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Theorem 4atexlemex2 36030
Description: Lemma for 4atexlem7 36034. 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 36028 . . 3 (𝜑𝐶𝐴)
1110adantr 472 . 2 ((𝜑𝐶𝑆) → 𝐶𝐴)
121, 2, 3, 4, 5, 6, 7, 8, 94atexlemnclw 36029 . . 3 (𝜑 → ¬ 𝐶 𝑊)
1312adantr 472 . 2 ((𝜑𝐶𝑆) → ¬ 𝐶 𝑊)
141, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 36027 . . . . 5 (𝜑 → ¬ 𝑇 (𝑃 𝑄))
15 id 22 . . . . . . . . . . 11 (𝐶 = 𝑃𝐶 = 𝑃)
169, 15syl5eqr 2813 . . . . . . . . . 10 (𝐶 = 𝑃 → ((𝑄 𝑇) (𝑃 𝑆)) = 𝑃)
1716adantl 473 . . . . . . . . 9 ((𝜑𝐶 = 𝑃) → ((𝑄 𝑇) (𝑃 𝑆)) = 𝑃)
1814atexlemkl 36016 . . . . . . . . . . . 12 (𝜑𝐾 ∈ Lat)
191, 3, 54atexlemqtb 36020 . . . . . . . . . . . 12 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
201, 3, 54atexlempsb 36019 . . . . . . . . . . . 12 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
21 eqid 2765 . . . . . . . . . . . . 13 (Base‘𝐾) = (Base‘𝐾)
2221, 2, 4latmle1 17345 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
2318, 19, 20, 22syl3anc 1490 . . . . . . . . . . 11 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
2414atexlemk 36006 . . . . . . . . . . . 12 (𝜑𝐾 ∈ HL)
2514atexlemq 36010 . . . . . . . . . . . 12 (𝜑𝑄𝐴)
2614atexlemt 36012 . . . . . . . . . . . 12 (𝜑𝑇𝐴)
273, 5hlatjcom 35327 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) = (𝑇 𝑄))
2824, 25, 26, 27syl3anc 1490 . . . . . . . . . . 11 (𝜑 → (𝑄 𝑇) = (𝑇 𝑄))
2923, 28breqtrd 4837 . . . . . . . . . 10 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑇 𝑄))
3029adantr 472 . . . . . . . . 9 ((𝜑𝐶 = 𝑃) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑇 𝑄))
3117, 30eqbrtrrd 4835 . . . . . . . 8 ((𝜑𝐶 = 𝑃) → 𝑃 (𝑇 𝑄))
3214atexlemkc 36017 . . . . . . . . . 10 (𝜑𝐾 ∈ CvLat)
3314atexlemp 36009 . . . . . . . . . 10 (𝜑𝑃𝐴)
3414atexlempnq 36014 . . . . . . . . . 10 (𝜑𝑃𝑄)
352, 3, 5cvlatexch2 35296 . . . . . . . . . 10 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑇𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 (𝑇 𝑄) → 𝑇 (𝑃 𝑄)))
3632, 33, 26, 25, 34, 35syl131anc 1502 . . . . . . . . 9 (𝜑 → (𝑃 (𝑇 𝑄) → 𝑇 (𝑃 𝑄)))
3736adantr 472 . . . . . . . 8 ((𝜑𝐶 = 𝑃) → (𝑃 (𝑇 𝑄) → 𝑇 (𝑃 𝑄)))
3831, 37mpd 15 . . . . . . 7 ((𝜑𝐶 = 𝑃) → 𝑇 (𝑃 𝑄))
3938ex 401 . . . . . 6 (𝜑 → (𝐶 = 𝑃𝑇 (𝑃 𝑄)))
4039necon3bd 2951 . . . . 5 (𝜑 → (¬ 𝑇 (𝑃 𝑄) → 𝐶𝑃))
4114, 40mpd 15 . . . 4 (𝜑𝐶𝑃)
4241adantr 472 . . 3 ((𝜑𝐶𝑆) → 𝐶𝑃)
43 simpr 477 . . 3 ((𝜑𝐶𝑆) → 𝐶𝑆)
4421, 2, 4latmle2 17346 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑆))
4518, 19, 20, 44syl3anc 1490 . . . . 5 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑆))
469, 45syl5eqbr 4846 . . . 4 (𝜑𝐶 (𝑃 𝑆))
4746adantr 472 . . 3 ((𝜑𝐶𝑆) → 𝐶 (𝑃 𝑆))
4814atexlems 36011 . . . . 5 (𝜑𝑆𝐴)
491, 2, 3, 54atexlempns 36021 . . . . 5 (𝜑𝑃𝑆)
505, 2, 3cvlsupr2 35302 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑆𝐴𝐶𝐴) ∧ 𝑃𝑆) → ((𝑃 𝐶) = (𝑆 𝐶) ↔ (𝐶𝑃𝐶𝑆𝐶 (𝑃 𝑆))))
5132, 33, 48, 10, 49, 50syl131anc 1502 . . . 4 (𝜑 → ((𝑃 𝐶) = (𝑆 𝐶) ↔ (𝐶𝑃𝐶𝑆𝐶 (𝑃 𝑆))))
5251adantr 472 . . 3 ((𝜑𝐶𝑆) → ((𝑃 𝐶) = (𝑆 𝐶) ↔ (𝐶𝑃𝐶𝑆𝐶 (𝑃 𝑆))))
5342, 43, 47, 52mpbir3and 1442 . 2 ((𝜑𝐶𝑆) → (𝑃 𝐶) = (𝑆 𝐶))
54 breq1 4814 . . . . 5 (𝑧 = 𝐶 → (𝑧 𝑊𝐶 𝑊))
5554notbid 309 . . . 4 (𝑧 = 𝐶 → (¬ 𝑧 𝑊 ↔ ¬ 𝐶 𝑊))
56 oveq2 6852 . . . . 5 (𝑧 = 𝐶 → (𝑃 𝑧) = (𝑃 𝐶))
57 oveq2 6852 . . . . 5 (𝑧 = 𝐶 → (𝑆 𝑧) = (𝑆 𝐶))
5856, 57eqeq12d 2780 . . . 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 3462 . 2 ((𝐶𝐴 ∧ (¬ 𝐶 𝑊 ∧ (𝑃 𝐶) = (𝑆 𝐶))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
6111, 13, 53, 60syl12anc 865 1 ((𝜑𝐶𝑆) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wrex 3056   class class class wbr 4811  cfv 6070  (class class class)co 6844  Basecbs 16133  lecple 16224  joincjn 17213  meetcmee 17214  Latclat 17314  Atomscatm 35222  CvLatclc 35224  HLchlt 35309  LHypclh 35943
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 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
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 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6805  df-ov 6847  df-oprab 6848  df-proset 17197  df-poset 17215  df-plt 17227  df-lub 17243  df-glb 17244  df-join 17245  df-meet 17246  df-p0 17308  df-p1 17309  df-lat 17315  df-clat 17377  df-oposet 35135  df-ol 35137  df-oml 35138  df-covers 35225  df-ats 35226  df-atl 35257  df-cvlat 35281  df-hlat 35310  df-llines 35457  df-lplanes 35458  df-lhyp 35947
This theorem is referenced by:  4atexlemex4  36032  4atexlemex6  36033
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