Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme27N Structured version   Visualization version   GIF version

Theorem cdleme27N 35476
Description: Part of proof of Lemma E in [Crawley] p. 113. Eliminate the 𝑠𝑡 antecedent in cdleme27a 35474. (Contributed by NM, 3-Feb-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme26.b 𝐵 = (Base‘𝐾)
cdleme26.l = (le‘𝐾)
cdleme26.j = (join‘𝐾)
cdleme26.m = (meet‘𝐾)
cdleme26.a 𝐴 = (Atoms‘𝐾)
cdleme26.h 𝐻 = (LHyp‘𝐾)
cdleme27.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme27.f 𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme27.z 𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
cdleme27.n 𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))
cdleme27.d 𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))
cdleme27.c 𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)
cdleme27.g 𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme27.o 𝑂 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))
cdleme27.e 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
cdleme27.y 𝑌 = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺)
Assertion
Ref Expression
cdleme27N ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐶 (𝑌 𝑉))
Distinct variable groups:   𝑡,𝑠,𝑢,𝑧,𝐴   𝐵,𝑠,𝑡,𝑢,𝑧   𝑢,𝐹   𝑢,𝐺   𝐻,𝑠,𝑡,𝑧   ,𝑠,𝑡,𝑢,𝑧   𝐾,𝑠,𝑡,𝑧   ,𝑠,𝑡,𝑢,𝑧   ,𝑠,𝑡,𝑢,𝑧   𝑡,𝑁,𝑢   𝑂,𝑠,𝑢   𝑃,𝑠,𝑡,𝑢,𝑧   𝑄,𝑠,𝑡,𝑢,𝑧   𝑈,𝑠,𝑡,𝑢,𝑧   𝑧,𝑉   𝑊,𝑠,𝑡,𝑢,𝑧
Allowed substitution hints:   𝐶(𝑧,𝑢,𝑡,𝑠)   𝐷(𝑧,𝑢,𝑡,𝑠)   𝐸(𝑧,𝑢,𝑡,𝑠)   𝐹(𝑧,𝑡,𝑠)   𝐺(𝑧,𝑡,𝑠)   𝐻(𝑢)   𝐾(𝑢)   𝑁(𝑧,𝑠)   𝑂(𝑧,𝑡)   𝑉(𝑢,𝑡,𝑠)   𝑌(𝑧,𝑢,𝑡,𝑠)   𝑍(𝑧,𝑢,𝑡,𝑠)

Proof of Theorem cdleme27N
StepHypRef Expression
1 cdleme26.b . . . . 5 𝐵 = (Base‘𝐾)
2 cdleme26.l . . . . 5 = (le‘𝐾)
3 cdleme26.j . . . . 5 = (join‘𝐾)
4 cdleme26.m . . . . 5 = (meet‘𝐾)
5 cdleme26.a . . . . 5 𝐴 = (Atoms‘𝐾)
6 cdleme26.h . . . . 5 𝐻 = (LHyp‘𝐾)
7 cdleme27.u . . . . 5 𝑈 = ((𝑃 𝑄) 𝑊)
8 cdleme27.f . . . . 5 𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
9 cdleme27.z . . . . 5 𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
10 cdleme27.n . . . . 5 𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))
11 cdleme27.d . . . . 5 𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))
12 cdleme27.c . . . . 5 𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)
13 cdleme27.g . . . . 5 𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
14 cdleme27.o . . . . 5 𝑂 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))
15 cdleme27.e . . . . 5 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
16 cdleme27.y . . . . 5 𝑌 = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺)
171, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16cdleme27b 35475 . . . 4 (𝑠 = 𝑡𝐶 = 𝑌)
1817adantl 482 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) ∧ 𝑠 = 𝑡) → 𝐶 = 𝑌)
19 simp11l 1170 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐾 ∈ HL)
20 hllat 34469 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2119, 20syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐾 ∈ Lat)
22 simp11r 1171 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑊𝐻)
23 simp21 1092 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
24 simp22 1093 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
25 simp23 1094 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑡𝐴 ∧ ¬ 𝑡 𝑊))
26 simp12 1090 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑃𝑄)
271, 2, 3, 4, 5, 6, 7, 13, 9, 14, 15, 16cdleme27cl 35473 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ 𝑃𝑄)) → 𝑌𝐵)
2819, 22, 23, 24, 25, 26, 27syl222anc 1340 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑌𝐵)
29 simp3rl 1132 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑉𝐴)
301, 5atbase 34395 . . . . . 6 (𝑉𝐴𝑉𝐵)
3129, 30syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑉𝐵)
321, 2, 3latlej1 17041 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑉𝐵) → 𝑌 (𝑌 𝑉))
3321, 28, 31, 32syl3anc 1324 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑌 (𝑌 𝑉))
3433adantr 481 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) ∧ 𝑠 = 𝑡) → 𝑌 (𝑌 𝑉))
3518, 34eqbrtrd 4666 . 2 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) ∧ 𝑠 = 𝑡) → 𝐶 (𝑌 𝑉))
36 simpl11 1134 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) ∧ 𝑠𝑡) → (𝐾 ∈ HL ∧ 𝑊𝐻))
37 simpl12 1135 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) ∧ 𝑠𝑡) → 𝑃𝑄)
38 simpl13 1136 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) ∧ 𝑠𝑡) → (𝑠𝐴 ∧ ¬ 𝑠 𝑊))
39 simpl21 1137 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) ∧ 𝑠𝑡) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
40 simpl22 1138 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) ∧ 𝑠𝑡) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
41 simpl23 1139 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) ∧ 𝑠𝑡) → (𝑡𝐴 ∧ ¬ 𝑡 𝑊))
42 simpr 477 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) ∧ 𝑠𝑡) → 𝑠𝑡)
43 simpl3l 1114 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) ∧ 𝑠𝑡) → 𝑠 (𝑡 𝑉))
4442, 43jca 554 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) ∧ 𝑠𝑡) → (𝑠𝑡𝑠 (𝑡 𝑉)))
45 simpl3r 1115 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) ∧ 𝑠𝑡) → (𝑉𝐴𝑉 𝑊))
461, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16cdleme27a 35474 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐶 (𝑌 𝑉))
4736, 37, 38, 39, 40, 41, 44, 45, 46syl332anc 1355 . 2 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) ∧ 𝑠𝑡) → 𝐶 (𝑌 𝑉))
4835, 47pm2.61dane 2878 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐶 (𝑌 𝑉))
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  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: (None)
  Copyright terms: Public domain W3C validator