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Theorem 4atexlemtlw 37083
Description: Lemma for 4atexlem7 37091. (Contributed by NM, 24-Nov-2012.)
Hypotheses
Ref Expression
4thatlem.ph (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))
4thatlem0.l = (le‘𝐾)
4thatlem0.j = (join‘𝐾)
4thatlem0.m = (meet‘𝐾)
4thatlem0.a 𝐴 = (Atoms‘𝐾)
4thatlem0.h 𝐻 = (LHyp‘𝐾)
4thatlem0.u 𝑈 = ((𝑃 𝑄) 𝑊)
4thatlem0.v 𝑉 = ((𝑃 𝑆) 𝑊)
Assertion
Ref Expression
4atexlemtlw (𝜑𝑇 𝑊)

Proof of Theorem 4atexlemtlw
StepHypRef Expression
1 eqid 2818 . 2 (Base‘𝐾) = (Base‘𝐾)
2 4thatlem0.l . 2 = (le‘𝐾)
3 4thatlem.ph . . 3 (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))
434atexlemkl 37073 . 2 (𝜑𝐾 ∈ Lat)
534atexlemt 37069 . . 3 (𝜑𝑇𝐴)
6 4thatlem0.a . . . 4 𝐴 = (Atoms‘𝐾)
71, 6atbase 36305 . . 3 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
85, 7syl 17 . 2 (𝜑𝑇 ∈ (Base‘𝐾))
934atexlemk 37063 . . 3 (𝜑𝐾 ∈ HL)
10 4thatlem0.j . . . 4 = (join‘𝐾)
11 4thatlem0.m . . . 4 = (meet‘𝐾)
12 4thatlem0.h . . . 4 𝐻 = (LHyp‘𝐾)
13 4thatlem0.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
143, 2, 10, 11, 6, 12, 134atexlemu 37080 . . 3 (𝜑𝑈𝐴)
15 4thatlem0.v . . . 4 𝑉 = ((𝑃 𝑆) 𝑊)
163, 2, 10, 11, 6, 12, 13, 154atexlemv 37081 . . 3 (𝜑𝑉𝐴)
171, 10, 6hlatjcl 36383 . . 3 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑉𝐴) → (𝑈 𝑉) ∈ (Base‘𝐾))
189, 14, 16, 17syl3anc 1363 . 2 (𝜑 → (𝑈 𝑉) ∈ (Base‘𝐾))
193, 124atexlemwb 37075 . 2 (𝜑𝑊 ∈ (Base‘𝐾))
2034atexlemkc 37074 . . 3 (𝜑𝐾 ∈ CvLat)
213, 2, 10, 11, 6, 12, 13, 154atexlemunv 37082 . . 3 (𝜑𝑈𝑉)
2234atexlemutvt 37070 . . 3 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
236, 2, 10cvlsupr4 36361 . . 3 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑉𝐴𝑇𝐴) ∧ (𝑈𝑉 ∧ (𝑈 𝑇) = (𝑉 𝑇))) → 𝑇 (𝑈 𝑉))
2420, 14, 16, 5, 21, 22, 23syl132anc 1380 . 2 (𝜑𝑇 (𝑈 𝑉))
2534atexlemp 37066 . . . . . 6 (𝜑𝑃𝐴)
2634atexlemq 37067 . . . . . 6 (𝜑𝑄𝐴)
271, 10, 6hlatjcl 36383 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
289, 25, 26, 27syl3anc 1363 . . . . 5 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
291, 2, 11latmle2 17675 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
304, 28, 19, 29syl3anc 1363 . . . 4 (𝜑 → ((𝑃 𝑄) 𝑊) 𝑊)
3113, 30eqbrtrid 5092 . . 3 (𝜑𝑈 𝑊)
323, 10, 64atexlempsb 37076 . . . . 5 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
331, 2, 11latmle2 17675 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) 𝑊)
344, 32, 19, 33syl3anc 1363 . . . 4 (𝜑 → ((𝑃 𝑆) 𝑊) 𝑊)
3515, 34eqbrtrid 5092 . . 3 (𝜑𝑉 𝑊)
361, 6atbase 36305 . . . . 5 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
3714, 36syl 17 . . . 4 (𝜑𝑈 ∈ (Base‘𝐾))
381, 6atbase 36305 . . . . 5 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
3916, 38syl 17 . . . 4 (𝜑𝑉 ∈ (Base‘𝐾))
401, 2, 10latjle12 17660 . . . 4 ((𝐾 ∈ Lat ∧ (𝑈 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑈 𝑊𝑉 𝑊) ↔ (𝑈 𝑉) 𝑊))
414, 37, 39, 19, 40syl13anc 1364 . . 3 (𝜑 → ((𝑈 𝑊𝑉 𝑊) ↔ (𝑈 𝑉) 𝑊))
4231, 35, 41mpbi2and 708 . 2 (𝜑 → (𝑈 𝑉) 𝑊)
431, 2, 4, 8, 18, 19, 24, 42lattrd 17656 1 (𝜑𝑇 𝑊)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  joincjn 17542  meetcmee 17543  Latclat 17643  Atomscatm 36279  CvLatclc 36281  HLchlt 36366  LHypclh 37000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-p1 17638  df-lat 17644  df-clat 17706  df-oposet 36192  df-ol 36194  df-oml 36195  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367  df-lhyp 37004
This theorem is referenced by:  4atexlemntlpq  37084  4atexlemnclw  37086  4atexlemcnd  37088
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