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Theorem 4atexlemnclw 39480
Description: Lemma for 4atexlem7 39485. (Contributed by NM, 24-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
4atexlemnclw (πœ‘ β†’ Β¬ 𝐢 ≀ π‘Š)

Proof of Theorem 4atexlemnclw
StepHypRef Expression
1 4thatlem0.c . . . 4 𝐢 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
2 4thatlem.ph . . . . . 6 (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))))
324atexlemkl 39467 . . . . 5 (πœ‘ β†’ 𝐾 ∈ Lat)
4 4thatlem0.j . . . . . 6 ∨ = (joinβ€˜πΎ)
5 4thatlem0.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
62, 4, 54atexlemqtb 39471 . . . . 5 (πœ‘ β†’ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
72, 4, 54atexlempsb 39470 . . . . 5 (πœ‘ β†’ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
8 eqid 2727 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
9 4thatlem0.l . . . . . 6 ≀ = (leβ€˜πΎ)
10 4thatlem0.m . . . . . 6 ∧ = (meetβ€˜πΎ)
118, 9, 10latmle1 18447 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ)) β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑄 ∨ 𝑇))
123, 6, 7, 11syl3anc 1369 . . . 4 (πœ‘ β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑄 ∨ 𝑇))
131, 12eqbrtrid 5177 . . 3 (πœ‘ β†’ 𝐢 ≀ (𝑄 ∨ 𝑇))
14 simp13r 1287 . . . . 5 ((((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))) β†’ Β¬ 𝑄 ≀ π‘Š)
152, 14sylbi 216 . . . 4 (πœ‘ β†’ Β¬ 𝑄 ≀ π‘Š)
1624atexlemkc 39468 . . . . . 6 (πœ‘ β†’ 𝐾 ∈ CvLat)
17 4thatlem0.h . . . . . . 7 𝐻 = (LHypβ€˜πΎ)
18 4thatlem0.u . . . . . . 7 π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
19 4thatlem0.v . . . . . . 7 𝑉 = ((𝑃 ∨ 𝑆) ∧ π‘Š)
202, 9, 4, 10, 5, 17, 18, 194atexlemv 39475 . . . . . 6 (πœ‘ β†’ 𝑉 ∈ 𝐴)
2124atexlemq 39461 . . . . . 6 (πœ‘ β†’ 𝑄 ∈ 𝐴)
2224atexlemt 39463 . . . . . 6 (πœ‘ β†’ 𝑇 ∈ 𝐴)
232, 9, 4, 10, 5, 17, 184atexlemu 39474 . . . . . . 7 (πœ‘ β†’ π‘ˆ ∈ 𝐴)
242, 9, 4, 10, 5, 17, 18, 194atexlemunv 39476 . . . . . . 7 (πœ‘ β†’ π‘ˆ β‰  𝑉)
2524atexlemutvt 39464 . . . . . . 7 (πœ‘ β†’ (π‘ˆ ∨ 𝑇) = (𝑉 ∨ 𝑇))
265, 4cvlsupr6 38756 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (π‘ˆ ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (π‘ˆ β‰  𝑉 ∧ (π‘ˆ ∨ 𝑇) = (𝑉 ∨ 𝑇))) β†’ 𝑇 β‰  𝑉)
2726necomd 2991 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (π‘ˆ ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (π‘ˆ β‰  𝑉 ∧ (π‘ˆ ∨ 𝑇) = (𝑉 ∨ 𝑇))) β†’ 𝑉 β‰  𝑇)
2816, 23, 20, 22, 24, 25, 27syl132anc 1386 . . . . . 6 (πœ‘ β†’ 𝑉 β‰  𝑇)
299, 4, 5cvlatexch2 38746 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑉 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ 𝑉 β‰  𝑇) β†’ (𝑉 ≀ (𝑄 ∨ 𝑇) β†’ 𝑄 ≀ (𝑉 ∨ 𝑇)))
3016, 20, 21, 22, 28, 29syl131anc 1381 . . . . 5 (πœ‘ β†’ (𝑉 ≀ (𝑄 ∨ 𝑇) β†’ 𝑄 ≀ (𝑉 ∨ 𝑇)))
312, 174atexlemwb 39469 . . . . . . . . 9 (πœ‘ β†’ π‘Š ∈ (Baseβ€˜πΎ))
328, 9, 10latmle2 18448 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ ((𝑃 ∨ 𝑆) ∧ π‘Š) ≀ π‘Š)
333, 7, 31, 32syl3anc 1369 . . . . . . . 8 (πœ‘ β†’ ((𝑃 ∨ 𝑆) ∧ π‘Š) ≀ π‘Š)
3419, 33eqbrtrid 5177 . . . . . . 7 (πœ‘ β†’ 𝑉 ≀ π‘Š)
352, 9, 4, 10, 5, 17, 18, 194atexlemtlw 39477 . . . . . . 7 (πœ‘ β†’ 𝑇 ≀ π‘Š)
368, 5atbase 38698 . . . . . . . . 9 (𝑉 ∈ 𝐴 β†’ 𝑉 ∈ (Baseβ€˜πΎ))
3720, 36syl 17 . . . . . . . 8 (πœ‘ β†’ 𝑉 ∈ (Baseβ€˜πΎ))
388, 5atbase 38698 . . . . . . . . 9 (𝑇 ∈ 𝐴 β†’ 𝑇 ∈ (Baseβ€˜πΎ))
3922, 38syl 17 . . . . . . . 8 (πœ‘ β†’ 𝑇 ∈ (Baseβ€˜πΎ))
408, 9, 4latjle12 18433 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑉 ∈ (Baseβ€˜πΎ) ∧ 𝑇 ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ))) β†’ ((𝑉 ≀ π‘Š ∧ 𝑇 ≀ π‘Š) ↔ (𝑉 ∨ 𝑇) ≀ π‘Š))
413, 37, 39, 31, 40syl13anc 1370 . . . . . . 7 (πœ‘ β†’ ((𝑉 ≀ π‘Š ∧ 𝑇 ≀ π‘Š) ↔ (𝑉 ∨ 𝑇) ≀ π‘Š))
4234, 35, 41mpbi2and 711 . . . . . 6 (πœ‘ β†’ (𝑉 ∨ 𝑇) ≀ π‘Š)
438, 5atbase 38698 . . . . . . . 8 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
4421, 43syl 17 . . . . . . 7 (πœ‘ β†’ 𝑄 ∈ (Baseβ€˜πΎ))
4524atexlemk 39457 . . . . . . . 8 (πœ‘ β†’ 𝐾 ∈ HL)
468, 4, 5hlatjcl 38776 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) β†’ (𝑉 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
4745, 20, 22, 46syl3anc 1369 . . . . . . 7 (πœ‘ β†’ (𝑉 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
488, 9lattr 18427 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Baseβ€˜πΎ) ∧ (𝑉 ∨ 𝑇) ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ))) β†’ ((𝑄 ≀ (𝑉 ∨ 𝑇) ∧ (𝑉 ∨ 𝑇) ≀ π‘Š) β†’ 𝑄 ≀ π‘Š))
493, 44, 47, 31, 48syl13anc 1370 . . . . . 6 (πœ‘ β†’ ((𝑄 ≀ (𝑉 ∨ 𝑇) ∧ (𝑉 ∨ 𝑇) ≀ π‘Š) β†’ 𝑄 ≀ π‘Š))
5042, 49mpan2d 693 . . . . 5 (πœ‘ β†’ (𝑄 ≀ (𝑉 ∨ 𝑇) β†’ 𝑄 ≀ π‘Š))
5130, 50syld 47 . . . 4 (πœ‘ β†’ (𝑉 ≀ (𝑄 ∨ 𝑇) β†’ 𝑄 ≀ π‘Š))
5215, 51mtod 197 . . 3 (πœ‘ β†’ Β¬ 𝑉 ≀ (𝑄 ∨ 𝑇))
53 nbrne2 5162 . . 3 ((𝐢 ≀ (𝑄 ∨ 𝑇) ∧ Β¬ 𝑉 ≀ (𝑄 ∨ 𝑇)) β†’ 𝐢 β‰  𝑉)
5413, 52, 53syl2anc 583 . 2 (πœ‘ β†’ 𝐢 β‰  𝑉)
5524atexlemw 39458 . . . 4 (πœ‘ β†’ π‘Š ∈ 𝐻)
5645, 55jca 511 . . 3 (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
5724atexlempw 39459 . . 3 (πœ‘ β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
5824atexlems 39462 . . 3 (πœ‘ β†’ 𝑆 ∈ 𝐴)
592, 9, 4, 10, 5, 17, 18, 19, 14atexlemc 39479 . . 3 (πœ‘ β†’ 𝐢 ∈ 𝐴)
602, 9, 4, 54atexlempns 39472 . . 3 (πœ‘ β†’ 𝑃 β‰  𝑆)
618, 9, 10latmle2 18448 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ)) β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑃 ∨ 𝑆))
623, 6, 7, 61syl3anc 1369 . . . 4 (πœ‘ β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑃 ∨ 𝑆))
631, 62eqbrtrid 5177 . . 3 (πœ‘ β†’ 𝐢 ≀ (𝑃 ∨ 𝑆))
649, 4, 10, 5, 17, 19lhpat3 39456 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) ∧ (𝑆 ∈ 𝐴 ∧ 𝐢 ∈ 𝐴) ∧ (𝑃 β‰  𝑆 ∧ 𝐢 ≀ (𝑃 ∨ 𝑆))) β†’ (Β¬ 𝐢 ≀ π‘Š ↔ 𝐢 β‰  𝑉))
6556, 57, 58, 59, 60, 63, 64syl222anc 1384 . 2 (πœ‘ β†’ (Β¬ 𝐢 ≀ π‘Š ↔ 𝐢 β‰  𝑉))
6654, 65mpbird 257 1 (πœ‘ β†’ Β¬ 𝐢 ≀ π‘Š)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2935   class class class wbr 5142  β€˜cfv 6542  (class class class)co 7414  Basecbs 17171  lecple 17231  joincjn 18294  meetcmee 18295  Latclat 18414  Atomscatm 38672  CvLatclc 38674  HLchlt 38759  LHypclh 39394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-proset 18278  df-poset 18296  df-plt 18313  df-lub 18329  df-glb 18330  df-join 18331  df-meet 18332  df-p0 18408  df-p1 18409  df-lat 18415  df-clat 18482  df-oposet 38585  df-ol 38587  df-oml 38588  df-covers 38675  df-ats 38676  df-atl 38707  df-cvlat 38731  df-hlat 38760  df-llines 38908  df-lplanes 38909  df-lhyp 39398
This theorem is referenced by:  4atexlemex2  39481  4atexlemcnd  39482
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