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Theorem llnexchb2lem 37109
 Description: Lemma for llnexchb2 37110. (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 1245 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝐾 ∈ HL)
2 simpl21 1248 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑃𝐴)
3 simpl12 1246 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑋𝑁)
4 eqid 2824 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
5 llnexch.n . . . . . . . 8 𝑁 = (LLines‘𝐾)
64, 5llnbase 36750 . . . . . . 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 36605 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝐾 ∈ Lat)
9 simpl13 1247 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑌𝑁)
104, 5llnbase 36750 . . . . . . . 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 17662 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 𝑌) ∈ (Base‘𝐾))
148, 7, 11, 13syl3anc 1368 . . . . . 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 17686 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 𝑌) 𝑋)
178, 7, 11, 16syl3anc 1368 . . . . . 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 37103 . . . . . 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 1380 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ((𝑋 𝑃) (𝑋 𝑌)) = (𝑋 (𝑃 (𝑋 𝑌))))
224, 19atbase 36530 . . . . . . . . 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 17685 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑋 𝑃) = (𝑃 𝑋))
258, 7, 23, 24syl3anc 1368 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑃) = (𝑃 𝑋))
26 simpl23 1250 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ¬ 𝑃 𝑋)
27 hlatl 36601 . . . . . . . . . 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 2824 . . . . . . . . . 10 (0.‘𝐾) = (0.‘𝐾)
304, 15, 12, 29, 19atnle 36558 . . . . . . . . 9 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋 ∈ (Base‘𝐾)) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = (0.‘𝐾)))
3128, 2, 7, 30syl3anc 1368 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = (0.‘𝐾)))
3226, 31mpbid 235 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑃 𝑋) = (0.‘𝐾))
3325, 32eqtrd 2859 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑃) = (0.‘𝐾))
3433oveq1d 7164 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ((𝑋 𝑃) (𝑋 𝑌)) = ((0.‘𝐾) (𝑋 𝑌)))
35 simpr 488 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) (𝑃 𝑄))
36 hlcvl 36600 . . . . . . . . 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 1190 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) ∈ 𝐴)
39 simpl22 1249 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → 𝑄𝐴)
40 breq1 5055 . . . . . . . . . . . 12 (𝑃 = (𝑋 𝑌) → (𝑃 𝑋 ↔ (𝑋 𝑌) 𝑋))
4117, 40syl5ibrcom 250 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑃 = (𝑋 𝑌) → 𝑃 𝑋))
4241necon3bd 3028 . . . . . . . . . 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 3069 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) ≠ 𝑃)
4515, 18, 19cvlatexchb1 36575 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ ((𝑋 𝑌) ∈ 𝐴𝑄𝐴𝑃𝐴) ∧ (𝑋 𝑌) ≠ 𝑃) → ((𝑋 𝑌) (𝑃 𝑄) ↔ (𝑃 (𝑋 𝑌)) = (𝑃 𝑄)))
4637, 38, 39, 2, 44, 45syl131anc 1380 . . . . . . 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 235 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑃 (𝑋 𝑌)) = (𝑃 𝑄))
4847oveq2d 7165 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 (𝑃 (𝑋 𝑌))) = (𝑋 (𝑃 𝑄)))
4921, 34, 483eqtr3rd 2868 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 (𝑃 𝑄)) = ((0.‘𝐾) (𝑋 𝑌)))
50 hlol 36602 . . . . . 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 36467 . . . . 5 ((𝐾 ∈ OL ∧ (𝑋 𝑌) ∈ (Base‘𝐾)) → ((0.‘𝐾) (𝑋 𝑌)) = (𝑋 𝑌))
5351, 14, 52syl2anc 587 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → ((0.‘𝐾) (𝑋 𝑌)) = (𝑋 𝑌))
5449, 53eqtr2d 2860 . . 3 ((((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) ∧ (𝑋 𝑌) (𝑃 𝑄)) → (𝑋 𝑌) = (𝑋 (𝑃 𝑄)))
5554ex 416 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → ((𝑋 𝑌) (𝑃 𝑄) → (𝑋 𝑌) = (𝑋 (𝑃 𝑄))))
56 simp11 1200 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝐾 ∈ HL)
5756hllatd 36605 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝐾 ∈ Lat)
58 simp12 1201 . . . . 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 1203 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝑃𝐴)
61 simp22 1204 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝑄𝐴)
624, 18, 19hlatjcl 36608 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
6356, 60, 61, 62syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
644, 15, 12latmle2 17687 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑋 (𝑃 𝑄)) (𝑃 𝑄))
6557, 59, 63, 64syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑋 (𝑃 𝑄)) (𝑃 𝑄))
66 breq1 5055 . . 3 ((𝑋 𝑌) = (𝑋 (𝑃 𝑄)) → ((𝑋 𝑌) (𝑃 𝑄) ↔ (𝑋 (𝑃 𝑄)) (𝑃 𝑄)))
6765, 66syl5ibrcom 250 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → ((𝑋 𝑌) = (𝑋 (𝑃 𝑄)) → (𝑋 𝑌) (𝑃 𝑄)))
6855, 67impbid 215 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ≠ wne 3014   class class class wbr 5052  ‘cfv 6343  (class class class)co 7149  Basecbs 16483  lecple 16572  joincjn 17554  meetcmee 17555  0.cp0 17647  Latclat 17655  OLcol 36415  Atomscatm 36504  AtLatcal 36505  CvLatclc 36506  HLchlt 36591  LLinesclln 36732 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-lat 17656  df-clat 17718  df-oposet 36417  df-ol 36419  df-oml 36420  df-covers 36507  df-ats 36508  df-atl 36539  df-cvlat 36563  df-hlat 36592  df-llines 36739  df-psubsp 36744  df-pmap 36745  df-padd 37037 This theorem is referenced by:  llnexchb2  37110
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