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Theorem 4atexlemunv 35954
Description: Lemma for 4atexlem7 35963. (Contributed by NM, 21-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
4atexlemunv (𝜑𝑈𝑉)

Proof of Theorem 4atexlemunv
StepHypRef Expression
1 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))))
214atexlemnslpq 35944 . 2 (𝜑 → ¬ 𝑆 (𝑃 𝑄))
314atexlemk 35935 . . . . . . 7 (𝜑𝐾 ∈ HL)
414atexlemp 35938 . . . . . . 7 (𝜑𝑃𝐴)
514atexlems 35940 . . . . . . 7 (𝜑𝑆𝐴)
6 4thatlem0.l . . . . . . . 8 = (le‘𝐾)
7 4thatlem0.j . . . . . . . 8 = (join‘𝐾)
8 4thatlem0.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
96, 7, 8hlatlej2 35264 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → 𝑆 (𝑃 𝑆))
103, 4, 5, 9syl3anc 1490 . . . . . 6 (𝜑𝑆 (𝑃 𝑆))
1110adantr 472 . . . . 5 ((𝜑𝑈 = 𝑉) → 𝑆 (𝑃 𝑆))
12 4thatlem0.v . . . . . . . . 9 𝑉 = ((𝑃 𝑆) 𝑊)
1314atexlemkl 35945 . . . . . . . . . 10 (𝜑𝐾 ∈ Lat)
141, 7, 84atexlempsb 35948 . . . . . . . . . 10 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
15 4thatlem0.h . . . . . . . . . . 11 𝐻 = (LHyp‘𝐾)
161, 154atexlemwb 35947 . . . . . . . . . 10 (𝜑𝑊 ∈ (Base‘𝐾))
17 eqid 2764 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
18 4thatlem0.m . . . . . . . . . . 11 = (meet‘𝐾)
1917, 6, 18latmle1 17343 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
2013, 14, 16, 19syl3anc 1490 . . . . . . . . 9 (𝜑 → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
2112, 20syl5eqbr 4843 . . . . . . . 8 (𝜑𝑉 (𝑃 𝑆))
2214atexlemkc 35946 . . . . . . . . 9 (𝜑𝐾 ∈ CvLat)
23 4thatlem0.u . . . . . . . . . 10 𝑈 = ((𝑃 𝑄) 𝑊)
241, 6, 7, 18, 8, 15, 23, 124atexlemv 35953 . . . . . . . . 9 (𝜑𝑉𝐴)
2517, 6, 18latmle2 17344 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) 𝑊)
2613, 14, 16, 25syl3anc 1490 . . . . . . . . . . 11 (𝜑 → ((𝑃 𝑆) 𝑊) 𝑊)
2712, 26syl5eqbr 4843 . . . . . . . . . 10 (𝜑𝑉 𝑊)
2814atexlempw 35937 . . . . . . . . . . 11 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
2928simprd 489 . . . . . . . . . 10 (𝜑 → ¬ 𝑃 𝑊)
30 nbrne2 4828 . . . . . . . . . 10 ((𝑉 𝑊 ∧ ¬ 𝑃 𝑊) → 𝑉𝑃)
3127, 29, 30syl2anc 579 . . . . . . . . 9 (𝜑𝑉𝑃)
326, 7, 8cvlatexchb1 35222 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑉𝐴𝑆𝐴𝑃𝐴) ∧ 𝑉𝑃) → (𝑉 (𝑃 𝑆) ↔ (𝑃 𝑉) = (𝑃 𝑆)))
3322, 24, 5, 4, 31, 32syl131anc 1502 . . . . . . . 8 (𝜑 → (𝑉 (𝑃 𝑆) ↔ (𝑃 𝑉) = (𝑃 𝑆)))
3421, 33mpbid 223 . . . . . . 7 (𝜑 → (𝑃 𝑉) = (𝑃 𝑆))
3534adantr 472 . . . . . 6 ((𝜑𝑈 = 𝑉) → (𝑃 𝑉) = (𝑃 𝑆))
36 oveq2 6849 . . . . . . . 8 (𝑈 = 𝑉 → (𝑃 𝑈) = (𝑃 𝑉))
3736eqcomd 2770 . . . . . . 7 (𝑈 = 𝑉 → (𝑃 𝑉) = (𝑃 𝑈))
3814atexlemq 35939 . . . . . . . . . . 11 (𝜑𝑄𝐴)
3917, 7, 8hlatjcl 35255 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
403, 4, 38, 39syl3anc 1490 . . . . . . . . . 10 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
4117, 6, 18latmle1 17343 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) (𝑃 𝑄))
4213, 40, 16, 41syl3anc 1490 . . . . . . . . 9 (𝜑 → ((𝑃 𝑄) 𝑊) (𝑃 𝑄))
4323, 42syl5eqbr 4843 . . . . . . . 8 (𝜑𝑈 (𝑃 𝑄))
441, 6, 7, 18, 8, 15, 234atexlemu 35952 . . . . . . . . 9 (𝜑𝑈𝐴)
4517, 6, 18latmle2 17344 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
4613, 40, 16, 45syl3anc 1490 . . . . . . . . . . 11 (𝜑 → ((𝑃 𝑄) 𝑊) 𝑊)
4723, 46syl5eqbr 4843 . . . . . . . . . 10 (𝜑𝑈 𝑊)
48 nbrne2 4828 . . . . . . . . . 10 ((𝑈 𝑊 ∧ ¬ 𝑃 𝑊) → 𝑈𝑃)
4947, 29, 48syl2anc 579 . . . . . . . . 9 (𝜑𝑈𝑃)
506, 7, 8cvlatexchb1 35222 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑄𝐴𝑃𝐴) ∧ 𝑈𝑃) → (𝑈 (𝑃 𝑄) ↔ (𝑃 𝑈) = (𝑃 𝑄)))
5122, 44, 38, 4, 49, 50syl131anc 1502 . . . . . . . 8 (𝜑 → (𝑈 (𝑃 𝑄) ↔ (𝑃 𝑈) = (𝑃 𝑄)))
5243, 51mpbid 223 . . . . . . 7 (𝜑 → (𝑃 𝑈) = (𝑃 𝑄))
5337, 52sylan9eqr 2820 . . . . . 6 ((𝜑𝑈 = 𝑉) → (𝑃 𝑉) = (𝑃 𝑄))
5435, 53eqtr3d 2800 . . . . 5 ((𝜑𝑈 = 𝑉) → (𝑃 𝑆) = (𝑃 𝑄))
5511, 54breqtrd 4834 . . . 4 ((𝜑𝑈 = 𝑉) → 𝑆 (𝑃 𝑄))
5655ex 401 . . 3 (𝜑 → (𝑈 = 𝑉𝑆 (𝑃 𝑄)))
5756necon3bd 2950 . 2 (𝜑 → (¬ 𝑆 (𝑃 𝑄) → 𝑈𝑉))
582, 57mpd 15 1 (𝜑𝑈𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2936   class class class wbr 4808  cfv 6067  (class class class)co 6841  Basecbs 16131  lecple 16222  joincjn 17211  meetcmee 17212  Latclat 17312  Atomscatm 35151  CvLatclc 35153  HLchlt 35238  LHypclh 35872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-proset 17195  df-poset 17213  df-plt 17225  df-lub 17241  df-glb 17242  df-join 17243  df-meet 17244  df-p0 17306  df-p1 17307  df-lat 17313  df-clat 17375  df-oposet 35064  df-ol 35066  df-oml 35067  df-covers 35154  df-ats 35155  df-atl 35186  df-cvlat 35210  df-hlat 35239  df-lhyp 35876
This theorem is referenced by:  4atexlemtlw  35955  4atexlemntlpq  35956  4atexlemc  35957  4atexlemnclw  35958
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