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Theorem 4atexlemex2 34375
 Description: Lemma for 4atexlem7 34379. Show that when 𝐶 ≠ 𝑆, 𝐶 satisfies the existence condition of the consequent. (Contributed by NM, 25-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)
4thatlem0.c 𝐶 = ((𝑄 𝑇) (𝑃 𝑆))
Assertion
Ref Expression
4atexlemex2 ((𝜑𝐶𝑆) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐶   𝑧,   𝑧,   𝑧,𝑃   𝑧,𝑆   𝑧,𝑊
Allowed substitution hints:   𝜑(𝑧)   𝑄(𝑧)   𝑅(𝑧)   𝑇(𝑧)   𝑈(𝑧)   𝐻(𝑧)   𝐾(𝑧)   (𝑧)   𝑉(𝑧)

Proof of Theorem 4atexlemex2
StepHypRef Expression
1 4thatlem.ph . . . 4 (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))))
2 4thatlem0.l . . . 4 = (le‘𝐾)
3 4thatlem0.j . . . 4 = (join‘𝐾)
4 4thatlem0.m . . . 4 = (meet‘𝐾)
5 4thatlem0.a . . . 4 𝐴 = (Atoms‘𝐾)
6 4thatlem0.h . . . 4 𝐻 = (LHyp‘𝐾)
7 4thatlem0.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
8 4thatlem0.v . . . 4 𝑉 = ((𝑃 𝑆) 𝑊)
9 4thatlem0.c . . . 4 𝐶 = ((𝑄 𝑇) (𝑃 𝑆))
101, 2, 3, 4, 5, 6, 7, 8, 94atexlemc 34373 . . 3 (𝜑𝐶𝐴)
1110adantr 480 . 2 ((𝜑𝐶𝑆) → 𝐶𝐴)
121, 2, 3, 4, 5, 6, 7, 8, 94atexlemnclw 34374 . . 3 (𝜑 → ¬ 𝐶 𝑊)
1312adantr 480 . 2 ((𝜑𝐶𝑆) → ¬ 𝐶 𝑊)
141, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 34372 . . . . 5 (𝜑 → ¬ 𝑇 (𝑃 𝑄))
15 id 22 . . . . . . . . . . 11 (𝐶 = 𝑃𝐶 = 𝑃)
169, 15syl5eqr 2658 . . . . . . . . . 10 (𝐶 = 𝑃 → ((𝑄 𝑇) (𝑃 𝑆)) = 𝑃)
1716adantl 481 . . . . . . . . 9 ((𝜑𝐶 = 𝑃) → ((𝑄 𝑇) (𝑃 𝑆)) = 𝑃)
1814atexlemkl 34361 . . . . . . . . . . . 12 (𝜑𝐾 ∈ Lat)
191, 3, 54atexlemqtb 34365 . . . . . . . . . . . 12 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
201, 3, 54atexlempsb 34364 . . . . . . . . . . . 12 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
21 eqid 2610 . . . . . . . . . . . . 13 (Base‘𝐾) = (Base‘𝐾)
2221, 2, 4latmle1 16899 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
2318, 19, 20, 22syl3anc 1318 . . . . . . . . . . 11 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
2414atexlemk 34351 . . . . . . . . . . . 12 (𝜑𝐾 ∈ HL)
2514atexlemq 34355 . . . . . . . . . . . 12 (𝜑𝑄𝐴)
2614atexlemt 34357 . . . . . . . . . . . 12 (𝜑𝑇𝐴)
273, 5hlatjcom 33672 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) = (𝑇 𝑄))
2824, 25, 26, 27syl3anc 1318 . . . . . . . . . . 11 (𝜑 → (𝑄 𝑇) = (𝑇 𝑄))
2923, 28breqtrd 4609 . . . . . . . . . 10 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑇 𝑄))
3029adantr 480 . . . . . . . . 9 ((𝜑𝐶 = 𝑃) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑇 𝑄))
3117, 30eqbrtrrd 4607 . . . . . . . 8 ((𝜑𝐶 = 𝑃) → 𝑃 (𝑇 𝑄))
3214atexlemkc 34362 . . . . . . . . . 10 (𝜑𝐾 ∈ CvLat)
3314atexlemp 34354 . . . . . . . . . 10 (𝜑𝑃𝐴)
3414atexlempnq 34359 . . . . . . . . . 10 (𝜑𝑃𝑄)
352, 3, 5cvlatexch2 33642 . . . . . . . . . 10 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑇𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 (𝑇 𝑄) → 𝑇 (𝑃 𝑄)))
3632, 33, 26, 25, 34, 35syl131anc 1331 . . . . . . . . 9 (𝜑 → (𝑃 (𝑇 𝑄) → 𝑇 (𝑃 𝑄)))
3736adantr 480 . . . . . . . 8 ((𝜑𝐶 = 𝑃) → (𝑃 (𝑇 𝑄) → 𝑇 (𝑃 𝑄)))
3831, 37mpd 15 . . . . . . 7 ((𝜑𝐶 = 𝑃) → 𝑇 (𝑃 𝑄))
3938ex 449 . . . . . 6 (𝜑 → (𝐶 = 𝑃𝑇 (𝑃 𝑄)))
4039necon3bd 2796 . . . . 5 (𝜑 → (¬ 𝑇 (𝑃 𝑄) → 𝐶𝑃))
4114, 40mpd 15 . . . 4 (𝜑𝐶𝑃)
4241adantr 480 . . 3 ((𝜑𝐶𝑆) → 𝐶𝑃)
43 simpr 476 . . 3 ((𝜑𝐶𝑆) → 𝐶𝑆)
4421, 2, 4latmle2 16900 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑆))
4518, 19, 20, 44syl3anc 1318 . . . . 5 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑆))
469, 45syl5eqbr 4618 . . . 4 (𝜑𝐶 (𝑃 𝑆))
4746adantr 480 . . 3 ((𝜑𝐶𝑆) → 𝐶 (𝑃 𝑆))
4814atexlems 34356 . . . . 5 (𝜑𝑆𝐴)
491, 2, 3, 54atexlempns 34366 . . . . 5 (𝜑𝑃𝑆)
505, 2, 3cvlsupr2 33648 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑆𝐴𝐶𝐴) ∧ 𝑃𝑆) → ((𝑃 𝐶) = (𝑆 𝐶) ↔ (𝐶𝑃𝐶𝑆𝐶 (𝑃 𝑆))))
5132, 33, 48, 10, 49, 50syl131anc 1331 . . . 4 (𝜑 → ((𝑃 𝐶) = (𝑆 𝐶) ↔ (𝐶𝑃𝐶𝑆𝐶 (𝑃 𝑆))))
5251adantr 480 . . 3 ((𝜑𝐶𝑆) → ((𝑃 𝐶) = (𝑆 𝐶) ↔ (𝐶𝑃𝐶𝑆𝐶 (𝑃 𝑆))))
5342, 43, 47, 52mpbir3and 1238 . 2 ((𝜑𝐶𝑆) → (𝑃 𝐶) = (𝑆 𝐶))
54 breq1 4586 . . . . 5 (𝑧 = 𝐶 → (𝑧 𝑊𝐶 𝑊))
5554notbid 307 . . . 4 (𝑧 = 𝐶 → (¬ 𝑧 𝑊 ↔ ¬ 𝐶 𝑊))
56 oveq2 6557 . . . . 5 (𝑧 = 𝐶 → (𝑃 𝑧) = (𝑃 𝐶))
57 oveq2 6557 . . . . 5 (𝑧 = 𝐶 → (𝑆 𝑧) = (𝑆 𝐶))
5856, 57eqeq12d 2625 . . . 4 (𝑧 = 𝐶 → ((𝑃 𝑧) = (𝑆 𝑧) ↔ (𝑃 𝐶) = (𝑆 𝐶)))
5955, 58anbi12d 743 . . 3 (𝑧 = 𝐶 → ((¬ 𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)) ↔ (¬ 𝐶 𝑊 ∧ (𝑃 𝐶) = (𝑆 𝐶))))
6059rspcev 3282 . 2 ((𝐶𝐴 ∧ (¬ 𝐶 𝑊 ∧ (𝑃 𝐶) = (𝑆 𝐶))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
6111, 13, 53, 60syl12anc 1316 1 ((𝜑𝐶𝑆) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Latclat 16868  Atomscatm 33568  CvLatclc 33570  HLchlt 33655  LHypclh 34288 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lhyp 34292 This theorem is referenced by:  4atexlemex4  34377  4atexlemex6  34378
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