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Theorem pmapglb2xN 38549
Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 38548, 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 38138 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OP)
2 pmapglb2.g . . . . . . . 8 𝐺 = (glb‘𝐾)
3 eqid 2733 . . . . . . . 8 (1.‘𝐾) = (1.‘𝐾)
42, 3glb0N 37969 . . . . . . 7 (𝐾 ∈ OP → (𝐺‘∅) = (1.‘𝐾))
54fveq2d 6885 . . . . . 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 38544 . . . . . 6 (𝐾 ∈ OP → (𝑀‘(1.‘𝐾)) = 𝐴)
95, 8eqtrd 2773 . . . . 5 (𝐾 ∈ OP → (𝑀‘(𝐺‘∅)) = 𝐴)
101, 9syl 17 . . . 4 (𝐾 ∈ HL → (𝑀‘(𝐺‘∅)) = 𝐴)
11 rexeq 3322 . . . . . . . . 9 (𝐼 = ∅ → (∃𝑖𝐼 𝑦 = 𝑆 ↔ ∃𝑖 ∈ ∅ 𝑦 = 𝑆))
1211abbidv 2802 . . . . . . . 8 (𝐼 = ∅ → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} = {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆})
13 rex0 4355 . . . . . . . . 9 ¬ ∃𝑖 ∈ ∅ 𝑦 = 𝑆
1413abf 4400 . . . . . . . 8 {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆} = ∅
1512, 14eqtrdi 2789 . . . . . . 7 (𝐼 = ∅ → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} = ∅)
1615fveq2d 6885 . . . . . 6 (𝐼 = ∅ → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) = (𝐺‘∅))
1716fveq2d 6885 . . . . 5 (𝐼 = ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝑀‘(𝐺‘∅)))
18 riin0 5081 . . . . 5 (𝐼 = ∅ → (𝐴 𝑖𝐼 (𝑀𝑆)) = 𝐴)
1917, 18eqeq12d 2749 . . . 4 (𝐼 = ∅ → ((𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆)) ↔ (𝑀‘(𝐺‘∅)) = 𝐴))
2010, 19syl5ibrcom 246 . . 3 (𝐾 ∈ HL → (𝐼 = ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆))))
2120adantr 482 . 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 38546 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 (𝑀𝑆))
24 nfv 1918 . . . . . . . . . 10 𝑖 𝐾 ∈ HL
25 nfra1 3282 . . . . . . . . . 10 𝑖𝑖𝐼 𝑆𝐵
2624, 25nfan 1903 . . . . . . . . 9 𝑖(𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵)
27 simpr 486 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → 𝑖𝐼)
28 simpll 766 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → 𝐾 ∈ HL)
29 rspa 3246 . . . . . . . . . . . . 13 ((∀𝑖𝐼 𝑆𝐵𝑖𝐼) → 𝑆𝐵)
3029adantll 713 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → 𝑆𝐵)
3122, 6, 7pmapssat 38536 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑀𝑆) ⊆ 𝐴)
3228, 30, 31syl2anc 585 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → (𝑀𝑆) ⊆ 𝐴)
3327, 32jca 513 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → (𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴))
3433ex 414 . . . . . . . . 9 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝑖𝐼 → (𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴)))
3526, 34eximd 2210 . . . . . . . 8 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (∃𝑖 𝑖𝐼 → ∃𝑖(𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴)))
36 n0 4344 . . . . . . . 8 (𝐼 ≠ ∅ ↔ ∃𝑖 𝑖𝐼)
37 df-rex 3072 . . . . . . . 8 (∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴 ↔ ∃𝑖(𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴))
3835, 36, 373imtr4g 296 . . . . . . 7 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐼 ≠ ∅ → ∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴))
39383impia 1118 . . . . . 6 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → ∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴)
40 iinss 5055 . . . . . 6 (∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴 𝑖𝐼 (𝑀𝑆) ⊆ 𝐴)
4139, 40syl 17 . . . . 5 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝑖𝐼 (𝑀𝑆) ⊆ 𝐴)
42 sseqin2 4213 . . . . 5 ( 𝑖𝐼 (𝑀𝑆) ⊆ 𝐴 ↔ (𝐴 𝑖𝐼 (𝑀𝑆)) = 𝑖𝐼 (𝑀𝑆))
4341, 42sylib 217 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝐴 𝑖𝐼 (𝑀𝑆)) = 𝑖𝐼 (𝑀𝑆))
4423, 43eqtr4d 2776 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆)))
45443expia 1122 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐼 ≠ ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆))))
4621, 45pm2.61dne 3029 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 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wne 2941  wral 3062  wrex 3071  cin 3945  wss 3946  c0 4320   ciin 4994  cfv 6535  Basecbs 17131  glbcglb 18250  1.cp1 18364  OPcops 37948  Atomscatm 38039  HLchlt 38126  pmapcpmap 38274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-iin 4996  df-br 5145  df-opab 5207  df-mpt 5228  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 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7352  df-ov 7399  df-oprab 7400  df-proset 18235  df-poset 18253  df-lub 18286  df-glb 18287  df-join 18288  df-meet 18289  df-p1 18366  df-lat 18372  df-clat 18439  df-oposet 37952  df-ol 37954  df-oml 37955  df-ats 38043  df-hlat 38127  df-pmap 38281
This theorem is referenced by:  polval2N  38683
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