Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemefs32sn1aw Structured version   Visualization version   GIF version

Theorem cdlemefs32sn1aw 39748
Description: Show that 𝑅 / 𝑠𝑁 is an atom not under 𝑊 when 𝑅 (𝑃 𝑄). (Contributed by NM, 24-Mar-2013.)
Hypotheses
Ref Expression
cdlemefs32.b 𝐵 = (Base‘𝐾)
cdlemefs32.l = (le‘𝐾)
cdlemefs32.j = (join‘𝐾)
cdlemefs32.m = (meet‘𝐾)
cdlemefs32.a 𝐴 = (Atoms‘𝐾)
cdlemefs32.h 𝐻 = (LHyp‘𝐾)
cdlemefs32.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdlemefs32.d 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdlemefs32.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
cdlemefs32.i 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))
cdlemefs32.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
cdlemefs32a1.y 𝑌 = ((𝑃 𝑄) (𝐷 ((𝑅 𝑡) 𝑊)))
cdlemefs32a1.z 𝑍 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌))
Assertion
Ref Expression
cdlemefs32sn1aw ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (𝑅 / 𝑠𝑁𝐴 ∧ ¬ 𝑅 / 𝑠𝑁 𝑊))
Distinct variable groups:   𝑡,𝑠,𝑦,𝐴   𝐵,𝑠,𝑡,𝑦   𝑦,𝐷   𝑦,𝐸   𝐻,𝑠,𝑡,𝑦   ,𝑠,𝑡,𝑦   𝐾,𝑠,𝑡,𝑦   ,𝑠,𝑡,𝑦   ,𝑠,𝑡,𝑦   𝑃,𝑠,𝑡,𝑦   𝑄,𝑠,𝑡,𝑦   𝑅,𝑠,𝑡,𝑦   𝑡,𝑈,𝑦   𝑊,𝑠,𝑡,𝑦   𝑦,𝑌   𝐷,𝑠
Allowed substitution hints:   𝐶(𝑦,𝑡,𝑠)   𝐷(𝑡)   𝑈(𝑠)   𝐸(𝑡,𝑠)   𝐼(𝑦,𝑡,𝑠)   𝑁(𝑦,𝑡,𝑠)   𝑌(𝑡,𝑠)   𝑍(𝑦,𝑡,𝑠)

Proof of Theorem cdlemefs32sn1aw
StepHypRef Expression
1 cdlemefs32.b . . . 4 𝐵 = (Base‘𝐾)
21fvexi 6905 . . 3 𝐵 ∈ V
3 nfv 1916 . . . 4 𝑡(((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄))
4 cdlemefs32a1.z . . . . . . . 8 𝑍 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌))
5 nfra1 3280 . . . . . . . . 9 𝑡𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌)
6 nfcv 2902 . . . . . . . . 9 𝑡𝐵
75, 6nfriota 7381 . . . . . . . 8 𝑡(𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌))
84, 7nfcxfr 2900 . . . . . . 7 𝑡𝑍
98nfel1 2918 . . . . . 6 𝑡 𝑍𝐴
10 nfcv 2902 . . . . . . . 8 𝑡
11 nfcv 2902 . . . . . . . 8 𝑡𝑊
128, 10, 11nfbr 5195 . . . . . . 7 𝑡 𝑍 𝑊
1312nfn 1859 . . . . . 6 𝑡 ¬ 𝑍 𝑊
149, 13nfan 1901 . . . . 5 𝑡(𝑍𝐴 ∧ ¬ 𝑍 𝑊)
1514a1i 11 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → Ⅎ𝑡(𝑍𝐴 ∧ ¬ 𝑍 𝑊))
164a1i 11 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → 𝑍 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌)))
17 eleq1 2820 . . . . . 6 (𝑌 = 𝑍 → (𝑌𝐴𝑍𝐴))
18 breq1 5151 . . . . . . 7 (𝑌 = 𝑍 → (𝑌 𝑊𝑍 𝑊))
1918notbid 318 . . . . . 6 (𝑌 = 𝑍 → (¬ 𝑌 𝑊 ↔ ¬ 𝑍 𝑊))
2017, 19anbi12d 630 . . . . 5 (𝑌 = 𝑍 → ((𝑌𝐴 ∧ ¬ 𝑌 𝑊) ↔ (𝑍𝐴 ∧ ¬ 𝑍 𝑊)))
2120adantl 481 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) ∧ 𝑌 = 𝑍) → ((𝑌𝐴 ∧ ¬ 𝑌 𝑊) ↔ (𝑍𝐴 ∧ ¬ 𝑍 𝑊)))
22 simpl1 1190 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ (¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)))) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
23 simpl2r 1226 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ (¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)))) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
24 simprl 768 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ (¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)))) → 𝑡𝐴)
25 simprrl 778 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ (¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)))) → ¬ 𝑡 𝑊)
2624, 25jca 511 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ (¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)))) → (𝑡𝐴 ∧ ¬ 𝑡 𝑊))
27 simpl2l 1225 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ (¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)))) → 𝑃𝑄)
28 simpl3 1192 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ (¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)))) → 𝑅 (𝑃 𝑄))
29 simprrr 779 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ (¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)))) → ¬ 𝑡 (𝑃 𝑄))
30 cdlemefs32.l . . . . . . . 8 = (le‘𝐾)
31 cdlemefs32.j . . . . . . . 8 = (join‘𝐾)
32 cdlemefs32.m . . . . . . . 8 = (meet‘𝐾)
33 cdlemefs32.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
34 cdlemefs32.h . . . . . . . 8 𝐻 = (LHyp‘𝐾)
35 cdlemefs32.u . . . . . . . 8 𝑈 = ((𝑃 𝑄) 𝑊)
36 cdlemefs32.d . . . . . . . 8 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
37 cdlemefs32a1.y . . . . . . . 8 𝑌 = ((𝑃 𝑄) (𝐷 ((𝑅 𝑡) 𝑊)))
3830, 31, 32, 33, 34, 35, 36, 37cdleme7ga 39582 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄))) → 𝑌𝐴)
3930, 31, 32, 33, 34, 35, 36, 37cdleme7 39583 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄))) → ¬ 𝑌 𝑊)
4038, 39jca 511 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄))) → (𝑌𝐴 ∧ ¬ 𝑌 𝑊))
4122, 23, 26, 27, 28, 29, 40syl123anc 1386 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ (¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)))) → (𝑌𝐴 ∧ ¬ 𝑌 𝑊))
4241ex 412 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → ((𝑡𝐴 ∧ (¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄))) → (𝑌𝐴 ∧ ¬ 𝑌 𝑊)))
43 simp1 1135 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
44 simp2rl 1241 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → 𝑅𝐴)
45 simp2rr 1242 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → ¬ 𝑅 𝑊)
46 simp2l 1198 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → 𝑃𝑄)
47 simp3 1137 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → 𝑅 (𝑃 𝑄))
481, 30, 31, 32, 33, 34, 35, 36, 37, 4cdleme25cl 39691 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → 𝑍𝐵)
4943, 44, 45, 46, 47, 48syl122anc 1378 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → 𝑍𝐵)
50 simp11 1202 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
51 simp12 1203 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
52 simp13 1204 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
5330, 31, 33, 34cdlemb2 39375 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → ∃𝑡𝐴𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)))
5450, 51, 52, 46, 53syl121anc 1374 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → ∃𝑡𝐴𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)))
553, 15, 16, 21, 42, 49, 54riotasv3d 38293 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) ∧ 𝐵 ∈ V) → (𝑍𝐴 ∧ ¬ 𝑍 𝑊))
562, 55mpan2 688 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (𝑍𝐴 ∧ ¬ 𝑍 𝑊))
57 cdlemefs32.e . . . . . 6 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
58 cdlemefs32.i . . . . . 6 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))
59 cdlemefs32.n . . . . . 6 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
6057, 58, 59, 37, 4cdleme31sn1c 39722 . . . . 5 ((𝑅𝐴𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = 𝑍)
6144, 47, 60syl2anc 583 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = 𝑍)
6261eleq1d 2817 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (𝑅 / 𝑠𝑁𝐴𝑍𝐴))
6361breq1d 5158 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (𝑅 / 𝑠𝑁 𝑊𝑍 𝑊))
6463notbid 318 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (¬ 𝑅 / 𝑠𝑁 𝑊 ↔ ¬ 𝑍 𝑊))
6562, 64anbi12d 630 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → ((𝑅 / 𝑠𝑁𝐴 ∧ ¬ 𝑅 / 𝑠𝑁 𝑊) ↔ (𝑍𝐴 ∧ ¬ 𝑍 𝑊)))
6656, 65mpbird 257 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (𝑅 / 𝑠𝑁𝐴 ∧ ¬ 𝑅 / 𝑠𝑁 𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wnf 1784  wcel 2105  wne 2939  wral 3060  wrex 3069  Vcvv 3473  csb 3893  ifcif 4528   class class class wbr 5148  cfv 6543  crio 7367  (class class class)co 7412  Basecbs 17151  lecple 17211  joincjn 18274  meetcmee 18275  Atomscatm 38596  HLchlt 38683  LHypclh 39318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-riotaBAD 38286
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-undef 8264  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-p1 18389  df-lat 18395  df-clat 18462  df-oposet 38509  df-ol 38511  df-oml 38512  df-covers 38599  df-ats 38600  df-atl 38631  df-cvlat 38655  df-hlat 38684  df-llines 38832  df-lplanes 38833  df-lvols 38834  df-lines 38835  df-psubsp 38837  df-pmap 38838  df-padd 39130  df-lhyp 39322
This theorem is referenced by:  cdlemefs32snb  39749  cdleme32sn1awN  39766  cdleme32snaw  39769
  Copyright terms: Public domain W3C validator