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Theorem cdleme32c 35550
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 19-Feb-2013.)
Hypotheses
Ref Expression
cdleme32.b 𝐵 = (Base‘𝐾)
cdleme32.l = (le‘𝐾)
cdleme32.j = (join‘𝐾)
cdleme32.m = (meet‘𝐾)
cdleme32.a 𝐴 = (Atoms‘𝐾)
cdleme32.h 𝐻 = (LHyp‘𝐾)
cdleme32.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme32.c 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme32.d 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme32.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
cdleme32.i 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))
cdleme32.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
cdleme32.o 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
cdleme32.f 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
Assertion
Ref Expression
cdleme32c ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝐹𝑋) (𝐹𝑌))
Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑧,𝐴   𝐵,𝑠,𝑡,𝑥,𝑦,𝑧   𝑦,𝐶   𝐷,𝑠,𝑦,𝑧   𝑦,𝐸   𝐻,𝑠,𝑡   ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,𝑠,𝑡   ,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   𝑥,𝑁,𝑧   𝑃,𝑠,𝑡,𝑥,𝑦,𝑧   𝑄,𝑠,𝑡,𝑥,𝑦,𝑧   𝑈,𝑠,𝑡,𝑥,𝑦,𝑧   𝑊,𝑠,𝑡,𝑥,𝑦,𝑧   𝑋,𝑠,𝑡,𝑥,𝑧   𝑦,𝐻   𝑦,𝐾   𝑦,𝑌   𝑧,𝐻   𝑧,𝐾   𝑌,𝑠,𝑡,𝑥,𝑧
Allowed substitution hints:   𝐶(𝑥,𝑧,𝑡,𝑠)   𝐷(𝑥,𝑡)   𝐸(𝑥,𝑧,𝑡,𝑠)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐻(𝑥)   𝐼(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐾(𝑥)   𝑁(𝑦,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑧,𝑡,𝑠)   𝑋(𝑦)

Proof of Theorem cdleme32c
StepHypRef Expression
1 simp33 1097 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → 𝑋 𝑌)
2 simp11l 1170 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → 𝐾 ∈ HL)
3 hllat 34469 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
42, 3syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → 𝐾 ∈ Lat)
5 simp21 1092 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → 𝑋𝐵)
6 simp22 1093 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → 𝑌𝐵)
7 simp11r 1171 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → 𝑊𝐻)
8 cdleme32.b . . . . . . 7 𝐵 = (Base‘𝐾)
9 cdleme32.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
108, 9lhpbase 35103 . . . . . 6 (𝑊𝐻𝑊𝐵)
117, 10syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → 𝑊𝐵)
12 cdleme32.l . . . . . 6 = (le‘𝐾)
13 cdleme32.m . . . . . 6 = (meet‘𝐾)
148, 12, 13latmlem1 17062 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑊𝐵)) → (𝑋 𝑌 → (𝑋 𝑊) (𝑌 𝑊)))
154, 5, 6, 11, 14syl13anc 1326 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝑋 𝑌 → (𝑋 𝑊) (𝑌 𝑊)))
161, 15mpd 15 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝑋 𝑊) (𝑌 𝑊))
178, 13latmcl 17033 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
184, 5, 11, 17syl3anc 1324 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝑋 𝑊) ∈ 𝐵)
198, 13latmcl 17033 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵) → (𝑌 𝑊) ∈ 𝐵)
204, 6, 11, 19syl3anc 1324 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝑌 𝑊) ∈ 𝐵)
21 simp12 1090 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
22 simp13 1091 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
23 simp31 1095 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝑠𝐴 ∧ ¬ 𝑠 𝑊))
24 simp23l 1180 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → 𝑃𝑄)
25 cdleme32.j . . . . . 6 = (join‘𝐾)
26 cdleme32.a . . . . . 6 𝐴 = (Atoms‘𝐾)
27 cdleme32.u . . . . . 6 𝑈 = ((𝑃 𝑄) 𝑊)
28 cdleme32.c . . . . . 6 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
29 cdleme32.d . . . . . 6 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
30 cdleme32.e . . . . . 6 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
31 cdleme32.i . . . . . 6 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))
32 cdleme32.n . . . . . 6 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
338, 12, 25, 13, 26, 9, 27, 28, 29, 30, 31, 32cdleme27cl 35473 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ 𝑃𝑄)) → 𝑁𝐵)
342, 7, 21, 22, 23, 24, 33syl222anc 1340 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → 𝑁𝐵)
358, 12, 25latjlej2 17047 . . . 4 ((𝐾 ∈ Lat ∧ ((𝑋 𝑊) ∈ 𝐵 ∧ (𝑌 𝑊) ∈ 𝐵𝑁𝐵)) → ((𝑋 𝑊) (𝑌 𝑊) → (𝑁 (𝑋 𝑊)) (𝑁 (𝑌 𝑊))))
364, 18, 20, 34, 35syl13anc 1326 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → ((𝑋 𝑊) (𝑌 𝑊) → (𝑁 (𝑋 𝑊)) (𝑁 (𝑌 𝑊))))
3716, 36mpd 15 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝑁 (𝑋 𝑊)) (𝑁 (𝑌 𝑊)))
38 simp1 1059 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
39 simp23 1094 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝑃𝑄 ∧ ¬ 𝑋 𝑊))
40 simp32 1096 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝑠 (𝑋 𝑊)) = 𝑋)
41 cdleme32.o . . . 4 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
42 cdleme32.f . . . 4 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
438, 12, 25, 13, 26, 9, 27, 28, 29, 30, 31, 32, 41, 42cdleme32a 35548 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋)) → (𝐹𝑋) = (𝑁 (𝑋 𝑊)))
4438, 5, 39, 23, 40, 43syl122anc 1333 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝐹𝑋) = (𝑁 (𝑋 𝑊)))
458, 12, 25, 13, 26, 9, 27, 28, 29, 30, 31, 32, 41, 42cdleme32b 35549 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝐹𝑌) = (𝑁 (𝑌 𝑊)))
4637, 44, 453brtr4d 4676 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝐹𝑋) (𝐹𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791  wral 2909  ifcif 4077   class class class wbr 4644  cmpt 4720  cfv 5876  crio 6595  (class class class)co 6635  Basecbs 15838  lecple 15929  joincjn 16925  meetcmee 16926  Latclat 17026  Atomscatm 34369  HLchlt 34456  LHypclh 35089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-riotaBAD 34058
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-undef 7384  df-preset 16909  df-poset 16927  df-plt 16939  df-lub 16955  df-glb 16956  df-join 16957  df-meet 16958  df-p0 17020  df-p1 17021  df-lat 17027  df-clat 17089  df-oposet 34282  df-ol 34284  df-oml 34285  df-covers 34372  df-ats 34373  df-atl 34404  df-cvlat 34428  df-hlat 34457  df-llines 34603  df-lplanes 34604  df-lvols 34605  df-lines 34606  df-psubsp 34608  df-pmap 34609  df-padd 34901  df-lhyp 35093
This theorem is referenced by:  cdleme32d  35551
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