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Theorem cvlsupr2 38201
Description: Two equivalent ways of expressing that 𝑅 is a superposition of 𝑃 and 𝑄. (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
cvlsupr2.a 𝐴 = (Atomsβ€˜πΎ)
cvlsupr2.l ≀ = (leβ€˜πΎ)
cvlsupr2.j ∨ = (joinβ€˜πΎ)
Assertion
Ref Expression
cvlsupr2 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))))
Proof of Theorem cvlsupr2
StepHypRef Expression
1 simpl3 1193 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ 𝑃 β‰  𝑄)
21necomd 2996 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ 𝑄 β‰  𝑃)
3 simplr 767 . . . . . . . . 9 ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) β†’ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
4 oveq2 7413 . . . . . . . . . . . 12 (𝑅 = 𝑃 β†’ (𝑃 ∨ 𝑅) = (𝑃 ∨ 𝑃))
5 oveq2 7413 . . . . . . . . . . . 12 (𝑅 = 𝑃 β†’ (𝑄 ∨ 𝑅) = (𝑄 ∨ 𝑃))
64, 5eqeq12d 2748 . . . . . . . . . . 11 (𝑅 = 𝑃 β†’ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑃) = (𝑄 ∨ 𝑃)))
7 eqcom 2739 . . . . . . . . . . 11 ((𝑃 ∨ 𝑃) = (𝑄 ∨ 𝑃) ↔ (𝑄 ∨ 𝑃) = (𝑃 ∨ 𝑃))
86, 7bitrdi 286 . . . . . . . . . 10 (𝑅 = 𝑃 β†’ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑄 ∨ 𝑃) = (𝑃 ∨ 𝑃)))
98adantl 482 . . . . . . . . 9 ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) β†’ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑄 ∨ 𝑃) = (𝑃 ∨ 𝑃)))
103, 9mpbid 231 . . . . . . . 8 ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) β†’ (𝑄 ∨ 𝑃) = (𝑃 ∨ 𝑃))
11 simpl1 1191 . . . . . . . . . . 11 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ 𝐾 ∈ CvLat)
12 cvllat 38184 . . . . . . . . . . 11 (𝐾 ∈ CvLat β†’ 𝐾 ∈ Lat)
1311, 12syl 17 . . . . . . . . . 10 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ 𝐾 ∈ Lat)
14 simpl21 1251 . . . . . . . . . . 11 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ 𝑃 ∈ 𝐴)
15 eqid 2732 . . . . . . . . . . . 12 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
16 cvlsupr2.a . . . . . . . . . . . 12 𝐴 = (Atomsβ€˜πΎ)
1715, 16atbase 38147 . . . . . . . . . . 11 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
1814, 17syl 17 . . . . . . . . . 10 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
19 cvlsupr2.j . . . . . . . . . . 11 ∨ = (joinβ€˜πΎ)
2015, 19latjidm 18411 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ)) β†’ (𝑃 ∨ 𝑃) = 𝑃)
2113, 18, 20syl2anc 584 . . . . . . . . 9 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ (𝑃 ∨ 𝑃) = 𝑃)
2221adantr 481 . . . . . . . 8 ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) β†’ (𝑃 ∨ 𝑃) = 𝑃)
2310, 22eqtrd 2772 . . . . . . 7 ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) β†’ (𝑄 ∨ 𝑃) = 𝑃)
2423ex 413 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ (𝑅 = 𝑃 β†’ (𝑄 ∨ 𝑃) = 𝑃))
25 simpl22 1252 . . . . . . . . 9 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ 𝑄 ∈ 𝐴)
2615, 16atbase 38147 . . . . . . . . 9 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
2725, 26syl 17 . . . . . . . 8 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
28 cvlsupr2.l . . . . . . . . 9 ≀ = (leβ€˜πΎ)
2915, 28, 19latleeqj1 18400 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Baseβ€˜πΎ) ∧ 𝑃 ∈ (Baseβ€˜πΎ)) β†’ (𝑄 ≀ 𝑃 ↔ (𝑄 ∨ 𝑃) = 𝑃))
3013, 27, 18, 29syl3anc 1371 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ (𝑄 ≀ 𝑃 ↔ (𝑄 ∨ 𝑃) = 𝑃))
31 cvlatl 38183 . . . . . . . . 9 (𝐾 ∈ CvLat β†’ 𝐾 ∈ AtLat)
3211, 31syl 17 . . . . . . . 8 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ 𝐾 ∈ AtLat)
3328, 16atcmp 38169 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) β†’ (𝑄 ≀ 𝑃 ↔ 𝑄 = 𝑃))
3432, 25, 14, 33syl3anc 1371 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ (𝑄 ≀ 𝑃 ↔ 𝑄 = 𝑃))
3530, 34bitr3d 280 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ ((𝑄 ∨ 𝑃) = 𝑃 ↔ 𝑄 = 𝑃))
3624, 35sylibd 238 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ (𝑅 = 𝑃 β†’ 𝑄 = 𝑃))
3736necon3d 2961 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ (𝑄 β‰  𝑃 β†’ 𝑅 β‰  𝑃))
382, 37mpd 15 . . 3 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ 𝑅 β‰  𝑃)
39 simplr 767 . . . . . . . . 9 ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) β†’ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
40 oveq2 7413 . . . . . . . . . . 11 (𝑅 = 𝑄 β†’ (𝑃 ∨ 𝑅) = (𝑃 ∨ 𝑄))
41 oveq2 7413 . . . . . . . . . . 11 (𝑅 = 𝑄 β†’ (𝑄 ∨ 𝑅) = (𝑄 ∨ 𝑄))
4240, 41eqeq12d 2748 . . . . . . . . . 10 (𝑅 = 𝑄 β†’ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄)))
4342adantl 482 . . . . . . . . 9 ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) β†’ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄)))
4439, 43mpbid 231 . . . . . . . 8 ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄))
4515, 19latjidm 18411 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ (𝑄 ∨ 𝑄) = 𝑄)
4613, 27, 45syl2anc 584 . . . . . . . . 9 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ (𝑄 ∨ 𝑄) = 𝑄)
4746adantr 481 . . . . . . . 8 ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) β†’ (𝑄 ∨ 𝑄) = 𝑄)
4844, 47eqtrd 2772 . . . . . . 7 ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) β†’ (𝑃 ∨ 𝑄) = 𝑄)
4948ex 413 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ (𝑅 = 𝑄 β†’ (𝑃 ∨ 𝑄) = 𝑄))
5015, 28, 19latleeqj1 18400 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ (𝑃 ≀ 𝑄 ↔ (𝑃 ∨ 𝑄) = 𝑄))
5113, 18, 27, 50syl3anc 1371 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ (𝑃 ≀ 𝑄 ↔ (𝑃 ∨ 𝑄) = 𝑄))
5228, 16atcmp 38169 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 ≀ 𝑄 ↔ 𝑃 = 𝑄))
5332, 14, 25, 52syl3anc 1371 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ (𝑃 ≀ 𝑄 ↔ 𝑃 = 𝑄))
5451, 53bitr3d 280 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ ((𝑃 ∨ 𝑄) = 𝑄 ↔ 𝑃 = 𝑄))
5549, 54sylibd 238 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ (𝑅 = 𝑄 β†’ 𝑃 = 𝑄))
5655necon3d 2961 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ (𝑃 β‰  𝑄 β†’ 𝑅 β‰  𝑄))
571, 56mpd 15 . . 3 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ 𝑅 β‰  𝑄)
58 simpl23 1253 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ 𝑅 ∈ 𝐴)
5915, 16atbase 38147 . . . . . . 7 (𝑅 ∈ 𝐴 β†’ 𝑅 ∈ (Baseβ€˜πΎ))
6058, 59syl 17 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ 𝑅 ∈ (Baseβ€˜πΎ))
6115, 28, 19latlej1 18397 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Baseβ€˜πΎ) ∧ 𝑅 ∈ (Baseβ€˜πΎ)) β†’ 𝑄 ≀ (𝑄 ∨ 𝑅))
6213, 27, 60, 61syl3anc 1371 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ 𝑄 ≀ (𝑄 ∨ 𝑅))
63 simpr 485 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
6462, 63breqtrrd 5175 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ 𝑄 ≀ (𝑃 ∨ 𝑅))
6528, 19, 16cvlatexch1 38194 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑄 β‰  𝑃) β†’ (𝑄 ≀ (𝑃 ∨ 𝑅) β†’ 𝑅 ≀ (𝑃 ∨ 𝑄)))
6611, 25, 58, 14, 2, 65syl131anc 1383 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ (𝑄 ≀ (𝑃 ∨ 𝑅) β†’ 𝑅 ≀ (𝑃 ∨ 𝑄)))
6764, 66mpd 15 . . 3 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ 𝑅 ≀ (𝑃 ∨ 𝑄))
6838, 57, 673jca 1128 . 2 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) β†’ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄)))
69 simpr3 1196 . . 3 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑅 ≀ (𝑃 ∨ 𝑄))
70 simpl1 1191 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐾 ∈ CvLat)
7170, 12syl 17 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐾 ∈ Lat)
72 simpl21 1251 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑃 ∈ 𝐴)
7372, 17syl 17 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
74 simpl22 1252 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑄 ∈ 𝐴)
7574, 26syl 17 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
7615, 19latjcom 18396 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
7771, 73, 75, 76syl3anc 1371 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
7877breq2d 5159 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑅 ≀ (𝑃 ∨ 𝑄) ↔ 𝑅 ≀ (𝑄 ∨ 𝑃)))
79 simpl23 1253 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑅 ∈ 𝐴)
80 simpr2 1195 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑅 β‰  𝑄)
8128, 19, 16cvlatexch1 38194 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑅 β‰  𝑄) β†’ (𝑅 ≀ (𝑄 ∨ 𝑃) β†’ 𝑃 ≀ (𝑄 ∨ 𝑅)))
8270, 79, 72, 74, 80, 81syl131anc 1383 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑅 ≀ (𝑄 ∨ 𝑃) β†’ 𝑃 ≀ (𝑄 ∨ 𝑅)))
83 simpr1 1194 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑅 β‰  𝑃)
8483necomd 2996 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑃 β‰  𝑅)
8528, 19, 16cvlatexchb2 38193 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑅) β†’ (𝑃 ≀ (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
8670, 72, 74, 79, 84, 85syl131anc 1383 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑃 ≀ (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
8782, 86sylibd 238 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑅 ≀ (𝑄 ∨ 𝑃) β†’ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
8878, 87sylbid 239 . . 3 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑅 ≀ (𝑃 ∨ 𝑄) β†’ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
8969, 88mpd 15 . 2 (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
9068, 89impbida 799 1 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  joincjn 18260  Latclat 18380  Atomscatm 38121  AtLatcal 38122  CvLatclc 38123
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-lat 18381  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180
This theorem is referenced by:  cvlsupr3  38202  cvlsupr4  38203  cvlsupr5  38204  cvlsupr6  38205  4atexlemex2  38930  4atex  38935  4atex3  38940  cdleme02N  39081  cdleme0ex2N  39083  cdleme0moN  39084  cdleme0nex  39149
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