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Theorem llnexchb2lem 36455
Description: Lemma for llnexchb2 36456. (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 1228 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝐾 ∈ HL)
2 simpl21 1231 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑃𝐴)
3 simpl12 1229 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑋𝑁)
4 eqid 2778 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
5 llnexch.n . . . . . . . 8 𝑁 = (LLines‘𝐾)
64, 5llnbase 36096 . . . . . . 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 35951 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝐾 ∈ Lat)
9 simpl13 1230 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑌𝑁)
104, 5llnbase 36096 . . . . . . . 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 17520 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 𝑌) ∈ (Base‘𝐾))
148, 7, 11, 13syl3anc 1351 . . . . . 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 17544 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 𝑌) 𝑋)
178, 7, 11, 16syl3anc 1351 . . . . . 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 36449 . . . . . 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 1363 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ((𝑋 𝑃) (𝑋 𝑌)) = (𝑋 (𝑃 (𝑋 𝑌))))
224, 19atbase 35876 . . . . . . . . 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 17543 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑋 𝑃) = (𝑃 𝑋))
258, 7, 23, 24syl3anc 1351 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑃) = (𝑃 𝑋))
26 simpl23 1233 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ¬ 𝑃 𝑋)
27 hlatl 35947 . . . . . . . . . 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 2778 . . . . . . . . . 10 (0.‘𝐾) = (0.‘𝐾)
304, 15, 12, 29, 19atnle 35904 . . . . . . . . 9 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋 ∈ (Base‘𝐾)) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = (0.‘𝐾)))
3128, 2, 7, 30syl3anc 1351 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = (0.‘𝐾)))
3226, 31mpbid 224 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑃 𝑋) = (0.‘𝐾))
3325, 32eqtrd 2814 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑃) = (0.‘𝐾))
3433oveq1d 6991 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ((𝑋 𝑃) (𝑋 𝑌)) = ((0.‘𝐾) (𝑋 𝑌)))
35 simpr 477 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) (𝑃 𝑄))
36 hlcvl 35946 . . . . . . . . 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 1173 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) ∈ 𝐴)
39 simpl22 1232 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑄𝐴)
40 breq1 4932 . . . . . . . . . . . 12 (𝑃 = (𝑋 𝑌) → (𝑃 𝑋 ↔ (𝑋 𝑌) 𝑋))
4117, 40syl5ibrcom 239 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑃 = (𝑋 𝑌) → 𝑃 𝑋))
4241necon3bd 2981 . . . . . . . . . 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 3022 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) ≠ 𝑃)
4515, 18, 19cvlatexchb1 35921 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ ((𝑋 𝑌) ∈ 𝐴𝑄𝐴𝑃𝐴) ∧ (𝑋 𝑌) ≠ 𝑃) → ((𝑋 𝑌) (𝑃 𝑄) ↔ (𝑃 (𝑋 𝑌)) = (𝑃 𝑄)))
4637, 38, 39, 2, 44, 45syl131anc 1363 . . . . . . 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 224 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑃 (𝑋 𝑌)) = (𝑃 𝑄))
4847oveq2d 6992 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 (𝑃 (𝑋 𝑌))) = (𝑋 (𝑃 𝑄)))
4921, 34, 483eqtr3rd 2823 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 (𝑃 𝑄)) = ((0.‘𝐾) (𝑋 𝑌)))
50 hlol 35948 . . . . . 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 35813 . . . . 5 ((𝐾 ∈ OL ∧ (𝑋 𝑌) ∈ (Base‘𝐾)) → ((0.‘𝐾) (𝑋 𝑌)) = (𝑋 𝑌))
5351, 14, 52syl2anc 576 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ((0.‘𝐾) (𝑋 𝑌)) = (𝑋 𝑌))
5449, 53eqtr2d 2815 . . 3 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) = (𝑋 (𝑃 𝑄)))
5554ex 405 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → ((𝑋 𝑌) (𝑃 𝑄) → (𝑋 𝑌) = (𝑋 (𝑃 𝑄))))
56 simp11 1183 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝐾 ∈ HL)
5756hllatd 35951 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝐾 ∈ Lat)
58 simp12 1184 . . . . 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 1186 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝑃𝐴)
61 simp22 1187 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝑄𝐴)
624, 18, 19hlatjcl 35954 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
6356, 60, 61, 62syl3anc 1351 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
644, 15, 12latmle2 17545 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑋 (𝑃 𝑄)) (𝑃 𝑄))
6557, 59, 63, 64syl3anc 1351 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑋 (𝑃 𝑄)) (𝑃 𝑄))
66 breq1 4932 . . 3 ((𝑋 𝑌) = (𝑋 (𝑃 𝑄)) → ((𝑋 𝑌) (𝑃 𝑄) ↔ (𝑋 (𝑃 𝑄)) (𝑃 𝑄)))
6765, 66syl5ibrcom 239 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → ((𝑋 𝑌) = (𝑋 (𝑃 𝑄)) → (𝑋 𝑌) (𝑃 𝑄)))
6855, 67impbid 204 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 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wne 2967   class class class wbr 4929  cfv 6188  (class class class)co 6976  Basecbs 16339  lecple 16428  joincjn 17412  meetcmee 17413  0.cp0 17505  Latclat 17513  OLcol 35761  Atomscatm 35850  AtLatcal 35851  CvLatclc 35852  HLchlt 35937  LLinesclln 36078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-iin 4795  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-1st 7501  df-2nd 7502  df-proset 17396  df-poset 17414  df-plt 17426  df-lub 17442  df-glb 17443  df-join 17444  df-meet 17445  df-p0 17507  df-lat 17514  df-clat 17576  df-oposet 35763  df-ol 35765  df-oml 35766  df-covers 35853  df-ats 35854  df-atl 35885  df-cvlat 35909  df-hlat 35938  df-llines 36085  df-psubsp 36090  df-pmap 36091  df-padd 36383
This theorem is referenced by:  llnexchb2  36456
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