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

Theorem 4atexlemcnd 39247
Description: Lemma for 4atexlem7 39250. (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))
4thatlem0.d 𝐷 = ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
Assertion
Ref Expression
4atexlemcnd (πœ‘ β†’ 𝐢 β‰  𝐷)

Proof of Theorem 4atexlemcnd
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)
91, 2, 3, 4, 5, 6, 7, 84atexlemtlw 39242 . . 3 (πœ‘ β†’ 𝑇 ≀ π‘Š)
10 4thatlem0.c . . . 4 𝐢 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
111, 2, 3, 4, 5, 6, 7, 8, 104atexlemnclw 39245 . . 3 (πœ‘ β†’ Β¬ 𝐢 ≀ π‘Š)
12 nbrne2 5168 . . 3 ((𝑇 ≀ π‘Š ∧ Β¬ 𝐢 ≀ π‘Š) β†’ 𝑇 β‰  𝐢)
139, 11, 12syl2anc 583 . 2 (πœ‘ β†’ 𝑇 β‰  𝐢)
1414atexlemk 39222 . . . . . . . . 9 (πœ‘ β†’ 𝐾 ∈ HL)
1514atexlemq 39226 . . . . . . . . 9 (πœ‘ β†’ 𝑄 ∈ 𝐴)
1614atexlemt 39228 . . . . . . . . 9 (πœ‘ β†’ 𝑇 ∈ 𝐴)
173, 5hlatjcom 38542 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) β†’ (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
1814, 15, 16, 17syl3anc 1370 . . . . . . . 8 (πœ‘ β†’ (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
19 simp221 1313 . . . . . . . . . 10 ((((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†’ 𝑅 ∈ 𝐴)
201, 19sylbi 216 . . . . . . . . 9 (πœ‘ β†’ 𝑅 ∈ 𝐴)
213, 5hlatjcom 38542 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) β†’ (𝑅 ∨ 𝑇) = (𝑇 ∨ 𝑅))
2214, 20, 16, 21syl3anc 1370 . . . . . . . 8 (πœ‘ β†’ (𝑅 ∨ 𝑇) = (𝑇 ∨ 𝑅))
2318, 22oveq12d 7430 . . . . . . 7 (πœ‘ β†’ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) = ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅)))
2414atexlemkc 39233 . . . . . . . . 9 (πœ‘ β†’ 𝐾 ∈ CvLat)
2514atexlemp 39225 . . . . . . . . 9 (πœ‘ β†’ 𝑃 ∈ 𝐴)
2614atexlempnq 39230 . . . . . . . . 9 (πœ‘ β†’ 𝑃 β‰  𝑄)
27 simp223 1315 . . . . . . . . . 10 ((((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†’ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
281, 27sylbi 216 . . . . . . . . 9 (πœ‘ β†’ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
295, 3cvlsupr6 38521 . . . . . . . . . 10 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) β†’ 𝑅 β‰  𝑄)
3029necomd 2995 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) β†’ 𝑄 β‰  𝑅)
3124, 25, 15, 20, 26, 28, 30syl132anc 1387 . . . . . . . 8 (πœ‘ β†’ 𝑄 β‰  𝑅)
321, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 39243 . . . . . . . . 9 (πœ‘ β†’ Β¬ 𝑇 ≀ (𝑃 ∨ 𝑄))
335, 3cvlsupr7 38522 . . . . . . . . . . . 12 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) β†’ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄))
3424, 25, 15, 20, 26, 28, 33syl132anc 1387 . . . . . . . . . . 11 (πœ‘ β†’ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄))
353, 5hlatjcom 38542 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) β†’ (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
3614, 15, 20, 35syl3anc 1370 . . . . . . . . . . 11 (πœ‘ β†’ (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
3734, 36eqtr4d 2774 . . . . . . . . . 10 (πœ‘ β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑅))
3837breq2d 5160 . . . . . . . . 9 (πœ‘ β†’ (𝑇 ≀ (𝑃 ∨ 𝑄) ↔ 𝑇 ≀ (𝑄 ∨ 𝑅)))
3932, 38mtbid 324 . . . . . . . 8 (πœ‘ β†’ Β¬ 𝑇 ≀ (𝑄 ∨ 𝑅))
402, 3, 4, 52llnma2 38964 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ Β¬ 𝑇 ≀ (𝑄 ∨ 𝑅))) β†’ ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅)) = 𝑇)
4114, 15, 20, 16, 31, 39, 40syl132anc 1387 . . . . . . 7 (πœ‘ β†’ ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅)) = 𝑇)
4223, 41eqtr2d 2772 . . . . . 6 (πœ‘ β†’ 𝑇 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
4342adantr 480 . . . . 5 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ 𝑇 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
4414atexlemkl 39232 . . . . . . . . . 10 (πœ‘ β†’ 𝐾 ∈ Lat)
451, 3, 54atexlemqtb 39236 . . . . . . . . . 10 (πœ‘ β†’ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
461, 3, 54atexlempsb 39235 . . . . . . . . . 10 (πœ‘ β†’ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
47 eqid 2731 . . . . . . . . . . 11 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
4847, 2, 4latmle1 18422 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ)) β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑄 ∨ 𝑇))
4944, 45, 46, 48syl3anc 1370 . . . . . . . . 9 (πœ‘ β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑄 ∨ 𝑇))
5010, 49eqbrtrid 5183 . . . . . . . 8 (πœ‘ β†’ 𝐢 ≀ (𝑄 ∨ 𝑇))
5150adantr 480 . . . . . . 7 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ 𝐢 ≀ (𝑄 ∨ 𝑇))
52 simpr 484 . . . . . . . 8 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ 𝐢 = 𝐷)
53 4thatlem0.d . . . . . . . . . 10 𝐷 = ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
5447, 3, 5hlatjcl 38541 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) β†’ (𝑅 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
5514, 20, 16, 54syl3anc 1370 . . . . . . . . . . 11 (πœ‘ β†’ (𝑅 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
5647, 2, 4latmle1 18422 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑅 ∨ 𝑇) ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ)) β†’ ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑅 ∨ 𝑇))
5744, 55, 46, 56syl3anc 1370 . . . . . . . . . 10 (πœ‘ β†’ ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑅 ∨ 𝑇))
5853, 57eqbrtrid 5183 . . . . . . . . 9 (πœ‘ β†’ 𝐷 ≀ (𝑅 ∨ 𝑇))
5958adantr 480 . . . . . . . 8 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ 𝐷 ≀ (𝑅 ∨ 𝑇))
6052, 59eqbrtrd 5170 . . . . . . 7 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ 𝐢 ≀ (𝑅 ∨ 𝑇))
611, 2, 3, 4, 5, 6, 7, 8, 104atexlemc 39244 . . . . . . . . . 10 (πœ‘ β†’ 𝐢 ∈ 𝐴)
6247, 5atbase 38463 . . . . . . . . . 10 (𝐢 ∈ 𝐴 β†’ 𝐢 ∈ (Baseβ€˜πΎ))
6361, 62syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐢 ∈ (Baseβ€˜πΎ))
6447, 2, 4latlem12 18424 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝐢 ∈ (Baseβ€˜πΎ) ∧ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ) ∧ (𝑅 ∨ 𝑇) ∈ (Baseβ€˜πΎ))) β†’ ((𝐢 ≀ (𝑄 ∨ 𝑇) ∧ 𝐢 ≀ (𝑅 ∨ 𝑇)) ↔ 𝐢 ≀ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
6544, 63, 45, 55, 64syl13anc 1371 . . . . . . . 8 (πœ‘ β†’ ((𝐢 ≀ (𝑄 ∨ 𝑇) ∧ 𝐢 ≀ (𝑅 ∨ 𝑇)) ↔ 𝐢 ≀ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
6665adantr 480 . . . . . . 7 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ ((𝐢 ≀ (𝑄 ∨ 𝑇) ∧ 𝐢 ≀ (𝑅 ∨ 𝑇)) ↔ 𝐢 ≀ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
6751, 60, 66mpbi2and 709 . . . . . 6 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ 𝐢 ≀ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
68 hlatl 38534 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
6914, 68syl 17 . . . . . . . 8 (πœ‘ β†’ 𝐾 ∈ AtLat)
7042, 16eqeltrrd 2833 . . . . . . . 8 (πœ‘ β†’ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ∈ 𝐴)
712, 5atcmp 38485 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝐢 ∈ 𝐴 ∧ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ∈ 𝐴) β†’ (𝐢 ≀ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ↔ 𝐢 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
7269, 61, 70, 71syl3anc 1370 . . . . . . 7 (πœ‘ β†’ (𝐢 ≀ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ↔ 𝐢 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
7372adantr 480 . . . . . 6 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ (𝐢 ≀ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ↔ 𝐢 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
7467, 73mpbid 231 . . . . 5 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ 𝐢 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
7543, 74eqtr4d 2774 . . . 4 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ 𝑇 = 𝐢)
7675ex 412 . . 3 (πœ‘ β†’ (𝐢 = 𝐷 β†’ 𝑇 = 𝐢))
7776necon3d 2960 . 2 (πœ‘ β†’ (𝑇 β‰  𝐢 β†’ 𝐢 β‰  𝐷))
7813, 77mpd 15 1 (πœ‘ β†’ 𝐢 β‰  𝐷)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7412  Basecbs 17149  lecple 17209  joincjn 18269  meetcmee 18270  Latclat 18389  Atomscatm 38437  AtLatcal 38438  CvLatclc 38439  HLchlt 38524  LHypclh 39159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  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 7368  df-ov 7415  df-oprab 7416  df-proset 18253  df-poset 18271  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-p1 18384  df-lat 18390  df-clat 18457  df-oposet 38350  df-ol 38352  df-oml 38353  df-covers 38440  df-ats 38441  df-atl 38472  df-cvlat 38496  df-hlat 38525  df-llines 38673  df-lplanes 38674  df-lhyp 39163
This theorem is referenced by:  4atexlemex4  39248
  Copyright terms: Public domain W3C validator