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Theorem cvlsupr2 35299
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 = (Atoms‘𝐾)
cvlsupr2.l = (le‘𝐾)
cvlsupr2.j = (join‘𝐾)
Assertion
Ref Expression
cvlsupr2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))

Proof of Theorem cvlsupr2
StepHypRef Expression
1 simpl3 1246 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑃𝑄)
21necomd 2992 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑄𝑃)
3 simplr 785 . . . . . . . . 9 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑃) → (𝑃 𝑅) = (𝑄 𝑅))
4 oveq2 6850 . . . . . . . . . . . 12 (𝑅 = 𝑃 → (𝑃 𝑅) = (𝑃 𝑃))
5 oveq2 6850 . . . . . . . . . . . 12 (𝑅 = 𝑃 → (𝑄 𝑅) = (𝑄 𝑃))
64, 5eqeq12d 2780 . . . . . . . . . . 11 (𝑅 = 𝑃 → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑃 𝑃) = (𝑄 𝑃)))
7 eqcom 2772 . . . . . . . . . . 11 ((𝑃 𝑃) = (𝑄 𝑃) ↔ (𝑄 𝑃) = (𝑃 𝑃))
86, 7syl6bb 278 . . . . . . . . . 10 (𝑅 = 𝑃 → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑄 𝑃) = (𝑃 𝑃)))
98adantl 473 . . . . . . . . 9 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑃) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑄 𝑃) = (𝑃 𝑃)))
103, 9mpbid 223 . . . . . . . 8 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑃) → (𝑄 𝑃) = (𝑃 𝑃))
11 simpl1 1242 . . . . . . . . . . 11 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝐾 ∈ CvLat)
12 cvllat 35282 . . . . . . . . . . 11 (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
1311, 12syl 17 . . . . . . . . . 10 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝐾 ∈ Lat)
14 simpl21 1335 . . . . . . . . . . 11 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑃𝐴)
15 eqid 2765 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘𝐾)
16 cvlsupr2.a . . . . . . . . . . . 12 𝐴 = (Atoms‘𝐾)
1715, 16atbase 35245 . . . . . . . . . . 11 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1814, 17syl 17 . . . . . . . . . 10 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑃 ∈ (Base‘𝐾))
19 cvlsupr2.j . . . . . . . . . . 11 = (join‘𝐾)
2015, 19latjidm 17340 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃 𝑃) = 𝑃)
2113, 18, 20syl2anc 579 . . . . . . . . 9 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑃 𝑃) = 𝑃)
2221adantr 472 . . . . . . . 8 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑃) → (𝑃 𝑃) = 𝑃)
2310, 22eqtrd 2799 . . . . . . 7 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑃) → (𝑄 𝑃) = 𝑃)
2423ex 401 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅 = 𝑃 → (𝑄 𝑃) = 𝑃))
25 simpl22 1337 . . . . . . . . 9 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑄𝐴)
2615, 16atbase 35245 . . . . . . . . 9 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
2725, 26syl 17 . . . . . . . 8 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑄 ∈ (Base‘𝐾))
28 cvlsupr2.l . . . . . . . . 9 = (le‘𝐾)
2915, 28, 19latleeqj1 17329 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑄 𝑃 ↔ (𝑄 𝑃) = 𝑃))
3013, 27, 18, 29syl3anc 1490 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑄 𝑃 ↔ (𝑄 𝑃) = 𝑃))
31 cvlatl 35281 . . . . . . . . 9 (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)
3211, 31syl 17 . . . . . . . 8 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝐾 ∈ AtLat)
3328, 16atcmp 35267 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑄𝐴𝑃𝐴) → (𝑄 𝑃𝑄 = 𝑃))
3432, 25, 14, 33syl3anc 1490 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑄 𝑃𝑄 = 𝑃))
3530, 34bitr3d 272 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → ((𝑄 𝑃) = 𝑃𝑄 = 𝑃))
3624, 35sylibd 230 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅 = 𝑃𝑄 = 𝑃))
3736necon3d 2958 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑄𝑃𝑅𝑃))
382, 37mpd 15 . . 3 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑅𝑃)
39 simplr 785 . . . . . . . . 9 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 𝑅) = (𝑄 𝑅))
40 oveq2 6850 . . . . . . . . . . 11 (𝑅 = 𝑄 → (𝑃 𝑅) = (𝑃 𝑄))
41 oveq2 6850 . . . . . . . . . . 11 (𝑅 = 𝑄 → (𝑄 𝑅) = (𝑄 𝑄))
4240, 41eqeq12d 2780 . . . . . . . . . 10 (𝑅 = 𝑄 → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑃 𝑄) = (𝑄 𝑄)))
4342adantl 473 . . . . . . . . 9 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑃 𝑄) = (𝑄 𝑄)))
4439, 43mpbid 223 . . . . . . . 8 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 𝑄) = (𝑄 𝑄))
4515, 19latjidm 17340 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑄 𝑄) = 𝑄)
4613, 27, 45syl2anc 579 . . . . . . . . 9 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑄 𝑄) = 𝑄)
4746adantr 472 . . . . . . . 8 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑄) → (𝑄 𝑄) = 𝑄)
4844, 47eqtrd 2799 . . . . . . 7 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 𝑄) = 𝑄)
4948ex 401 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅 = 𝑄 → (𝑃 𝑄) = 𝑄))
5015, 28, 19latleeqj1 17329 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄 ↔ (𝑃 𝑄) = 𝑄))
5113, 18, 27, 50syl3anc 1490 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑃 𝑄 ↔ (𝑃 𝑄) = 𝑄))
5228, 16atcmp 35267 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄𝑃 = 𝑄))
5332, 14, 25, 52syl3anc 1490 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑃 𝑄𝑃 = 𝑄))
5451, 53bitr3d 272 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → ((𝑃 𝑄) = 𝑄𝑃 = 𝑄))
5549, 54sylibd 230 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅 = 𝑄𝑃 = 𝑄))
5655necon3d 2958 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑃𝑄𝑅𝑄))
571, 56mpd 15 . . 3 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑅𝑄)
58 simpl23 1339 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑅𝐴)
5915, 16atbase 35245 . . . . . . 7 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
6058, 59syl 17 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑅 ∈ (Base‘𝐾))
6115, 28, 19latlej1 17326 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑄 (𝑄 𝑅))
6213, 27, 60, 61syl3anc 1490 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑄 (𝑄 𝑅))
63 simpr 477 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑃 𝑅) = (𝑄 𝑅))
6462, 63breqtrrd 4837 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑄 (𝑃 𝑅))
6528, 19, 16cvlatexch1 35292 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 𝑅) → 𝑅 (𝑃 𝑄)))
6611, 25, 58, 14, 2, 65syl131anc 1502 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑄 (𝑃 𝑅) → 𝑅 (𝑃 𝑄)))
6764, 66mpd 15 . . 3 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑅 (𝑃 𝑄))
6838, 57, 673jca 1158 . 2 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)))
69 simpr3 1252 . . 3 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑅 (𝑃 𝑄))
70 simpl1 1242 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝐾 ∈ CvLat)
7170, 12syl 17 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝐾 ∈ Lat)
72 simpl21 1335 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑃𝐴)
7372, 17syl 17 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑃 ∈ (Base‘𝐾))
74 simpl22 1337 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑄𝐴)
7574, 26syl 17 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑄 ∈ (Base‘𝐾))
7615, 19latjcom 17325 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) = (𝑄 𝑃))
7771, 73, 75, 76syl3anc 1490 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑃 𝑄) = (𝑄 𝑃))
7877breq2d 4821 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑅 (𝑃 𝑄) ↔ 𝑅 (𝑄 𝑃)))
79 simpl23 1339 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑅𝐴)
80 simpr2 1250 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑅𝑄)
8128, 19, 16cvlatexch1 35292 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑅𝐴𝑃𝐴𝑄𝐴) ∧ 𝑅𝑄) → (𝑅 (𝑄 𝑃) → 𝑃 (𝑄 𝑅)))
8270, 79, 72, 74, 80, 81syl131anc 1502 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑅 (𝑄 𝑃) → 𝑃 (𝑄 𝑅)))
83 simpr1 1248 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑅𝑃)
8483necomd 2992 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑃𝑅)
8528, 19, 16cvlatexchb2 35291 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑄 𝑅) ↔ (𝑃 𝑅) = (𝑄 𝑅)))
8670, 72, 74, 79, 84, 85syl131anc 1502 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑃 (𝑄 𝑅) ↔ (𝑃 𝑅) = (𝑄 𝑅)))
8782, 86sylibd 230 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑅 (𝑄 𝑃) → (𝑃 𝑅) = (𝑄 𝑅)))
8878, 87sylbid 231 . . 3 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑅 (𝑃 𝑄) → (𝑃 𝑅) = (𝑄 𝑅)))
8969, 88mpd 15 . 2 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑃 𝑅) = (𝑄 𝑅))
9068, 89impbida 835 1 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937   class class class wbr 4809  cfv 6068  (class class class)co 6842  Basecbs 16130  lecple 16221  joincjn 17210  Latclat 17311  Atomscatm 35219  AtLatcal 35220  CvLatclc 35221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-proset 17194  df-poset 17212  df-plt 17224  df-lub 17240  df-glb 17241  df-join 17242  df-meet 17243  df-p0 17305  df-lat 17312  df-covers 35222  df-ats 35223  df-atl 35254  df-cvlat 35278
This theorem is referenced by:  cvlsupr3  35300  cvlsupr4  35301  cvlsupr5  35302  cvlsupr6  35303  4atexlemex2  36027  4atex  36032  4atex3  36037  cdleme02N  36178  cdleme0ex2N  36180  cdleme0moN  36181  cdleme0nex  36246
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