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Theorem llnexchb2lem 35675
Description: Lemma for llnexchb2 35676. (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 1314 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝐾 ∈ HL)
2 simpl21 1320 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑃𝐴)
3 simpl12 1316 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑋𝑁)
4 eqid 2771 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
5 llnexch.n . . . . . . . 8 𝑁 = (LLines‘𝐾)
64, 5llnbase 35316 . . . . . . 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 35171 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝐾 ∈ Lat)
9 simpl13 1318 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑌𝑁)
104, 5llnbase 35316 . . . . . . . 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 17260 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 𝑌) ∈ (Base‘𝐾))
148, 7, 11, 13syl3anc 1476 . . . . . 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 17284 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 𝑌) 𝑋)
178, 7, 11, 16syl3anc 1476 . . . . . 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 35669 . . . . . 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 1489 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ((𝑋 𝑃) (𝑋 𝑌)) = (𝑋 (𝑃 (𝑋 𝑌))))
224, 19atbase 35096 . . . . . . . . 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 17283 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑋 𝑃) = (𝑃 𝑋))
258, 7, 23, 24syl3anc 1476 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑃) = (𝑃 𝑋))
26 simpl23 1324 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ¬ 𝑃 𝑋)
27 hlatl 35167 . . . . . . . . . 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 2771 . . . . . . . . . 10 (0.‘𝐾) = (0.‘𝐾)
304, 15, 12, 29, 19atnle 35124 . . . . . . . . 9 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋 ∈ (Base‘𝐾)) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = (0.‘𝐾)))
3128, 2, 7, 30syl3anc 1476 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = (0.‘𝐾)))
3226, 31mpbid 222 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑃 𝑋) = (0.‘𝐾))
3325, 32eqtrd 2805 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑃) = (0.‘𝐾))
3433oveq1d 6811 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ((𝑋 𝑃) (𝑋 𝑌)) = ((0.‘𝐾) (𝑋 𝑌)))
35 simpr 471 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) (𝑃 𝑄))
36 hlcvl 35166 . . . . . . . . 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 1231 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) ∈ 𝐴)
39 simpl22 1322 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑄𝐴)
40 breq1 4790 . . . . . . . . . . . 12 (𝑃 = (𝑋 𝑌) → (𝑃 𝑋 ↔ (𝑋 𝑌) 𝑋))
4117, 40syl5ibrcom 237 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑃 = (𝑋 𝑌) → 𝑃 𝑋))
4241necon3bd 2957 . . . . . . . . . 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 2998 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) ≠ 𝑃)
4515, 18, 19cvlatexchb1 35141 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ ((𝑋 𝑌) ∈ 𝐴𝑄𝐴𝑃𝐴) ∧ (𝑋 𝑌) ≠ 𝑃) → ((𝑋 𝑌) (𝑃 𝑄) ↔ (𝑃 (𝑋 𝑌)) = (𝑃 𝑄)))
4637, 38, 39, 2, 44, 45syl131anc 1489 . . . . . . 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 222 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑃 (𝑋 𝑌)) = (𝑃 𝑄))
4847oveq2d 6812 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 (𝑃 (𝑋 𝑌))) = (𝑋 (𝑃 𝑄)))
4921, 34, 483eqtr3rd 2814 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 (𝑃 𝑄)) = ((0.‘𝐾) (𝑋 𝑌)))
50 hlol 35168 . . . . . 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 35033 . . . . 5 ((𝐾 ∈ OL ∧ (𝑋 𝑌) ∈ (Base‘𝐾)) → ((0.‘𝐾) (𝑋 𝑌)) = (𝑋 𝑌))
5351, 14, 52syl2anc 573 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ((0.‘𝐾) (𝑋 𝑌)) = (𝑋 𝑌))
5449, 53eqtr2d 2806 . . 3 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) = (𝑋 (𝑃 𝑄)))
5554ex 397 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → ((𝑋 𝑌) (𝑃 𝑄) → (𝑋 𝑌) = (𝑋 (𝑃 𝑄))))
56 simp11 1245 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝐾 ∈ HL)
5756hllatd 35171 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝐾 ∈ Lat)
58 simp12 1246 . . . . 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 1248 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝑃𝐴)
61 simp22 1249 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝑄𝐴)
624, 18, 19hlatjcl 35174 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
6356, 60, 61, 62syl3anc 1476 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
644, 15, 12latmle2 17285 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑋 (𝑃 𝑄)) (𝑃 𝑄))
6557, 59, 63, 64syl3anc 1476 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑋 (𝑃 𝑄)) (𝑃 𝑄))
66 breq1 4790 . . 3 ((𝑋 𝑌) = (𝑋 (𝑃 𝑄)) → ((𝑋 𝑌) (𝑃 𝑄) ↔ (𝑋 (𝑃 𝑄)) (𝑃 𝑄)))
6765, 66syl5ibrcom 237 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → ((𝑋 𝑌) = (𝑋 (𝑃 𝑄)) → (𝑋 𝑌) (𝑃 𝑄)))
6855, 67impbid 202 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 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943   class class class wbr 4787  cfv 6030  (class class class)co 6796  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  0.cp0 17245  Latclat 17253  OLcol 34981  Atomscatm 35070  AtLatcal 35071  CvLatclc 35072  HLchlt 35157  LLinesclln 35298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-clat 17316  df-oposet 34983  df-ol 34985  df-oml 34986  df-covers 35073  df-ats 35074  df-atl 35105  df-cvlat 35129  df-hlat 35158  df-llines 35305  df-psubsp 35310  df-pmap 35311  df-padd 35603
This theorem is referenced by:  llnexchb2  35676
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