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Theorem 4atexlemex2 36139
Description: Lemma for 4atexlem7 36143. 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 36137 . . 3 (𝜑𝐶𝐴)
1110adantr 474 . 2 ((𝜑𝐶𝑆) → 𝐶𝐴)
121, 2, 3, 4, 5, 6, 7, 8, 94atexlemnclw 36138 . . 3 (𝜑 → ¬ 𝐶 𝑊)
1312adantr 474 . 2 ((𝜑𝐶𝑆) → ¬ 𝐶 𝑊)
141, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 36136 . . . . 5 (𝜑 → ¬ 𝑇 (𝑃 𝑄))
15 id 22 . . . . . . . . . . 11 (𝐶 = 𝑃𝐶 = 𝑃)
169, 15syl5eqr 2875 . . . . . . . . . 10 (𝐶 = 𝑃 → ((𝑄 𝑇) (𝑃 𝑆)) = 𝑃)
1716adantl 475 . . . . . . . . 9 ((𝜑𝐶 = 𝑃) → ((𝑄 𝑇) (𝑃 𝑆)) = 𝑃)
1814atexlemkl 36125 . . . . . . . . . . . 12 (𝜑𝐾 ∈ Lat)
191, 3, 54atexlemqtb 36129 . . . . . . . . . . . 12 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
201, 3, 54atexlempsb 36128 . . . . . . . . . . . 12 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
21 eqid 2825 . . . . . . . . . . . . 13 (Base‘𝐾) = (Base‘𝐾)
2221, 2, 4latmle1 17429 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
2318, 19, 20, 22syl3anc 1494 . . . . . . . . . . 11 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
2414atexlemk 36115 . . . . . . . . . . . 12 (𝜑𝐾 ∈ HL)
2514atexlemq 36119 . . . . . . . . . . . 12 (𝜑𝑄𝐴)
2614atexlemt 36121 . . . . . . . . . . . 12 (𝜑𝑇𝐴)
273, 5hlatjcom 35436 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) = (𝑇 𝑄))
2824, 25, 26, 27syl3anc 1494 . . . . . . . . . . 11 (𝜑 → (𝑄 𝑇) = (𝑇 𝑄))
2923, 28breqtrd 4899 . . . . . . . . . 10 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑇 𝑄))
3029adantr 474 . . . . . . . . 9 ((𝜑𝐶 = 𝑃) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑇 𝑄))
3117, 30eqbrtrrd 4897 . . . . . . . 8 ((𝜑𝐶 = 𝑃) → 𝑃 (𝑇 𝑄))
3214atexlemkc 36126 . . . . . . . . . 10 (𝜑𝐾 ∈ CvLat)
3314atexlemp 36118 . . . . . . . . . 10 (𝜑𝑃𝐴)
3414atexlempnq 36123 . . . . . . . . . 10 (𝜑𝑃𝑄)
352, 3, 5cvlatexch2 35405 . . . . . . . . . 10 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑇𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 (𝑇 𝑄) → 𝑇 (𝑃 𝑄)))
3632, 33, 26, 25, 34, 35syl131anc 1506 . . . . . . . . 9 (𝜑 → (𝑃 (𝑇 𝑄) → 𝑇 (𝑃 𝑄)))
3736adantr 474 . . . . . . . 8 ((𝜑𝐶 = 𝑃) → (𝑃 (𝑇 𝑄) → 𝑇 (𝑃 𝑄)))
3831, 37mpd 15 . . . . . . 7 ((𝜑𝐶 = 𝑃) → 𝑇 (𝑃 𝑄))
3938ex 403 . . . . . 6 (𝜑 → (𝐶 = 𝑃𝑇 (𝑃 𝑄)))
4039necon3bd 3013 . . . . 5 (𝜑 → (¬ 𝑇 (𝑃 𝑄) → 𝐶𝑃))
4114, 40mpd 15 . . . 4 (𝜑𝐶𝑃)
4241adantr 474 . . 3 ((𝜑𝐶𝑆) → 𝐶𝑃)
43 simpr 479 . . 3 ((𝜑𝐶𝑆) → 𝐶𝑆)
4421, 2, 4latmle2 17430 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑆))
4518, 19, 20, 44syl3anc 1494 . . . . 5 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑆))
469, 45syl5eqbr 4908 . . . 4 (𝜑𝐶 (𝑃 𝑆))
4746adantr 474 . . 3 ((𝜑𝐶𝑆) → 𝐶 (𝑃 𝑆))
4814atexlems 36120 . . . . 5 (𝜑𝑆𝐴)
491, 2, 3, 54atexlempns 36130 . . . . 5 (𝜑𝑃𝑆)
505, 2, 3cvlsupr2 35411 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑆𝐴𝐶𝐴) ∧ 𝑃𝑆) → ((𝑃 𝐶) = (𝑆 𝐶) ↔ (𝐶𝑃𝐶𝑆𝐶 (𝑃 𝑆))))
5132, 33, 48, 10, 49, 50syl131anc 1506 . . . 4 (𝜑 → ((𝑃 𝐶) = (𝑆 𝐶) ↔ (𝐶𝑃𝐶𝑆𝐶 (𝑃 𝑆))))
5251adantr 474 . . 3 ((𝜑𝐶𝑆) → ((𝑃 𝐶) = (𝑆 𝐶) ↔ (𝐶𝑃𝐶𝑆𝐶 (𝑃 𝑆))))
5342, 43, 47, 52mpbir3and 1446 . 2 ((𝜑𝐶𝑆) → (𝑃 𝐶) = (𝑆 𝐶))
54 breq1 4876 . . . . 5 (𝑧 = 𝐶 → (𝑧 𝑊𝐶 𝑊))
5554notbid 310 . . . 4 (𝑧 = 𝐶 → (¬ 𝑧 𝑊 ↔ ¬ 𝐶 𝑊))
56 oveq2 6913 . . . . 5 (𝑧 = 𝐶 → (𝑃 𝑧) = (𝑃 𝐶))
57 oveq2 6913 . . . . 5 (𝑧 = 𝐶 → (𝑆 𝑧) = (𝑆 𝐶))
5856, 57eqeq12d 2840 . . . 4 (𝑧 = 𝐶 → ((𝑃 𝑧) = (𝑆 𝑧) ↔ (𝑃 𝐶) = (𝑆 𝐶)))
5955, 58anbi12d 624 . . 3 (𝑧 = 𝐶 → ((¬ 𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)) ↔ (¬ 𝐶 𝑊 ∧ (𝑃 𝐶) = (𝑆 𝐶))))
6059rspcev 3526 . 2 ((𝐶𝐴 ∧ (¬ 𝐶 𝑊 ∧ (𝑃 𝐶) = (𝑆 𝐶))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
6111, 13, 53, 60syl12anc 870 1 ((𝜑𝐶𝑆) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wne 2999  wrex 3118   class class class wbr 4873  cfv 6123  (class class class)co 6905  Basecbs 16222  lecple 16312  joincjn 17297  meetcmee 17298  Latclat 17398  Atomscatm 35331  CvLatclc 35333  HLchlt 35418  LHypclh 36052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-proset 17281  df-poset 17299  df-plt 17311  df-lub 17327  df-glb 17328  df-join 17329  df-meet 17330  df-p0 17392  df-p1 17393  df-lat 17399  df-clat 17461  df-oposet 35244  df-ol 35246  df-oml 35247  df-covers 35334  df-ats 35335  df-atl 35366  df-cvlat 35390  df-hlat 35419  df-llines 35566  df-lplanes 35567  df-lhyp 36056
This theorem is referenced by:  4atexlemex4  36141  4atexlemex6  36142
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