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Theorem 4atexlemcnd 38943
Description: Lemma for 4atexlem7 38946. (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 38938 . . 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 38941 . . 3 (πœ‘ β†’ Β¬ 𝐢 ≀ π‘Š)
12 nbrne2 5169 . . 3 ((𝑇 ≀ π‘Š ∧ Β¬ 𝐢 ≀ π‘Š) β†’ 𝑇 β‰  𝐢)
139, 11, 12syl2anc 585 . 2 (πœ‘ β†’ 𝑇 β‰  𝐢)
1414atexlemk 38918 . . . . . . . . 9 (πœ‘ β†’ 𝐾 ∈ HL)
1514atexlemq 38922 . . . . . . . . 9 (πœ‘ β†’ 𝑄 ∈ 𝐴)
1614atexlemt 38924 . . . . . . . . 9 (πœ‘ β†’ 𝑇 ∈ 𝐴)
173, 5hlatjcom 38238 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) β†’ (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
1814, 15, 16, 17syl3anc 1372 . . . . . . . 8 (πœ‘ β†’ (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
19 simp221 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†’ 𝑅 ∈ 𝐴)
201, 19sylbi 216 . . . . . . . . 9 (πœ‘ β†’ 𝑅 ∈ 𝐴)
213, 5hlatjcom 38238 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) β†’ (𝑅 ∨ 𝑇) = (𝑇 ∨ 𝑅))
2214, 20, 16, 21syl3anc 1372 . . . . . . . 8 (πœ‘ β†’ (𝑅 ∨ 𝑇) = (𝑇 ∨ 𝑅))
2318, 22oveq12d 7427 . . . . . . 7 (πœ‘ β†’ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) = ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅)))
2414atexlemkc 38929 . . . . . . . . 9 (πœ‘ β†’ 𝐾 ∈ CvLat)
2514atexlemp 38921 . . . . . . . . 9 (πœ‘ β†’ 𝑃 ∈ 𝐴)
2614atexlempnq 38926 . . . . . . . . 9 (πœ‘ β†’ 𝑃 β‰  𝑄)
27 simp223 1317 . . . . . . . . . 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 38217 . . . . . . . . . 10 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) β†’ 𝑅 β‰  𝑄)
3029necomd 2997 . . . . . . . . 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 1389 . . . . . . . 8 (πœ‘ β†’ 𝑄 β‰  𝑅)
321, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 38939 . . . . . . . . 9 (πœ‘ β†’ Β¬ 𝑇 ≀ (𝑃 ∨ 𝑄))
335, 3cvlsupr7 38218 . . . . . . . . . . . 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 1389 . . . . . . . . . . 11 (πœ‘ β†’ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄))
353, 5hlatjcom 38238 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) β†’ (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
3614, 15, 20, 35syl3anc 1372 . . . . . . . . . . 11 (πœ‘ β†’ (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
3734, 36eqtr4d 2776 . . . . . . . . . 10 (πœ‘ β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑅))
3837breq2d 5161 . . . . . . . . 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 38660 . . . . . . . 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 1389 . . . . . . 7 (πœ‘ β†’ ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅)) = 𝑇)
4223, 41eqtr2d 2774 . . . . . 6 (πœ‘ β†’ 𝑇 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
4342adantr 482 . . . . 5 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ 𝑇 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
4414atexlemkl 38928 . . . . . . . . . 10 (πœ‘ β†’ 𝐾 ∈ Lat)
451, 3, 54atexlemqtb 38932 . . . . . . . . . 10 (πœ‘ β†’ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
461, 3, 54atexlempsb 38931 . . . . . . . . . 10 (πœ‘ β†’ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
47 eqid 2733 . . . . . . . . . . 11 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
4847, 2, 4latmle1 18417 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ)) β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑄 ∨ 𝑇))
4944, 45, 46, 48syl3anc 1372 . . . . . . . . 9 (πœ‘ β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑄 ∨ 𝑇))
5010, 49eqbrtrid 5184 . . . . . . . 8 (πœ‘ β†’ 𝐢 ≀ (𝑄 ∨ 𝑇))
5150adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ 𝐢 ≀ (𝑄 ∨ 𝑇))
52 simpr 486 . . . . . . . 8 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ 𝐢 = 𝐷)
53 4thatlem0.d . . . . . . . . . 10 𝐷 = ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
5447, 3, 5hlatjcl 38237 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) β†’ (𝑅 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
5514, 20, 16, 54syl3anc 1372 . . . . . . . . . . 11 (πœ‘ β†’ (𝑅 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
5647, 2, 4latmle1 18417 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑅 ∨ 𝑇) ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ)) β†’ ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑅 ∨ 𝑇))
5744, 55, 46, 56syl3anc 1372 . . . . . . . . . 10 (πœ‘ β†’ ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≀ (𝑅 ∨ 𝑇))
5853, 57eqbrtrid 5184 . . . . . . . . 9 (πœ‘ β†’ 𝐷 ≀ (𝑅 ∨ 𝑇))
5958adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ 𝐷 ≀ (𝑅 ∨ 𝑇))
6052, 59eqbrtrd 5171 . . . . . . 7 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ 𝐢 ≀ (𝑅 ∨ 𝑇))
611, 2, 3, 4, 5, 6, 7, 8, 104atexlemc 38940 . . . . . . . . . 10 (πœ‘ β†’ 𝐢 ∈ 𝐴)
6247, 5atbase 38159 . . . . . . . . . 10 (𝐢 ∈ 𝐴 β†’ 𝐢 ∈ (Baseβ€˜πΎ))
6361, 62syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐢 ∈ (Baseβ€˜πΎ))
6447, 2, 4latlem12 18419 . . . . . . . . 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 1373 . . . . . . . 8 (πœ‘ β†’ ((𝐢 ≀ (𝑄 ∨ 𝑇) ∧ 𝐢 ≀ (𝑅 ∨ 𝑇)) ↔ 𝐢 ≀ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
6665adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ ((𝐢 ≀ (𝑄 ∨ 𝑇) ∧ 𝐢 ≀ (𝑅 ∨ 𝑇)) ↔ 𝐢 ≀ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
6751, 60, 66mpbi2and 711 . . . . . 6 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ 𝐢 ≀ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
68 hlatl 38230 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
6914, 68syl 17 . . . . . . . 8 (πœ‘ β†’ 𝐾 ∈ AtLat)
7042, 16eqeltrrd 2835 . . . . . . . 8 (πœ‘ β†’ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ∈ 𝐴)
712, 5atcmp 38181 . . . . . . . 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 1372 . . . . . . 7 (πœ‘ β†’ (𝐢 ≀ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ↔ 𝐢 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
7372adantr 482 . . . . . 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 2776 . . . 4 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ 𝑇 = 𝐢)
7675ex 414 . . 3 (πœ‘ β†’ (𝐢 = 𝐷 β†’ 𝑇 = 𝐢))
7776necon3d 2962 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204  joincjn 18264  meetcmee 18265  Latclat 18384  Atomscatm 38133  AtLatcal 38134  CvLatclc 38135  HLchlt 38220  LHypclh 38855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18385  df-clat 18452  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-llines 38369  df-lplanes 38370  df-lhyp 38859
This theorem is referenced by:  4atexlemex4  38944
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