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Theorem pmapglb2xN 38235
Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 38234, 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 37824 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OP)
2 pmapglb2.g . . . . . . . 8 𝐺 = (glb‘𝐾)
3 eqid 2736 . . . . . . . 8 (1.‘𝐾) = (1.‘𝐾)
42, 3glb0N 37655 . . . . . . 7 (𝐾 ∈ OP → (𝐺‘∅) = (1.‘𝐾))
54fveq2d 6846 . . . . . 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 38230 . . . . . 6 (𝐾 ∈ OP → (𝑀‘(1.‘𝐾)) = 𝐴)
95, 8eqtrd 2776 . . . . 5 (𝐾 ∈ OP → (𝑀‘(𝐺‘∅)) = 𝐴)
101, 9syl 17 . . . 4 (𝐾 ∈ HL → (𝑀‘(𝐺‘∅)) = 𝐴)
11 rexeq 3310 . . . . . . . . 9 (𝐼 = ∅ → (∃𝑖𝐼 𝑦 = 𝑆 ↔ ∃𝑖 ∈ ∅ 𝑦 = 𝑆))
1211abbidv 2805 . . . . . . . 8 (𝐼 = ∅ → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} = {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆})
13 rex0 4317 . . . . . . . . 9 ¬ ∃𝑖 ∈ ∅ 𝑦 = 𝑆
1413abf 4362 . . . . . . . 8 {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆} = ∅
1512, 14eqtrdi 2792 . . . . . . 7 (𝐼 = ∅ → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} = ∅)
1615fveq2d 6846 . . . . . 6 (𝐼 = ∅ → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) = (𝐺‘∅))
1716fveq2d 6846 . . . . 5 (𝐼 = ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝑀‘(𝐺‘∅)))
18 riin0 5042 . . . . 5 (𝐼 = ∅ → (𝐴 𝑖𝐼 (𝑀𝑆)) = 𝐴)
1917, 18eqeq12d 2752 . . . 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 481 . 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 38232 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 (𝑀𝑆))
24 nfv 1917 . . . . . . . . . 10 𝑖 𝐾 ∈ HL
25 nfra1 3267 . . . . . . . . . 10 𝑖𝑖𝐼 𝑆𝐵
2624, 25nfan 1902 . . . . . . . . 9 𝑖(𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵)
27 simpr 485 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → 𝑖𝐼)
28 simpll 765 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → 𝐾 ∈ HL)
29 rspa 3231 . . . . . . . . . . . . 13 ((∀𝑖𝐼 𝑆𝐵𝑖𝐼) → 𝑆𝐵)
3029adantll 712 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → 𝑆𝐵)
3122, 6, 7pmapssat 38222 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑀𝑆) ⊆ 𝐴)
3228, 30, 31syl2anc 584 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → (𝑀𝑆) ⊆ 𝐴)
3327, 32jca 512 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → (𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴))
3433ex 413 . . . . . . . . 9 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝑖𝐼 → (𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴)))
3526, 34eximd 2209 . . . . . . . 8 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (∃𝑖 𝑖𝐼 → ∃𝑖(𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴)))
36 n0 4306 . . . . . . . 8 (𝐼 ≠ ∅ ↔ ∃𝑖 𝑖𝐼)
37 df-rex 3074 . . . . . . . 8 (∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴 ↔ ∃𝑖(𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴))
3835, 36, 373imtr4g 295 . . . . . . 7 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐼 ≠ ∅ → ∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴))
39383impia 1117 . . . . . 6 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → ∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴)
40 iinss 5016 . . . . . 6 (∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴 𝑖𝐼 (𝑀𝑆) ⊆ 𝐴)
4139, 40syl 17 . . . . 5 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝑖𝐼 (𝑀𝑆) ⊆ 𝐴)
42 sseqin2 4175 . . . . 5 ( 𝑖𝐼 (𝑀𝑆) ⊆ 𝐴 ↔ (𝐴 𝑖𝐼 (𝑀𝑆)) = 𝑖𝐼 (𝑀𝑆))
4341, 42sylib 217 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝐴 𝑖𝐼 (𝑀𝑆)) = 𝑖𝐼 (𝑀𝑆))
4423, 43eqtr4d 2779 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆)))
45443expia 1121 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐼 ≠ ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆))))
4621, 45pm2.61dne 3031 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 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  cin 3909  wss 3910  c0 4282   ciin 4955  cfv 6496  Basecbs 17083  glbcglb 18199  1.cp1 18313  OPcops 37634  Atomscatm 37725  HLchlt 37812  pmapcpmap 37960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p1 18315  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-ats 37729  df-hlat 37813  df-pmap 37967
This theorem is referenced by:  polval2N  38369
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