Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  4atexlemtlw Structured version   Visualization version   GIF version

Theorem 4atexlemtlw 39451
Description: Lemma for 4atexlem7 39459. (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)
Assertion
Ref Expression
4atexlemtlw (πœ‘ β†’ 𝑇 ≀ π‘Š)

Proof of Theorem 4atexlemtlw
StepHypRef Expression
1 eqid 2726 . 2 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 4thatlem0.l . 2 ≀ = (leβ€˜πΎ)
3 4thatlem.ph . . 3 (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))))
434atexlemkl 39441 . 2 (πœ‘ β†’ 𝐾 ∈ Lat)
534atexlemt 39437 . . 3 (πœ‘ β†’ 𝑇 ∈ 𝐴)
6 4thatlem0.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
71, 6atbase 38672 . . 3 (𝑇 ∈ 𝐴 β†’ 𝑇 ∈ (Baseβ€˜πΎ))
85, 7syl 17 . 2 (πœ‘ β†’ 𝑇 ∈ (Baseβ€˜πΎ))
934atexlemk 39431 . . 3 (πœ‘ β†’ 𝐾 ∈ HL)
10 4thatlem0.j . . . 4 ∨ = (joinβ€˜πΎ)
11 4thatlem0.m . . . 4 ∧ = (meetβ€˜πΎ)
12 4thatlem0.h . . . 4 𝐻 = (LHypβ€˜πΎ)
13 4thatlem0.u . . . 4 π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
143, 2, 10, 11, 6, 12, 134atexlemu 39448 . . 3 (πœ‘ β†’ π‘ˆ ∈ 𝐴)
15 4thatlem0.v . . . 4 𝑉 = ((𝑃 ∨ 𝑆) ∧ π‘Š)
163, 2, 10, 11, 6, 12, 13, 154atexlemv 39449 . . 3 (πœ‘ β†’ 𝑉 ∈ 𝐴)
171, 10, 6hlatjcl 38750 . . 3 ((𝐾 ∈ HL ∧ π‘ˆ ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) β†’ (π‘ˆ ∨ 𝑉) ∈ (Baseβ€˜πΎ))
189, 14, 16, 17syl3anc 1368 . 2 (πœ‘ β†’ (π‘ˆ ∨ 𝑉) ∈ (Baseβ€˜πΎ))
193, 124atexlemwb 39443 . 2 (πœ‘ β†’ π‘Š ∈ (Baseβ€˜πΎ))
2034atexlemkc 39442 . . 3 (πœ‘ β†’ 𝐾 ∈ CvLat)
213, 2, 10, 11, 6, 12, 13, 154atexlemunv 39450 . . 3 (πœ‘ β†’ π‘ˆ β‰  𝑉)
2234atexlemutvt 39438 . . 3 (πœ‘ β†’ (π‘ˆ ∨ 𝑇) = (𝑉 ∨ 𝑇))
236, 2, 10cvlsupr4 38728 . . 3 ((𝐾 ∈ CvLat ∧ (π‘ˆ ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (π‘ˆ β‰  𝑉 ∧ (π‘ˆ ∨ 𝑇) = (𝑉 ∨ 𝑇))) β†’ 𝑇 ≀ (π‘ˆ ∨ 𝑉))
2420, 14, 16, 5, 21, 22, 23syl132anc 1385 . 2 (πœ‘ β†’ 𝑇 ≀ (π‘ˆ ∨ 𝑉))
2534atexlemp 39434 . . . . . 6 (πœ‘ β†’ 𝑃 ∈ 𝐴)
2634atexlemq 39435 . . . . . 6 (πœ‘ β†’ 𝑄 ∈ 𝐴)
271, 10, 6hlatjcl 38750 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
289, 25, 26, 27syl3anc 1368 . . . . 5 (πœ‘ β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
291, 2, 11latmle2 18430 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ ((𝑃 ∨ 𝑄) ∧ π‘Š) ≀ π‘Š)
304, 28, 19, 29syl3anc 1368 . . . 4 (πœ‘ β†’ ((𝑃 ∨ 𝑄) ∧ π‘Š) ≀ π‘Š)
3113, 30eqbrtrid 5176 . . 3 (πœ‘ β†’ π‘ˆ ≀ π‘Š)
323, 10, 64atexlempsb 39444 . . . . 5 (πœ‘ β†’ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
331, 2, 11latmle2 18430 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ ((𝑃 ∨ 𝑆) ∧ π‘Š) ≀ π‘Š)
344, 32, 19, 33syl3anc 1368 . . . 4 (πœ‘ β†’ ((𝑃 ∨ 𝑆) ∧ π‘Š) ≀ π‘Š)
3515, 34eqbrtrid 5176 . . 3 (πœ‘ β†’ 𝑉 ≀ π‘Š)
361, 6atbase 38672 . . . . 5 (π‘ˆ ∈ 𝐴 β†’ π‘ˆ ∈ (Baseβ€˜πΎ))
3714, 36syl 17 . . . 4 (πœ‘ β†’ π‘ˆ ∈ (Baseβ€˜πΎ))
381, 6atbase 38672 . . . . 5 (𝑉 ∈ 𝐴 β†’ 𝑉 ∈ (Baseβ€˜πΎ))
3916, 38syl 17 . . . 4 (πœ‘ β†’ 𝑉 ∈ (Baseβ€˜πΎ))
401, 2, 10latjle12 18415 . . . 4 ((𝐾 ∈ Lat ∧ (π‘ˆ ∈ (Baseβ€˜πΎ) ∧ 𝑉 ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ))) β†’ ((π‘ˆ ≀ π‘Š ∧ 𝑉 ≀ π‘Š) ↔ (π‘ˆ ∨ 𝑉) ≀ π‘Š))
414, 37, 39, 19, 40syl13anc 1369 . . 3 (πœ‘ β†’ ((π‘ˆ ≀ π‘Š ∧ 𝑉 ≀ π‘Š) ↔ (π‘ˆ ∨ 𝑉) ≀ π‘Š))
4231, 35, 41mpbi2and 709 . 2 (πœ‘ β†’ (π‘ˆ ∨ 𝑉) ≀ π‘Š)
431, 2, 4, 8, 18, 19, 24, 42lattrd 18411 1 (πœ‘ β†’ 𝑇 ≀ π‘Š)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  lecple 17213  joincjn 18276  meetcmee 18277  Latclat 18396  Atomscatm 38646  CvLatclc 38648  HLchlt 38733  LHypclh 39368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-p1 18391  df-lat 18397  df-clat 18464  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734  df-lhyp 39372
This theorem is referenced by:  4atexlemntlpq  39452  4atexlemnclw  39454  4atexlemcnd  39456
  Copyright terms: Public domain W3C validator