Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihglblem2aN Structured version   Visualization version   GIF version
Theorem dihglblem2aN 40256
Description: Lemma for isomorphism H of a GLB. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihglblem.b 𝐡 = (Baseβ€˜πΎ)
dihglblem.l ≀ = (leβ€˜πΎ)
dihglblem.m ∧ = (meetβ€˜πΎ)
dihglblem.g 𝐺 = (glbβ€˜πΎ)
dihglblem.h 𝐻 = (LHypβ€˜πΎ)
dihglblem.t 𝑇 = {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)}
Assertion
Ref Expression
dihglblem2aN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) β†’ 𝑇 β‰  βˆ…)
Distinct variable groups:   𝑣,𝑒, ∧   𝑒,𝐡   𝑒,𝑆,𝑣   𝑒,π‘Š,𝑣
Allowed substitution hints:   𝐡(𝑣)   𝑇(𝑣,𝑒)   𝐺(𝑣,𝑒)   𝐻(𝑣,𝑒)   𝐾(𝑣,𝑒)   ≀ (𝑣,𝑒)
Proof of Theorem dihglblem2aN
Dummy variables 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dihglblem.t . . 3 𝑇 = {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)}
21a1i 11 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) β†’ 𝑇 = {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)})
3 simprr 771 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) β†’ 𝑆 β‰  βˆ…)
4 n0 4346 . . . 4 (𝑆 β‰  βˆ… ↔ βˆƒπ‘§ 𝑧 ∈ 𝑆)
53, 4sylib 217 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) β†’ βˆƒπ‘§ 𝑧 ∈ 𝑆)
6 hllat 38325 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
76ad3antrrr 728 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ 𝐾 ∈ Lat)
8 simplrl 775 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ 𝑆 βŠ† 𝐡)
9 simpr 485 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ 𝑧 ∈ 𝑆)
108, 9sseldd 3983 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ 𝑧 ∈ 𝐡)
11 dihglblem.b . . . . . . . 8 𝐡 = (Baseβ€˜πΎ)
12 dihglblem.h . . . . . . . 8 𝐻 = (LHypβ€˜πΎ)
1311, 12lhpbase 38961 . . . . . . 7 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ 𝐡)
1413ad3antlr 729 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ π‘Š ∈ 𝐡)
15 dihglblem.m . . . . . . 7 ∧ = (meetβ€˜πΎ)
1611, 15latmcl 18395 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑧 ∈ 𝐡 ∧ π‘Š ∈ 𝐡) β†’ (𝑧 ∧ π‘Š) ∈ 𝐡)
177, 10, 14, 16syl3anc 1371 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ (𝑧 ∧ π‘Š) ∈ 𝐡)
18 eqidd 2733 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ (𝑧 ∧ π‘Š) = (𝑧 ∧ π‘Š))
19 oveq1 7418 . . . . . . 7 (𝑣 = 𝑧 β†’ (𝑣 ∧ π‘Š) = (𝑧 ∧ π‘Š))
2019rspceeqv 3633 . . . . . 6 ((𝑧 ∈ 𝑆 ∧ (𝑧 ∧ π‘Š) = (𝑧 ∧ π‘Š)) β†’ βˆƒπ‘£ ∈ 𝑆 (𝑧 ∧ π‘Š) = (𝑣 ∧ π‘Š))
219, 18, 20syl2anc 584 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ βˆƒπ‘£ ∈ 𝑆 (𝑧 ∧ π‘Š) = (𝑣 ∧ π‘Š))
22 ovex 7444 . . . . . 6 (𝑧 ∧ π‘Š) ∈ V
23 eleq1 2821 . . . . . . 7 (𝑀 = (𝑧 ∧ π‘Š) β†’ (𝑀 ∈ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)} ↔ (𝑧 ∧ π‘Š) ∈ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)}))
24 eqeq1 2736 . . . . . . . . 9 (𝑒 = (𝑧 ∧ π‘Š) β†’ (𝑒 = (𝑣 ∧ π‘Š) ↔ (𝑧 ∧ π‘Š) = (𝑣 ∧ π‘Š)))
2524rexbidv 3178 . . . . . . . 8 (𝑒 = (𝑧 ∧ π‘Š) β†’ (βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š) ↔ βˆƒπ‘£ ∈ 𝑆 (𝑧 ∧ π‘Š) = (𝑣 ∧ π‘Š)))
2625elrab 3683 . . . . . . 7 ((𝑧 ∧ π‘Š) ∈ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)} ↔ ((𝑧 ∧ π‘Š) ∈ 𝐡 ∧ βˆƒπ‘£ ∈ 𝑆 (𝑧 ∧ π‘Š) = (𝑣 ∧ π‘Š)))
2723, 26bitrdi 286 . . . . . 6 (𝑀 = (𝑧 ∧ π‘Š) β†’ (𝑀 ∈ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)} ↔ ((𝑧 ∧ π‘Š) ∈ 𝐡 ∧ βˆƒπ‘£ ∈ 𝑆 (𝑧 ∧ π‘Š) = (𝑣 ∧ π‘Š))))
2822, 27spcev 3596 . . . . 5 (((𝑧 ∧ π‘Š) ∈ 𝐡 ∧ βˆƒπ‘£ ∈ 𝑆 (𝑧 ∧ π‘Š) = (𝑣 ∧ π‘Š)) β†’ βˆƒπ‘€ 𝑀 ∈ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)})
2917, 21, 28syl2anc 584 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ βˆƒπ‘€ 𝑀 ∈ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)})
30 n0 4346 . . . 4 ({𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)} β‰  βˆ… ↔ βˆƒπ‘€ 𝑀 ∈ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)})
3129, 30sylibr 233 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)} β‰  βˆ…)
325, 31exlimddv 1938 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) β†’ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)} β‰  βˆ…)
332, 32eqnetrd 3008 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) β†’ 𝑇 β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070  {crab 3432   βŠ† wss 3948  βˆ…c0 4322  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  lecple 17206  glbcglb 18265  meetcmee 18267  Latclat 18386  HLchlt 38312  LHypclh 38947
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-lat 18387  df-atl 38260  df-cvlat 38284  df-hlat 38313  df-lhyp 38951
This theorem is referenced by:  dihglblem3N  40258
  Copyright terms: Public domain W3C validator