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Theorem 4atexlemex2 37206
Description: Lemma for 4atexlem7 37210. Show that when 𝐶𝑆, 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 37204 . . 3 (𝜑𝐶𝐴)
1110adantr 483 . 2 ((𝜑𝐶𝑆) → 𝐶𝐴)
121, 2, 3, 4, 5, 6, 7, 8, 94atexlemnclw 37205 . . 3 (𝜑 → ¬ 𝐶 𝑊)
1312adantr 483 . 2 ((𝜑𝐶𝑆) → ¬ 𝐶 𝑊)
141, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 37203 . . . . 5 (𝜑 → ¬ 𝑇 (𝑃 𝑄))
15 id 22 . . . . . . . . . . 11 (𝐶 = 𝑃𝐶 = 𝑃)
169, 15syl5eqr 2870 . . . . . . . . . 10 (𝐶 = 𝑃 → ((𝑄 𝑇) (𝑃 𝑆)) = 𝑃)
1716adantl 484 . . . . . . . . 9 ((𝜑𝐶 = 𝑃) → ((𝑄 𝑇) (𝑃 𝑆)) = 𝑃)
1814atexlemkl 37192 . . . . . . . . . . . 12 (𝜑𝐾 ∈ Lat)
191, 3, 54atexlemqtb 37196 . . . . . . . . . . . 12 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
201, 3, 54atexlempsb 37195 . . . . . . . . . . . 12 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
21 eqid 2821 . . . . . . . . . . . . 13 (Base‘𝐾) = (Base‘𝐾)
2221, 2, 4latmle1 17685 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
2318, 19, 20, 22syl3anc 1367 . . . . . . . . . . 11 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
2414atexlemk 37182 . . . . . . . . . . . 12 (𝜑𝐾 ∈ HL)
2514atexlemq 37186 . . . . . . . . . . . 12 (𝜑𝑄𝐴)
2614atexlemt 37188 . . . . . . . . . . . 12 (𝜑𝑇𝐴)
273, 5hlatjcom 36503 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) = (𝑇 𝑄))
2824, 25, 26, 27syl3anc 1367 . . . . . . . . . . 11 (𝜑 → (𝑄 𝑇) = (𝑇 𝑄))
2923, 28breqtrd 5091 . . . . . . . . . 10 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑇 𝑄))
3029adantr 483 . . . . . . . . 9 ((𝜑𝐶 = 𝑃) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑇 𝑄))
3117, 30eqbrtrrd 5089 . . . . . . . 8 ((𝜑𝐶 = 𝑃) → 𝑃 (𝑇 𝑄))
3214atexlemkc 37193 . . . . . . . . . 10 (𝜑𝐾 ∈ CvLat)
3314atexlemp 37185 . . . . . . . . . 10 (𝜑𝑃𝐴)
3414atexlempnq 37190 . . . . . . . . . 10 (𝜑𝑃𝑄)
352, 3, 5cvlatexch2 36472 . . . . . . . . . 10 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑇𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 (𝑇 𝑄) → 𝑇 (𝑃 𝑄)))
3632, 33, 26, 25, 34, 35syl131anc 1379 . . . . . . . . 9 (𝜑 → (𝑃 (𝑇 𝑄) → 𝑇 (𝑃 𝑄)))
3736adantr 483 . . . . . . . 8 ((𝜑𝐶 = 𝑃) → (𝑃 (𝑇 𝑄) → 𝑇 (𝑃 𝑄)))
3831, 37mpd 15 . . . . . . 7 ((𝜑𝐶 = 𝑃) → 𝑇 (𝑃 𝑄))
3938ex 415 . . . . . 6 (𝜑 → (𝐶 = 𝑃𝑇 (𝑃 𝑄)))
4039necon3bd 3030 . . . . 5 (𝜑 → (¬ 𝑇 (𝑃 𝑄) → 𝐶𝑃))
4114, 40mpd 15 . . . 4 (𝜑𝐶𝑃)
4241adantr 483 . . 3 ((𝜑𝐶𝑆) → 𝐶𝑃)
43 simpr 487 . . 3 ((𝜑𝐶𝑆) → 𝐶𝑆)
4421, 2, 4latmle2 17686 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑆))
4518, 19, 20, 44syl3anc 1367 . . . . 5 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑆))
469, 45eqbrtrid 5100 . . . 4 (𝜑𝐶 (𝑃 𝑆))
4746adantr 483 . . 3 ((𝜑𝐶𝑆) → 𝐶 (𝑃 𝑆))
4814atexlems 37187 . . . . 5 (𝜑𝑆𝐴)
491, 2, 3, 54atexlempns 37197 . . . . 5 (𝜑𝑃𝑆)
505, 2, 3cvlsupr2 36478 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑆𝐴𝐶𝐴) ∧ 𝑃𝑆) → ((𝑃 𝐶) = (𝑆 𝐶) ↔ (𝐶𝑃𝐶𝑆𝐶 (𝑃 𝑆))))
5132, 33, 48, 10, 49, 50syl131anc 1379 . . . 4 (𝜑 → ((𝑃 𝐶) = (𝑆 𝐶) ↔ (𝐶𝑃𝐶𝑆𝐶 (𝑃 𝑆))))
5251adantr 483 . . 3 ((𝜑𝐶𝑆) → ((𝑃 𝐶) = (𝑆 𝐶) ↔ (𝐶𝑃𝐶𝑆𝐶 (𝑃 𝑆))))
5342, 43, 47, 52mpbir3and 1338 . 2 ((𝜑𝐶𝑆) → (𝑃 𝐶) = (𝑆 𝐶))
54 breq1 5068 . . . . 5 (𝑧 = 𝐶 → (𝑧 𝑊𝐶 𝑊))
5554notbid 320 . . . 4 (𝑧 = 𝐶 → (¬ 𝑧 𝑊 ↔ ¬ 𝐶 𝑊))
56 oveq2 7163 . . . . 5 (𝑧 = 𝐶 → (𝑃 𝑧) = (𝑃 𝐶))
57 oveq2 7163 . . . . 5 (𝑧 = 𝐶 → (𝑆 𝑧) = (𝑆 𝐶))
5856, 57eqeq12d 2837 . . . 4 (𝑧 = 𝐶 → ((𝑃 𝑧) = (𝑆 𝑧) ↔ (𝑃 𝐶) = (𝑆 𝐶)))
5955, 58anbi12d 632 . . 3 (𝑧 = 𝐶 → ((¬ 𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)) ↔ (¬ 𝐶 𝑊 ∧ (𝑃 𝐶) = (𝑆 𝐶))))
6059rspcev 3622 . 2 ((𝐶𝐴 ∧ (¬ 𝐶 𝑊 ∧ (𝑃 𝐶) = (𝑆 𝐶))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
6111, 13, 53, 60syl12anc 834 1 ((𝜑𝐶𝑆) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wcel 2110  wne 3016  wrex 3139   class class class wbr 5065  cfv 6354  (class class class)co 7155  Basecbs 16482  lecple 16571  joincjn 17553  meetcmee 17554  Latclat 17654  Atomscatm 36398  CvLatclc 36400  HLchlt 36485  LHypclh 37119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-proset 17537  df-poset 17555  df-plt 17567  df-lub 17583  df-glb 17584  df-join 17585  df-meet 17586  df-p0 17648  df-p1 17649  df-lat 17655  df-clat 17717  df-oposet 36311  df-ol 36313  df-oml 36314  df-covers 36401  df-ats 36402  df-atl 36433  df-cvlat 36457  df-hlat 36486  df-llines 36633  df-lplanes 36634  df-lhyp 37123
This theorem is referenced by:  4atexlemex4  37208  4atexlemex6  37209
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