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Theorem llnexchb2lem 36886
Description: Lemma for llnexchb2 36887. (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 1240 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝐾 ∈ HL)
2 simpl21 1243 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑃𝐴)
3 simpl12 1241 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑋𝑁)
4 eqid 2821 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
5 llnexch.n . . . . . . . 8 𝑁 = (LLines‘𝐾)
64, 5llnbase 36527 . . . . . . 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 36382 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝐾 ∈ Lat)
9 simpl13 1242 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑌𝑁)
104, 5llnbase 36527 . . . . . . . 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 17652 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 𝑌) ∈ (Base‘𝐾))
148, 7, 11, 13syl3anc 1363 . . . . . 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 17676 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 𝑌) 𝑋)
178, 7, 11, 16syl3anc 1363 . . . . . 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 36880 . . . . . 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 1375 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ((𝑋 𝑃) (𝑋 𝑌)) = (𝑋 (𝑃 (𝑋 𝑌))))
224, 19atbase 36307 . . . . . . . . 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 17675 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑋 𝑃) = (𝑃 𝑋))
258, 7, 23, 24syl3anc 1363 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑃) = (𝑃 𝑋))
26 simpl23 1245 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ¬ 𝑃 𝑋)
27 hlatl 36378 . . . . . . . . . 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 2821 . . . . . . . . . 10 (0.‘𝐾) = (0.‘𝐾)
304, 15, 12, 29, 19atnle 36335 . . . . . . . . 9 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋 ∈ (Base‘𝐾)) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = (0.‘𝐾)))
3128, 2, 7, 30syl3anc 1363 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = (0.‘𝐾)))
3226, 31mpbid 233 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑃 𝑋) = (0.‘𝐾))
3325, 32eqtrd 2856 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑃) = (0.‘𝐾))
3433oveq1d 7160 . . . . 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 36377 . . . . . . . . 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 1185 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) ∈ 𝐴)
39 simpl22 1244 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑄𝐴)
40 breq1 5061 . . . . . . . . . . . 12 (𝑃 = (𝑋 𝑌) → (𝑃 𝑋 ↔ (𝑋 𝑌) 𝑋))
4117, 40syl5ibrcom 248 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑃 = (𝑋 𝑌) → 𝑃 𝑋))
4241necon3bd 3030 . . . . . . . . . 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 3071 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) ≠ 𝑃)
4515, 18, 19cvlatexchb1 36352 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ ((𝑋 𝑌) ∈ 𝐴𝑄𝐴𝑃𝐴) ∧ (𝑋 𝑌) ≠ 𝑃) → ((𝑋 𝑌) (𝑃 𝑄) ↔ (𝑃 (𝑋 𝑌)) = (𝑃 𝑄)))
4637, 38, 39, 2, 44, 45syl131anc 1375 . . . . . . 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 233 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑃 (𝑋 𝑌)) = (𝑃 𝑄))
4847oveq2d 7161 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 (𝑃 (𝑋 𝑌))) = (𝑋 (𝑃 𝑄)))
4921, 34, 483eqtr3rd 2865 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 (𝑃 𝑄)) = ((0.‘𝐾) (𝑋 𝑌)))
50 hlol 36379 . . . . . 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 36244 . . . . 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 2857 . . 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 1195 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝐾 ∈ HL)
5756hllatd 36382 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝐾 ∈ Lat)
58 simp12 1196 . . . . 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 1198 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝑃𝐴)
61 simp22 1199 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝑄𝐴)
624, 18, 19hlatjcl 36385 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
6356, 60, 61, 62syl3anc 1363 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
644, 15, 12latmle2 17677 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑋 (𝑃 𝑄)) (𝑃 𝑄))
6557, 59, 63, 64syl3anc 1363 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑋 (𝑃 𝑄)) (𝑃 𝑄))
66 breq1 5061 . . 3 ((𝑋 𝑌) = (𝑋 (𝑃 𝑄)) → ((𝑋 𝑌) (𝑃 𝑄) ↔ (𝑋 (𝑃 𝑄)) (𝑃 𝑄)))
6765, 66syl5ibrcom 248 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → ((𝑋 𝑌) = (𝑋 (𝑃 𝑄)) → (𝑋 𝑌) (𝑃 𝑄)))
6855, 67impbid 213 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3016   class class class wbr 5058  cfv 6349  (class class class)co 7145  Basecbs 16473  lecple 16562  joincjn 17544  meetcmee 17545  0.cp0 17637  Latclat 17645  OLcol 36192  Atomscatm 36281  AtLatcal 36282  CvLatclc 36283  HLchlt 36368  LLinesclln 36509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-iun 4914  df-iin 4915  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7680  df-2nd 7681  df-proset 17528  df-poset 17546  df-plt 17558  df-lub 17574  df-glb 17575  df-join 17576  df-meet 17577  df-p0 17639  df-lat 17646  df-clat 17708  df-oposet 36194  df-ol 36196  df-oml 36197  df-covers 36284  df-ats 36285  df-atl 36316  df-cvlat 36340  df-hlat 36369  df-llines 36516  df-psubsp 36521  df-pmap 36522  df-padd 36814
This theorem is referenced by:  llnexchb2  36887
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