∈ HL)
29 rspa 2959 . . . . . . . . . . . . 13 ((∀𝑖𝐼 𝑆𝐵𝑖𝐼) → 𝑆𝐵)
3029adantll 750 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → 𝑆𝐵)
3122, 6, 7pmapssat 35363 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑀𝑆) ⊆ 𝐴)
3228, 30, 31syl2anc 694 . . . . . . . . . . 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 2123 . . . . . . . 8 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (∃𝑖 𝑖𝐼 → ∃𝑖(𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴)))
36 n0 3964 . . . . . . . 8 (𝐼 ≠ ∅ ↔ ∃𝑖 𝑖𝐼)
37 df-rex 2947 . . . . . . . 8 (∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴 ↔ ∃𝑖(𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴))
3835, 36, 373imtr4g 285 . . . . . . 7 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐼 ≠ ∅ → ∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴))
39383impia 1280 . . . . . 6 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → ∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴)
40 iinss 4603 . . . . . 6 (∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴 𝑖𝐼 (𝑀𝑆) ⊆ 𝐴)
4139, 40syl 17 . . . . 5 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝑖𝐼 (𝑀𝑆) ⊆ 𝐴)
42 sseqin2 3850 . . . . 5 ( 𝑖𝐼 (𝑀𝑆) ⊆ 𝐴 ↔ (𝐴 𝑖𝐼 (𝑀𝑆)) = 𝑖𝐼 (𝑀𝑆))
4341, 42sylib 208 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝐴 𝑖𝐼 (𝑀𝑆)) = 𝑖𝐼 (𝑀𝑆))
4423, 43eqtr4d 2688 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆)))
45443expia 1286 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐼 ≠ ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆))))
4621, 45pm2.61dne 2909 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 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  cin 3606  wss 3607  c0 3948   ciin 4553  cfv 5926  Basecbs 15904  glbcglb 16990  1.cp1 17085  OPcops 34777  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-preset 16975  df-poset 16993  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p1 17087  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-ats 34872  df-hlat 34956  df-pmap 35108
This theorem is referenced by:  polval2N  35510
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