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Theorem cvlsupr2 39325
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 1192 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑃𝑄)
21necomd 2994 . . . 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 2751 . . . . . . . . . . 11 (𝑅 = 𝑃 → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑃 𝑃) = (𝑄 𝑃)))
7 eqcom 2742 . . . . . . . . . . 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 1190 . . . . . . . . . . 11 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝐾 ∈ CvLat)
12 cvllat 39308 . . . . . . . . . . 11 (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
1311, 12syl 17 . . . . . . . . . 10 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝐾 ∈ Lat)
14 simpl21 1250 . . . . . . . . . . 11 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑃𝐴)
15 eqid 2735 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘𝐾)
16 cvlsupr2.a . . . . . . . . . . . 12 𝐴 = (Atoms‘𝐾)
1715, 16atbase 39271 . . . . . . . . . . 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 18520 . . . . . . . . . 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 2775 . . . . . . 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 1251 . . . . . . . . 9 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑄𝐴)
2615, 16atbase 39271 . . . . . . . . 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 18509 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑄 𝑃 ↔ (𝑄 𝑃) = 𝑃))
3013, 27, 18, 29syl3anc 1370 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑄 𝑃 ↔ (𝑄 𝑃) = 𝑃))
31 cvlatl 39307 . . . . . . . . 9 (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)
3211, 31syl 17 . . . . . . . 8 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝐾 ∈ AtLat)
3328, 16atcmp 39293 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑄𝐴𝑃𝐴) → (𝑄 𝑃𝑄 = 𝑃))
3432, 25, 14, 33syl3anc 1370 . . . . . . 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 2959 . . . 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 2751 . . . . . . . . . 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 18520 . . . . . . . . . 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 2775 . . . . . . 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 18509 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄 ↔ (𝑃 𝑄) = 𝑄))
5113, 18, 27, 50syl3anc 1370 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑃 𝑄 ↔ (𝑃 𝑄) = 𝑄))
5228, 16atcmp 39293 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄𝑃 = 𝑄))
5332, 14, 25, 52syl3anc 1370 . . . . . . 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 2959 . . . 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 1252 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑅𝐴)
5915, 16atbase 39271 . . . . . . 7 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
6058, 59syl 17 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑅 ∈ (Base‘𝐾))
6115, 28, 19latlej1 18506 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑄 (𝑄 𝑅))
6213, 27, 60, 61syl3anc 1370 . . . . 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 5176 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑄 (𝑃 𝑅))
6528, 19, 16cvlatexch1 39318 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 𝑅) → 𝑅 (𝑃 𝑄)))
6611, 25, 58, 14, 2, 65syl131anc 1382 . . . 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 1127 . 2 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)))
69 simpr3 1195 . . 3 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑅 (𝑃 𝑄))
70 simpl1 1190 . . . . . . 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 1250 . . . . . . 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 1251 . . . . . . 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 18505 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) = (𝑄 𝑃))
7771, 73, 75, 76syl3anc 1370 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑃 𝑄) = (𝑄 𝑃))
7877breq2d 5160 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑅 (𝑃 𝑄) ↔ 𝑅 (𝑄 𝑃)))
79 simpl23 1252 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑅𝐴)
80 simpr2 1194 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑅𝑄)
8128, 19, 16cvlatexch1 39318 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑅𝐴𝑃𝐴𝑄𝐴) ∧ 𝑅𝑄) → (𝑅 (𝑄 𝑃) → 𝑃 (𝑄 𝑅)))
8270, 79, 72, 74, 80, 81syl131anc 1382 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑅 (𝑄 𝑃) → 𝑃 (𝑄 𝑅)))
83 simpr1 1193 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑅𝑃)
8483necomd 2994 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑃𝑅)
8528, 19, 16cvlatexchb2 39317 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑄 𝑅) ↔ (𝑃 𝑅) = (𝑄 𝑅)))
8670, 72, 74, 79, 84, 85syl131anc 1382 . . . . 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 1086   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  Latclat 18489  Atomscatm 39245  AtLatcal 39246  CvLatclc 39247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304
This theorem is referenced by:  cvlsupr3  39326  cvlsupr4  39327  cvlsupr5  39328  cvlsupr6  39329  4atexlemex2  40054  4atex  40059  4atex3  40064  cdleme02N  40205  cdleme0ex2N  40207  cdleme0moN  40208  cdleme0nex  40273
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