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Theorem cvlsupr2 35151
 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 1232 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑃𝑄)
21necomd 2987 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑄𝑃)
3 simplr 809 . . . . . . . . 9 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑃) → (𝑃 𝑅) = (𝑄 𝑅))
4 oveq2 6822 . . . . . . . . . . . 12 (𝑅 = 𝑃 → (𝑃 𝑅) = (𝑃 𝑃))
5 oveq2 6822 . . . . . . . . . . . 12 (𝑅 = 𝑃 → (𝑄 𝑅) = (𝑄 𝑃))
64, 5eqeq12d 2775 . . . . . . . . . . 11 (𝑅 = 𝑃 → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑃 𝑃) = (𝑄 𝑃)))
7 eqcom 2767 . . . . . . . . . . 11 ((𝑃 𝑃) = (𝑄 𝑃) ↔ (𝑄 𝑃) = (𝑃 𝑃))
86, 7syl6bb 276 . . . . . . . . . 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 222 . . . . . . . 8 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑃) → (𝑄 𝑃) = (𝑃 𝑃))
11 simpl1 1228 . . . . . . . . . . 11 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝐾 ∈ CvLat)
12 cvllat 35134 . . . . . . . . . . 11 (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
1311, 12syl 17 . . . . . . . . . 10 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝐾 ∈ Lat)
14 simpl21 1321 . . . . . . . . . . 11 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑃𝐴)
15 eqid 2760 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘𝐾)
16 cvlsupr2.a . . . . . . . . . . . 12 𝐴 = (Atoms‘𝐾)
1715, 16atbase 35097 . . . . . . . . . . 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 17295 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃 𝑃) = 𝑃)
2113, 18, 20syl2anc 696 . . . . . . . . 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 2794 . . . . . . 7 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑃) → (𝑄 𝑃) = 𝑃)
2423ex 449 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅 = 𝑃 → (𝑄 𝑃) = 𝑃))
25 simpl22 1323 . . . . . . . . 9 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑄𝐴)
2615, 16atbase 35097 . . . . . . . . 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 17284 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑄 𝑃 ↔ (𝑄 𝑃) = 𝑃))
3013, 27, 18, 29syl3anc 1477 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑄 𝑃 ↔ (𝑄 𝑃) = 𝑃))
31 cvlatl 35133 . . . . . . . . 9 (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)
3211, 31syl 17 . . . . . . . 8 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝐾 ∈ AtLat)
3328, 16atcmp 35119 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑄𝐴𝑃𝐴) → (𝑄 𝑃𝑄 = 𝑃))
3432, 25, 14, 33syl3anc 1477 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑄 𝑃𝑄 = 𝑃))
3530, 34bitr3d 270 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → ((𝑄 𝑃) = 𝑃𝑄 = 𝑃))
3624, 35sylibd 229 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅 = 𝑃𝑄 = 𝑃))
3736necon3d 2953 . . . 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 809 . . . . . . . . 9 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 𝑅) = (𝑄 𝑅))
40 oveq2 6822 . . . . . . . . . . 11 (𝑅 = 𝑄 → (𝑃 𝑅) = (𝑃 𝑄))
41 oveq2 6822 . . . . . . . . . . 11 (𝑅 = 𝑄 → (𝑄 𝑅) = (𝑄 𝑄))
4240, 41eqeq12d 2775 . . . . . . . . . 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 222 . . . . . . . 8 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 𝑄) = (𝑄 𝑄))
4515, 19latjidm 17295 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑄 𝑄) = 𝑄)
4613, 27, 45syl2anc 696 . . . . . . . . 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 2794 . . . . . . 7 ((((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 𝑄) = 𝑄)
4948ex 449 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅 = 𝑄 → (𝑃 𝑄) = 𝑄))
5015, 28, 19latleeqj1 17284 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄 ↔ (𝑃 𝑄) = 𝑄))
5113, 18, 27, 50syl3anc 1477 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑃 𝑄 ↔ (𝑃 𝑄) = 𝑄))
5228, 16atcmp 35119 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄𝑃 = 𝑄))
5332, 14, 25, 52syl3anc 1477 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑃 𝑄𝑃 = 𝑄))
5451, 53bitr3d 270 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → ((𝑃 𝑄) = 𝑄𝑃 = 𝑄))
5549, 54sylibd 229 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅 = 𝑄𝑃 = 𝑄))
5655necon3d 2953 . . . 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 1325 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑅𝐴)
5915, 16atbase 35097 . . . . . . 7 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
6058, 59syl 17 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑅 ∈ (Base‘𝐾))
6115, 28, 19latlej1 17281 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑄 (𝑄 𝑅))
6213, 27, 60, 61syl3anc 1477 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑄 (𝑄 𝑅))
63 simpr 479 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑃 𝑅) = (𝑄 𝑅))
6462, 63breqtrrd 4832 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → 𝑄 (𝑃 𝑅))
6528, 19, 16cvlatexch1 35144 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 𝑅) → 𝑅 (𝑃 𝑄)))
6611, 25, 58, 14, 2, 65syl131anc 1490 . . . 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 1123 . 2 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)))
69 simpr3 1238 . . 3 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑅 (𝑃 𝑄))
70 simpl1 1228 . . . . . . 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 1321 . . . . . . 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 1323 . . . . . . 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 17280 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) = (𝑄 𝑃))
7771, 73, 75, 76syl3anc 1477 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑃 𝑄) = (𝑄 𝑃))
7877breq2d 4816 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑅 (𝑃 𝑄) ↔ 𝑅 (𝑄 𝑃)))
79 simpl23 1325 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑅𝐴)
80 simpr2 1236 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑅𝑄)
8128, 19, 16cvlatexch1 35144 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑅𝐴𝑃𝐴𝑄𝐴) ∧ 𝑅𝑄) → (𝑅 (𝑄 𝑃) → 𝑃 (𝑄 𝑅)))
8270, 79, 72, 74, 80, 81syl131anc 1490 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑅 (𝑄 𝑃) → 𝑃 (𝑄 𝑅)))
83 simpr1 1234 . . . . . . 7 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑅𝑃)
8483necomd 2987 . . . . . 6 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → 𝑃𝑅)
8528, 19, 16cvlatexchb2 35143 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑄 𝑅) ↔ (𝑃 𝑅) = (𝑄 𝑅)))
8670, 72, 74, 79, 84, 85syl131anc 1490 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑃 (𝑄 𝑅) ↔ (𝑃 𝑅) = (𝑄 𝑅)))
8782, 86sylibd 229 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) → (𝑅 (𝑄 𝑃) → (𝑃 𝑅) = (𝑄 𝑅)))
8878, 87sylbid 230 . . 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 913 1 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139   ≠ wne 2932   class class class wbr 4804  ‘cfv 6049  (class class class)co 6814  Basecbs 16079  lecple 16170  joincjn 17165  Latclat 17266  Atomscatm 35071  AtLatcal 35072  CvLatclc 35073 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-preset 17149  df-poset 17167  df-plt 17179  df-lub 17195  df-glb 17196  df-join 17197  df-meet 17198  df-p0 17260  df-lat 17267  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130 This theorem is referenced by:  cvlsupr3  35152  cvlsupr4  35153  cvlsupr5  35154  cvlsupr6  35155  4atexlemex2  35878  4atex  35883  4atex3  35888  cdleme02N  36030  cdleme0ex2N  36032  cdleme0moN  36033  cdleme0nex  36098
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