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Theorem pmapglb2xN 35579
Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 35578, 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 35170 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OP)
2 pmapglb2.g . . . . . . . 8 𝐺 = (glb‘𝐾)
3 eqid 2760 . . . . . . . 8 (1.‘𝐾) = (1.‘𝐾)
42, 3glb0N 35001 . . . . . . 7 (𝐾 ∈ OP → (𝐺‘∅) = (1.‘𝐾))
54fveq2d 6357 . . . . . 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 35574 . . . . . 6 (𝐾 ∈ OP → (𝑀‘(1.‘𝐾)) = 𝐴)
95, 8eqtrd 2794 . . . . 5 (𝐾 ∈ OP → (𝑀‘(𝐺‘∅)) = 𝐴)
101, 9syl 17 . . . 4 (𝐾 ∈ HL → (𝑀‘(𝐺‘∅)) = 𝐴)
11 rexeq 3278 . . . . . . . . 9 (𝐼 = ∅ → (∃𝑖𝐼 𝑦 = 𝑆 ↔ ∃𝑖 ∈ ∅ 𝑦 = 𝑆))
1211abbidv 2879 . . . . . . . 8 (𝐼 = ∅ → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} = {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆})
13 rex0 4081 . . . . . . . . 9 ¬ ∃𝑖 ∈ ∅ 𝑦 = 𝑆
1413abf 4121 . . . . . . . 8 {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆} = ∅
1512, 14syl6eq 2810 . . . . . . 7 (𝐼 = ∅ → {𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆} = ∅)
1615fveq2d 6357 . . . . . 6 (𝐼 = ∅ → (𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆}) = (𝐺‘∅))
1716fveq2d 6357 . . . . 5 (𝐼 = ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝑀‘(𝐺‘∅)))
18 riin0 4746 . . . . 5 (𝐼 = ∅ → (𝐴 𝑖𝐼 (𝑀𝑆)) = 𝐴)
1917, 18eqeq12d 2775 . . . 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 472 . 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 35576 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 (𝑀𝑆))
24 nfv 1992 . . . . . . . . . 10 𝑖 𝐾 ∈ HL
25 nfra1 3079 . . . . . . . . . 10 𝑖𝑖𝐼 𝑆𝐵
2624, 25nfan 1977 . . . . . . . . 9 𝑖(𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵)
27 simpr 479 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → 𝑖𝐼)
28 simpll 807 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → 𝐾 ∈ HL)
29 rspa 3068 . . . . . . . . . . . . 13 ((∀𝑖𝐼 𝑆𝐵𝑖𝐼) → 𝑆𝐵)
3029adantll 752 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → 𝑆𝐵)
3122, 6, 7pmapssat 35566 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑀𝑆) ⊆ 𝐴)
3228, 30, 31syl2anc 696 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → (𝑀𝑆) ⊆ 𝐴)
3327, 32jca 555 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → (𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴))
3433ex 449 . . . . . . . . 9 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝑖𝐼 → (𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴)))
3526, 34eximd 2232 . . . . . . . 8 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (∃𝑖 𝑖𝐼 → ∃𝑖(𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴)))
36 n0 4074 . . . . . . . 8 (𝐼 ≠ ∅ ↔ ∃𝑖 𝑖𝐼)
37 df-rex 3056 . . . . . . . 8 (∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴 ↔ ∃𝑖(𝑖𝐼 ∧ (𝑀𝑆) ⊆ 𝐴))
3835, 36, 373imtr4g 285 . . . . . . 7 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐼 ≠ ∅ → ∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴))
39383impia 1110 . . . . . 6 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → ∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴)
40 iinss 4723 . . . . . 6 (∃𝑖𝐼 (𝑀𝑆) ⊆ 𝐴 𝑖𝐼 (𝑀𝑆) ⊆ 𝐴)
4139, 40syl 17 . . . . 5 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → 𝑖𝐼 (𝑀𝑆) ⊆ 𝐴)
42 sseqin2 3960 . . . . 5 ( 𝑖𝐼 (𝑀𝑆) ⊆ 𝐴 ↔ (𝐴 𝑖𝐼 (𝑀𝑆)) = 𝑖𝐼 (𝑀𝑆))
4341, 42sylib 208 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝐴 𝑖𝐼 (𝑀𝑆)) = 𝑖𝐼 (𝑀𝑆))
4423, 43eqtr4d 2797 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆)))
45443expia 1115 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐼 ≠ ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆))))
4621, 45pm2.61dne 3018 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 1072   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wne 2932  wral 3050  wrex 3051  cin 3714  wss 3715  c0 4058   ciin 4673  cfv 6049  Basecbs 16079  glbcglb 17164  1.cp1 17259  OPcops 34980  Atomscatm 35071  HLchlt 35158  pmapcpmap 35304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-preset 17149  df-poset 17167  df-lub 17195  df-glb 17196  df-join 17197  df-meet 17198  df-p1 17261  df-lat 17267  df-clat 17329  df-oposet 34984  df-ol 34986  df-oml 34987  df-ats 35075  df-hlat 35159  df-pmap 35311
This theorem is referenced by:  polval2N  35713
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