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Theorem cvlsupr2 39344
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 1194 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑃𝑄)
21necomd 2996 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑄𝑃)
3 simplr 769 . . . . . . . . 9 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑃) → (𝑃 𝑅) = (𝑄 𝑅))
4 oveq2 7439 . . . . . . . . . . . 12 (𝑅 = 𝑃 → (𝑃 𝑅) = (𝑃 𝑃))
5 oveq2 7439 . . . . . . . . . . . 12 (𝑅 = 𝑃 → (𝑄 𝑅) = (𝑄 𝑃))
64, 5eqeq12d 2753 . . . . . . . . . . 11 (𝑅 = 𝑃 → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑃 𝑃) = (𝑄 𝑃)))
7 eqcom 2744 . . . . . . . . . . 11 ((𝑃 𝑃) = (𝑄 𝑃) ↔ (𝑄 𝑃) = (𝑃 𝑃))
86, 7bitrdi 287 . . . . . . . . . 10 (𝑅 = 𝑃 → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑄 𝑃) = (𝑃 𝑃)))
98adantl 481 . . . . . . . . 9 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑃) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑄 𝑃) = (𝑃 𝑃)))
103, 9mpbid 232 . . . . . . . 8 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑃) → (𝑄 𝑃) = (𝑃 𝑃))
11 simpl1 1192 . . . . . . . . . . 11 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝐾 ∈ CvLat)
12 cvllat 39327 . . . . . . . . . . 11 (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
1311, 12syl 17 . . . . . . . . . 10 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝐾 ∈ Lat)
14 simpl21 1252 . . . . . . . . . . 11 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑃𝐴)
15 eqid 2737 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘𝐾)
16 cvlsupr2.a . . . . . . . . . . . 12 𝐴 = (Atoms‘𝐾)
1715, 16atbase 39290 . . . . . . . . . . 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 18507 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃 𝑃) = 𝑃)
2113, 18, 20syl2anc 584 . . . . . . . . 9 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑃 𝑃) = 𝑃)
2221adantr 480 . . . . . . . 8 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑃) → (𝑃 𝑃) = 𝑃)
2310, 22eqtrd 2777 . . . . . . 7 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑃) → (𝑄 𝑃) = 𝑃)
2423ex 412 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅 = 𝑃 → (𝑄 𝑃) = 𝑃))
25 simpl22 1253 . . . . . . . . 9 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑄𝐴)
2615, 16atbase 39290 . . . . . . . . 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 18496 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑄 𝑃 ↔ (𝑄 𝑃) = 𝑃))
3013, 27, 18, 29syl3anc 1373 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑄 𝑃 ↔ (𝑄 𝑃) = 𝑃))
31 cvlatl 39326 . . . . . . . . 9 (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)
3211, 31syl 17 . . . . . . . 8 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝐾 ∈ AtLat)
3328, 16atcmp 39312 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑄𝐴𝑃𝐴) → (𝑄 𝑃𝑄 = 𝑃))
3432, 25, 14, 33syl3anc 1373 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑄 𝑃𝑄 = 𝑃))
3530, 34bitr3d 281 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → ((𝑄 𝑃) = 𝑃𝑄 = 𝑃))
3624, 35sylibd 239 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅 = 𝑃𝑄 = 𝑃))
3736necon3d 2961 . . . 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 769 . . . . . . . . 9 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 𝑅) = (𝑄 𝑅))
40 oveq2 7439 . . . . . . . . . . 11 (𝑅 = 𝑄 → (𝑃 𝑅) = (𝑃 𝑄))
41 oveq2 7439 . . . . . . . . . . 11 (𝑅 = 𝑄 → (𝑄 𝑅) = (𝑄 𝑄))
4240, 41eqeq12d 2753 . . . . . . . . . 10 (𝑅 = 𝑄 → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑃 𝑄) = (𝑄 𝑄)))
4342adantl 481 . . . . . . . . 9 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑃 𝑄) = (𝑄 𝑄)))
4439, 43mpbid 232 . . . . . . . 8 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 𝑄) = (𝑄 𝑄))
4515, 19latjidm 18507 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑄 𝑄) = 𝑄)
4613, 27, 45syl2anc 584 . . . . . . . . 9 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑄 𝑄) = 𝑄)
4746adantr 480 . . . . . . . 8 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑄) → (𝑄 𝑄) = 𝑄)
4844, 47eqtrd 2777 . . . . . . 7 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 𝑄) = 𝑄)
4948ex 412 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅 = 𝑄 → (𝑃 𝑄) = 𝑄))
5015, 28, 19latleeqj1 18496 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄 ↔ (𝑃 𝑄) = 𝑄))
5113, 18, 27, 50syl3anc 1373 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑃 𝑄 ↔ (𝑃 𝑄) = 𝑄))
5228, 16atcmp 39312 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄𝑃 = 𝑄))
5332, 14, 25, 52syl3anc 1373 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑃 𝑄𝑃 = 𝑄))
5451, 53bitr3d 281 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → ((𝑃 𝑄) = 𝑄𝑃 = 𝑄))
5549, 54sylibd 239 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅 = 𝑄𝑃 = 𝑄))
5655necon3d 2961 . . . 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 1254 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑅𝐴)
5915, 16atbase 39290 . . . . . . 7 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
6058, 59syl 17 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑅 ∈ (Base‘𝐾))
6115, 28, 19latlej1 18493 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑄 (𝑄 𝑅))
6213, 27, 60, 61syl3anc 1373 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑄 (𝑄 𝑅))
63 simpr 484 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑃 𝑅) = (𝑄 𝑅))
6462, 63breqtrrd 5171 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑄 (𝑃 𝑅))
6528, 19, 16cvlatexch1 39337 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 𝑅) → 𝑅 (𝑃 𝑄)))
6611, 25, 58, 14, 2, 65syl131anc 1385 . . . 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 1129 . 2 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)))
69 simpr3 1197 . . 3 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑅 (𝑃 𝑄))
70 simpl1 1192 . . . . . . 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 1252 . . . . . . 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 1253 . . . . . . 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 18492 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) = (𝑄 𝑃))
7771, 73, 75, 76syl3anc 1373 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑃 𝑄) = (𝑄 𝑃))
7877breq2d 5155 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑅 (𝑃 𝑄) ↔ 𝑅 (𝑄 𝑃)))
79 simpl23 1254 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑅𝐴)
80 simpr2 1196 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑅𝑄)
8128, 19, 16cvlatexch1 39337 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑅𝐴𝑃𝐴𝑄𝐴) ∧ 𝑅𝑄) → (𝑅 (𝑄 𝑃) → 𝑃 (𝑄 𝑅)))
8270, 79, 72, 74, 80, 81syl131anc 1385 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑅 (𝑄 𝑃) → 𝑃 (𝑄 𝑅)))
83 simpr1 1195 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑅𝑃)
8483necomd 2996 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑃𝑅)
8528, 19, 16cvlatexchb2 39336 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑄 𝑅) ↔ (𝑃 𝑅) = (𝑄 𝑅)))
8670, 72, 74, 79, 84, 85syl131anc 1385 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑃 (𝑄 𝑅) ↔ (𝑃 𝑅) = (𝑄 𝑅)))
8782, 86sylibd 239 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑅 (𝑄 𝑃) → (𝑃 𝑅) = (𝑄 𝑅)))
8878, 87sylbid 240 . . 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 801 1 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  Latclat 18476  Atomscatm 39264  AtLatcal 39265  CvLatclc 39266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323
This theorem is referenced by:  cvlsupr3  39345  cvlsupr4  39346  cvlsupr5  39347  cvlsupr6  39348  4atexlemex2  40073  4atex  40078  4atex3  40083  cdleme02N  40224  cdleme0ex2N  40226  cdleme0moN  40227  cdleme0nex  40292
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