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Theorem llnexchb2lem 38129
Description: Lemma for llnexchb2 38130. (Contributed by NM, 17-Nov-2012.)
Hypotheses
Ref Expression
llnexch.l = (le‘𝐾)
llnexch.j = (join‘𝐾)
llnexch.m = (meet‘𝐾)
llnexch.a 𝐴 = (Atoms‘𝐾)
llnexch.n 𝑁 = (LLines‘𝐾)
Assertion
Ref Expression
llnexchb2lem (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → ((𝑋 𝑌) (𝑃 𝑄) ↔ (𝑋 𝑌) = (𝑋 (𝑃 𝑄))))

Proof of Theorem llnexchb2lem
StepHypRef Expression
1 simpl11 1247 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝐾 ∈ HL)
2 simpl21 1250 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑃𝐴)
3 simpl12 1248 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑋𝑁)
4 eqid 2736 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
5 llnexch.n . . . . . . . 8 𝑁 = (LLines‘𝐾)
64, 5llnbase 37770 . . . . . . 7 (𝑋𝑁𝑋 ∈ (Base‘𝐾))
73, 6syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑋 ∈ (Base‘𝐾))
81hllatd 37624 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝐾 ∈ Lat)
9 simpl13 1249 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑌𝑁)
104, 5llnbase 37770 . . . . . . . 8 (𝑌𝑁𝑌 ∈ (Base‘𝐾))
119, 10syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑌 ∈ (Base‘𝐾))
12 llnexch.m . . . . . . . 8 = (meet‘𝐾)
134, 12latmcl 18247 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 𝑌) ∈ (Base‘𝐾))
148, 7, 11, 13syl3anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) ∈ (Base‘𝐾))
15 llnexch.l . . . . . . . 8 = (le‘𝐾)
164, 15, 12latmle1 18271 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 𝑌) 𝑋)
178, 7, 11, 16syl3anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) 𝑋)
18 llnexch.j . . . . . . 7 = (join‘𝐾)
19 llnexch.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
204, 15, 18, 12, 19atmod2i2 38123 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋 ∈ (Base‘𝐾) ∧ (𝑋 𝑌) ∈ (Base‘𝐾)) ∧ (𝑋 𝑌) 𝑋) → ((𝑋 𝑃) (𝑋 𝑌)) = (𝑋 (𝑃 (𝑋 𝑌))))
211, 2, 7, 14, 17, 20syl131anc 1382 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ((𝑋 𝑃) (𝑋 𝑌)) = (𝑋 (𝑃 (𝑋 𝑌))))
224, 19atbase 37549 . . . . . . . . 9 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
232, 22syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑃 ∈ (Base‘𝐾))
244, 12latmcom 18270 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑋 𝑃) = (𝑃 𝑋))
258, 7, 23, 24syl3anc 1370 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑃) = (𝑃 𝑋))
26 simpl23 1252 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ¬ 𝑃 𝑋)
27 hlatl 37620 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
281, 27syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝐾 ∈ AtLat)
29 eqid 2736 . . . . . . . . . 10 (0.‘𝐾) = (0.‘𝐾)
304, 15, 12, 29, 19atnle 37577 . . . . . . . . 9 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋 ∈ (Base‘𝐾)) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = (0.‘𝐾)))
3128, 2, 7, 30syl3anc 1370 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = (0.‘𝐾)))
3226, 31mpbid 231 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑃 𝑋) = (0.‘𝐾))
3325, 32eqtrd 2776 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑃) = (0.‘𝐾))
3433oveq1d 7344 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ((𝑋 𝑃) (𝑋 𝑌)) = ((0.‘𝐾) (𝑋 𝑌)))
35 simpr 485 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) (𝑃 𝑄))
36 hlcvl 37619 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ CvLat)
371, 36syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝐾 ∈ CvLat)
38 simpl3 1192 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) ∈ 𝐴)
39 simpl22 1251 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑄𝐴)
40 breq1 5092 . . . . . . . . . . . 12 (𝑃 = (𝑋 𝑌) → (𝑃 𝑋 ↔ (𝑋 𝑌) 𝑋))
4117, 40syl5ibrcom 246 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑃 = (𝑋 𝑌) → 𝑃 𝑋))
4241necon3bd 2954 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (¬ 𝑃 𝑋𝑃 ≠ (𝑋 𝑌)))
4326, 42mpd 15 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑃 ≠ (𝑋 𝑌))
4443necomd 2996 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) ≠ 𝑃)
4515, 18, 19cvlatexchb1 37594 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ ((𝑋 𝑌) ∈ 𝐴𝑄𝐴𝑃𝐴) ∧ (𝑋 𝑌) ≠ 𝑃) → ((𝑋 𝑌) (𝑃 𝑄) ↔ (𝑃 (𝑋 𝑌)) = (𝑃 𝑄)))
4637, 38, 39, 2, 44, 45syl131anc 1382 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ((𝑋 𝑌) (𝑃 𝑄) ↔ (𝑃 (𝑋 𝑌)) = (𝑃 𝑄)))
4735, 46mpbid 231 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑃 (𝑋 𝑌)) = (𝑃 𝑄))
4847oveq2d 7345 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 (𝑃 (𝑋 𝑌))) = (𝑋 (𝑃 𝑄)))
4921, 34, 483eqtr3rd 2785 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 (𝑃 𝑄)) = ((0.‘𝐾) (𝑋 𝑌)))
50 hlol 37621 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ OL)
511, 50syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝐾 ∈ OL)
524, 18, 29olj02 37486 . . . . 5 ((𝐾 ∈ OL ∧ (𝑋 𝑌) ∈ (Base‘𝐾)) → ((0.‘𝐾) (𝑋 𝑌)) = (𝑋 𝑌))
5351, 14, 52syl2anc 584 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ((0.‘𝐾) (𝑋 𝑌)) = (𝑋 𝑌))
5449, 53eqtr2d 2777 . . 3 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) = (𝑋 (𝑃 𝑄)))
5554ex 413 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → ((𝑋 𝑌) (𝑃 𝑄) → (𝑋 𝑌) = (𝑋 (𝑃 𝑄))))
56 simp11 1202 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝐾 ∈ HL)
5756hllatd 37624 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝐾 ∈ Lat)
58 simp12 1203 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝑋𝑁)
5958, 6syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝑋 ∈ (Base‘𝐾))
60 simp21 1205 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝑃𝐴)
61 simp22 1206 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝑄𝐴)
624, 18, 19hlatjcl 37627 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
6356, 60, 61, 62syl3anc 1370 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
644, 15, 12latmle2 18272 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑋 (𝑃 𝑄)) (𝑃 𝑄))
6557, 59, 63, 64syl3anc 1370 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑋 (𝑃 𝑄)) (𝑃 𝑄))
66 breq1 5092 . . 3 ((𝑋 𝑌) = (𝑋 (𝑃 𝑄)) → ((𝑋 𝑌) (𝑃 𝑄) ↔ (𝑋 (𝑃 𝑄)) (𝑃 𝑄)))
6765, 66syl5ibrcom 246 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → ((𝑋 𝑌) = (𝑋 (𝑃 𝑄)) → (𝑋 𝑌) (𝑃 𝑄)))
6855, 67impbid 211 1 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → ((𝑋 𝑌) (𝑃 𝑄) ↔ (𝑋 𝑌) = (𝑋 (𝑃 𝑄))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wne 2940   class class class wbr 5089  cfv 6473  (class class class)co 7329  Basecbs 17001  lecple 17058  joincjn 18118  meetcmee 18119  0.cp0 18230  Latclat 18238  OLcol 37434  Atomscatm 37523  AtLatcal 37524  CvLatclc 37525  HLchlt 37610  LLinesclln 37752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-iin 4941  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-1st 7891  df-2nd 7892  df-proset 18102  df-poset 18120  df-plt 18137  df-lub 18153  df-glb 18154  df-join 18155  df-meet 18156  df-p0 18232  df-lat 18239  df-clat 18306  df-oposet 37436  df-ol 37438  df-oml 37439  df-covers 37526  df-ats 37527  df-atl 37558  df-cvlat 37582  df-hlat 37611  df-llines 37759  df-psubsp 37764  df-pmap 37765  df-padd 38057
This theorem is referenced by:  llnexchb2  38130
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