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Theorem 4atexlemex2 39034
Description: Lemma for 4atexlem7 39038. Show that when 𝐢 β‰  𝑆, 𝐢 satisfies the existence condition of the consequent. (Contributed by NM, 25-Nov-2012.)
Hypotheses
Ref Expression
4thatlem.ph (πœ‘ ↔ (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (π‘ˆ ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))))
4thatlem0.l ≀ = (leβ€˜πΎ)
4thatlem0.j ∨ = (joinβ€˜πΎ)
4thatlem0.m ∧ = (meetβ€˜πΎ)
4thatlem0.a 𝐴 = (Atomsβ€˜πΎ)
4thatlem0.h 𝐻 = (LHypβ€˜πΎ)
4thatlem0.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
4thatlem0.v 𝑉 = ((𝑃 ∨ 𝑆) ∧ π‘Š)
4thatlem0.c 𝐢 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
Assertion
Ref Expression
4atexlemex2 ((πœ‘ ∧ 𝐢 β‰  𝑆) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐢   𝑧, ∨   𝑧, ≀   𝑧,𝑃   𝑧,𝑆   𝑧,π‘Š
Allowed substitution hints:   πœ‘(𝑧)   𝑄(𝑧)   𝑅(𝑧)   𝑇(𝑧)   π‘ˆ(𝑧)   𝐻(𝑧)   𝐾(𝑧)   ∧ (𝑧)   𝑉(𝑧)

Proof of Theorem 4atexlemex2
StepHypRef Expression
1 4thatlem.ph . . . 4 (πœ‘ ↔ (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (π‘ˆ ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))))
2 4thatlem0.l . . . 4 ≀ = (leβ€˜πΎ)
3 4thatlem0.j . . . 4 ∨ = (joinβ€˜πΎ)
4 4thatlem0.m . . . 4 ∧ = (meetβ€˜πΎ)
5 4thatlem0.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
6 4thatlem0.h . . . 4 𝐻 = (LHypβ€˜πΎ)
7 4thatlem0.u . . . 4 π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
8 4thatlem0.v . . . 4 𝑉 = ((𝑃 ∨ 𝑆) ∧ π‘Š)
9 4thatlem0.c . . . 4 𝐢 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
101, 2, 3, 4, 5, 6, 7, 8, 94atexlemc 39032 . . 3 (πœ‘ β†’ 𝐢 ∈ 𝐴)
1110adantr 481 . 2 ((πœ‘ ∧ 𝐢 β‰  𝑆) β†’ 𝐢 ∈ 𝐴)
121, 2, 3, 4, 5, 6, 7, 8, 94atexlemnclw 39033 . . 3 (πœ‘ β†’ Β¬ 𝐢 ≀ π‘Š)
1312adantr 481 . 2 ((πœ‘ ∧ 𝐢 β‰  𝑆) β†’ Β¬ 𝐢 ≀ π‘Š)
141, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 39031 . . . . 5 (πœ‘ β†’ Β¬ 𝑇 ≀ (𝑃 ∨ 𝑄))
15 id 22 . . . . . . . . . . 11 (𝐢 = 𝑃 β†’ 𝐢 = 𝑃)
169, 15eqtr3id 2786 . . . . . . . . . 10 (𝐢 = 𝑃 β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = 𝑃)
1716adantl 482 . . . . . . . . 9 ((πœ‘ ∧ 𝐢 = 𝑃) β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = 𝑃)
1814atexlemkl 39020 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐾 ∈ Lat)
191, 3, 54atexlemqtb 39024 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
201, 3, 54atexlempsb 39023 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
21 eqid 2732 . . . . . . . . . . . . 13 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2221, 2, 4latmle1 18419 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ)) β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑄 ∨ 𝑇))
2318, 19, 20, 22syl3anc 1371 . . . . . . . . . . 11 (πœ‘ β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑄 ∨ 𝑇))
2414atexlemk 39010 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐾 ∈ HL)
2514atexlemq 39014 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑄 ∈ 𝐴)
2614atexlemt 39016 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑇 ∈ 𝐴)
273, 5hlatjcom 38330 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) β†’ (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
2824, 25, 26, 27syl3anc 1371 . . . . . . . . . . 11 (πœ‘ β†’ (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
2923, 28breqtrd 5174 . . . . . . . . . 10 (πœ‘ β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑇 ∨ 𝑄))
3029adantr 481 . . . . . . . . 9 ((πœ‘ ∧ 𝐢 = 𝑃) β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑇 ∨ 𝑄))
3117, 30eqbrtrrd 5172 . . . . . . . 8 ((πœ‘ ∧ 𝐢 = 𝑃) β†’ 𝑃 ≀ (𝑇 ∨ 𝑄))
3214atexlemkc 39021 . . . . . . . . . 10 (πœ‘ β†’ 𝐾 ∈ CvLat)
3314atexlemp 39013 . . . . . . . . . 10 (πœ‘ β†’ 𝑃 ∈ 𝐴)
3414atexlempnq 39018 . . . . . . . . . 10 (πœ‘ β†’ 𝑃 β‰  𝑄)
352, 3, 5cvlatexch2 38299 . . . . . . . . . 10 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ (𝑃 ≀ (𝑇 ∨ 𝑄) β†’ 𝑇 ≀ (𝑃 ∨ 𝑄)))
3632, 33, 26, 25, 34, 35syl131anc 1383 . . . . . . . . 9 (πœ‘ β†’ (𝑃 ≀ (𝑇 ∨ 𝑄) β†’ 𝑇 ≀ (𝑃 ∨ 𝑄)))
3736adantr 481 . . . . . . . 8 ((πœ‘ ∧ 𝐢 = 𝑃) β†’ (𝑃 ≀ (𝑇 ∨ 𝑄) β†’ 𝑇 ≀ (𝑃 ∨ 𝑄)))
3831, 37mpd 15 . . . . . . 7 ((πœ‘ ∧ 𝐢 = 𝑃) β†’ 𝑇 ≀ (𝑃 ∨ 𝑄))
3938ex 413 . . . . . 6 (πœ‘ β†’ (𝐢 = 𝑃 β†’ 𝑇 ≀ (𝑃 ∨ 𝑄)))
4039necon3bd 2954 . . . . 5 (πœ‘ β†’ (Β¬ 𝑇 ≀ (𝑃 ∨ 𝑄) β†’ 𝐢 β‰  𝑃))
4114, 40mpd 15 . . . 4 (πœ‘ β†’ 𝐢 β‰  𝑃)
4241adantr 481 . . 3 ((πœ‘ ∧ 𝐢 β‰  𝑆) β†’ 𝐢 β‰  𝑃)
43 simpr 485 . . 3 ((πœ‘ ∧ 𝐢 β‰  𝑆) β†’ 𝐢 β‰  𝑆)
4421, 2, 4latmle2 18420 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ)) β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑃 ∨ 𝑆))
4518, 19, 20, 44syl3anc 1371 . . . . 5 (πœ‘ β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑃 ∨ 𝑆))
469, 45eqbrtrid 5183 . . . 4 (πœ‘ β†’ 𝐢 ≀ (𝑃 ∨ 𝑆))
4746adantr 481 . . 3 ((πœ‘ ∧ 𝐢 β‰  𝑆) β†’ 𝐢 ≀ (𝑃 ∨ 𝑆))
4814atexlems 39015 . . . . 5 (πœ‘ β†’ 𝑆 ∈ 𝐴)
491, 2, 3, 54atexlempns 39025 . . . . 5 (πœ‘ β†’ 𝑃 β‰  𝑆)
505, 2, 3cvlsupr2 38305 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝐢 ∈ 𝐴) ∧ 𝑃 β‰  𝑆) β†’ ((𝑃 ∨ 𝐢) = (𝑆 ∨ 𝐢) ↔ (𝐢 β‰  𝑃 ∧ 𝐢 β‰  𝑆 ∧ 𝐢 ≀ (𝑃 ∨ 𝑆))))
5132, 33, 48, 10, 49, 50syl131anc 1383 . . . 4 (πœ‘ β†’ ((𝑃 ∨ 𝐢) = (𝑆 ∨ 𝐢) ↔ (𝐢 β‰  𝑃 ∧ 𝐢 β‰  𝑆 ∧ 𝐢 ≀ (𝑃 ∨ 𝑆))))
5251adantr 481 . . 3 ((πœ‘ ∧ 𝐢 β‰  𝑆) β†’ ((𝑃 ∨ 𝐢) = (𝑆 ∨ 𝐢) ↔ (𝐢 β‰  𝑃 ∧ 𝐢 β‰  𝑆 ∧ 𝐢 ≀ (𝑃 ∨ 𝑆))))
5342, 43, 47, 52mpbir3and 1342 . 2 ((πœ‘ ∧ 𝐢 β‰  𝑆) β†’ (𝑃 ∨ 𝐢) = (𝑆 ∨ 𝐢))
54 breq1 5151 . . . . 5 (𝑧 = 𝐢 β†’ (𝑧 ≀ π‘Š ↔ 𝐢 ≀ π‘Š))
5554notbid 317 . . . 4 (𝑧 = 𝐢 β†’ (Β¬ 𝑧 ≀ π‘Š ↔ Β¬ 𝐢 ≀ π‘Š))
56 oveq2 7419 . . . . 5 (𝑧 = 𝐢 β†’ (𝑃 ∨ 𝑧) = (𝑃 ∨ 𝐢))
57 oveq2 7419 . . . . 5 (𝑧 = 𝐢 β†’ (𝑆 ∨ 𝑧) = (𝑆 ∨ 𝐢))
5856, 57eqeq12d 2748 . . . 4 (𝑧 = 𝐢 β†’ ((𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧) ↔ (𝑃 ∨ 𝐢) = (𝑆 ∨ 𝐢)))
5955, 58anbi12d 631 . . 3 (𝑧 = 𝐢 β†’ ((Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ (Β¬ 𝐢 ≀ π‘Š ∧ (𝑃 ∨ 𝐢) = (𝑆 ∨ 𝐢))))
6059rspcev 3612 . 2 ((𝐢 ∈ 𝐴 ∧ (Β¬ 𝐢 ≀ π‘Š ∧ (𝑃 ∨ 𝐢) = (𝑆 ∨ 𝐢))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
6111, 13, 53, 60syl12anc 835 1 ((πœ‘ ∧ 𝐢 β‰  𝑆) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  lecple 17206  joincjn 18266  meetcmee 18267  Latclat 18386  Atomscatm 38225  CvLatclc 38227  HLchlt 38312  LHypclh 38947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-proset 18250  df-poset 18268  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-p1 18381  df-lat 18387  df-clat 18454  df-oposet 38138  df-ol 38140  df-oml 38141  df-covers 38228  df-ats 38229  df-atl 38260  df-cvlat 38284  df-hlat 38313  df-llines 38461  df-lplanes 38462  df-lhyp 38951
This theorem is referenced by:  4atexlemex4  39036  4atexlemex6  39037
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