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Theorem 4atexlemex2 35183
Description: Lemma for 4atexlem7 35187. 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 35181 . . 3 (𝜑𝐶𝐴)
1110adantr 481 . 2 ((𝜑𝐶𝑆) → 𝐶𝐴)
121, 2, 3, 4, 5, 6, 7, 8, 94atexlemnclw 35182 . . 3 (𝜑 → ¬ 𝐶 𝑊)
1312adantr 481 . 2 ((𝜑𝐶𝑆) → ¬ 𝐶 𝑊)
141, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 35180 . . . . 5 (𝜑 → ¬ 𝑇 (𝑃 𝑄))
15 id 22 . . . . . . . . . . 11 (𝐶 = 𝑃𝐶 = 𝑃)
169, 15syl5eqr 2669 . . . . . . . . . 10 (𝐶 = 𝑃 → ((𝑄 𝑇) (𝑃 𝑆)) = 𝑃)
1716adantl 482 . . . . . . . . 9 ((𝜑𝐶 = 𝑃) → ((𝑄 𝑇) (𝑃 𝑆)) = 𝑃)
1814atexlemkl 35169 . . . . . . . . . . . 12 (𝜑𝐾 ∈ Lat)
191, 3, 54atexlemqtb 35173 . . . . . . . . . . . 12 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
201, 3, 54atexlempsb 35172 . . . . . . . . . . . 12 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
21 eqid 2621 . . . . . . . . . . . . 13 (Base‘𝐾) = (Base‘𝐾)
2221, 2, 4latmle1 17070 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
2318, 19, 20, 22syl3anc 1325 . . . . . . . . . . 11 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
2414atexlemk 35159 . . . . . . . . . . . 12 (𝜑𝐾 ∈ HL)
2514atexlemq 35163 . . . . . . . . . . . 12 (𝜑𝑄𝐴)
2614atexlemt 35165 . . . . . . . . . . . 12 (𝜑𝑇𝐴)
273, 5hlatjcom 34480 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) = (𝑇 𝑄))
2824, 25, 26, 27syl3anc 1325 . . . . . . . . . . 11 (𝜑 → (𝑄 𝑇) = (𝑇 𝑄))
2923, 28breqtrd 4677 . . . . . . . . . 10 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑇 𝑄))
3029adantr 481 . . . . . . . . 9 ((𝜑𝐶 = 𝑃) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑇 𝑄))
3117, 30eqbrtrrd 4675 . . . . . . . 8 ((𝜑𝐶 = 𝑃) → 𝑃 (𝑇 𝑄))
3214atexlemkc 35170 . . . . . . . . . 10 (𝜑𝐾 ∈ CvLat)
3314atexlemp 35162 . . . . . . . . . 10 (𝜑𝑃𝐴)
3414atexlempnq 35167 . . . . . . . . . 10 (𝜑𝑃𝑄)
352, 3, 5cvlatexch2 34450 . . . . . . . . . 10 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑇𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 (𝑇 𝑄) → 𝑇 (𝑃 𝑄)))
3632, 33, 26, 25, 34, 35syl131anc 1338 . . . . . . . . 9 (𝜑 → (𝑃 (𝑇 𝑄) → 𝑇 (𝑃 𝑄)))
3736adantr 481 . . . . . . . 8 ((𝜑𝐶 = 𝑃) → (𝑃 (𝑇 𝑄) → 𝑇 (𝑃 𝑄)))
3831, 37mpd 15 . . . . . . 7 ((𝜑𝐶 = 𝑃) → 𝑇 (𝑃 𝑄))
3938ex 450 . . . . . 6 (𝜑 → (𝐶 = 𝑃𝑇 (𝑃 𝑄)))
4039necon3bd 2807 . . . . 5 (𝜑 → (¬ 𝑇 (𝑃 𝑄) → 𝐶𝑃))
4114, 40mpd 15 . . . 4 (𝜑𝐶𝑃)
4241adantr 481 . . 3 ((𝜑𝐶𝑆) → 𝐶𝑃)
43 simpr 477 . . 3 ((𝜑𝐶𝑆) → 𝐶𝑆)
4421, 2, 4latmle2 17071 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑆))
4518, 19, 20, 44syl3anc 1325 . . . . 5 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑆))
469, 45syl5eqbr 4686 . . . 4 (𝜑𝐶 (𝑃 𝑆))
4746adantr 481 . . 3 ((𝜑𝐶𝑆) → 𝐶 (𝑃 𝑆))
4814atexlems 35164 . . . . 5 (𝜑𝑆𝐴)
491, 2, 3, 54atexlempns 35174 . . . . 5 (𝜑𝑃𝑆)
505, 2, 3cvlsupr2 34456 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑆𝐴𝐶𝐴) ∧ 𝑃𝑆) → ((𝑃 𝐶) = (𝑆 𝐶) ↔ (𝐶𝑃𝐶𝑆𝐶 (𝑃 𝑆))))
5132, 33, 48, 10, 49, 50syl131anc 1338 . . . 4 (𝜑 → ((𝑃 𝐶) = (𝑆 𝐶) ↔ (𝐶𝑃𝐶𝑆𝐶 (𝑃 𝑆))))
5251adantr 481 . . 3 ((𝜑𝐶𝑆) → ((𝑃 𝐶) = (𝑆 𝐶) ↔ (𝐶𝑃𝐶𝑆𝐶 (𝑃 𝑆))))
5342, 43, 47, 52mpbir3and 1244 . 2 ((𝜑𝐶𝑆) → (𝑃 𝐶) = (𝑆 𝐶))
54 breq1 4654 . . . . 5 (𝑧 = 𝐶 → (𝑧 𝑊𝐶 𝑊))
5554notbid 308 . . . 4 (𝑧 = 𝐶 → (¬ 𝑧 𝑊 ↔ ¬ 𝐶 𝑊))
56 oveq2 6655 . . . . 5 (𝑧 = 𝐶 → (𝑃 𝑧) = (𝑃 𝐶))
57 oveq2 6655 . . . . 5 (𝑧 = 𝐶 → (𝑆 𝑧) = (𝑆 𝐶))
5856, 57eqeq12d 2636 . . . 4 (𝑧 = 𝐶 → ((𝑃 𝑧) = (𝑆 𝑧) ↔ (𝑃 𝐶) = (𝑆 𝐶)))
5955, 58anbi12d 747 . . 3 (𝑧 = 𝐶 → ((¬ 𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)) ↔ (¬ 𝐶 𝑊 ∧ (𝑃 𝐶) = (𝑆 𝐶))))
6059rspcev 3307 . 2 ((𝐶𝐴 ∧ (¬ 𝐶 𝑊 ∧ (𝑃 𝐶) = (𝑆 𝐶))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
6111, 13, 53, 60syl12anc 1323 1 ((𝜑𝐶𝑆) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1037   = wceq 1482  wcel 1989  wne 2793  wrex 2912   class class class wbr 4651  cfv 5886  (class class class)co 6647  Basecbs 15851  lecple 15942  joincjn 16938  meetcmee 16939  Latclat 17039  Atomscatm 34376  CvLatclc 34378  HLchlt 34463  LHypclh 35096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-preset 16922  df-poset 16940  df-plt 16952  df-lub 16968  df-glb 16969  df-join 16970  df-meet 16971  df-p0 17033  df-p1 17034  df-lat 17040  df-clat 17102  df-oposet 34289  df-ol 34291  df-oml 34292  df-covers 34379  df-ats 34380  df-atl 34411  df-cvlat 34435  df-hlat 34464  df-llines 34610  df-lplanes 34611  df-lhyp 35100
This theorem is referenced by:  4atexlemex4  35185  4atexlemex6  35186
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