Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme0moN Structured version   Visualization version   GIF version

Theorem cdleme0moN 34324
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme0.l = (le‘𝐾)
cdleme0.j = (join‘𝐾)
cdleme0.m = (meet‘𝐾)
cdleme0.a 𝐴 = (Atoms‘𝐾)
cdleme0.h 𝐻 = (LHyp‘𝐾)
cdleme0.u 𝑈 = ((𝑃 𝑄) 𝑊)
Assertion
Ref Expression
cdleme0moN (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑅 = 𝑃𝑅 = 𝑄))
Distinct variable groups:   𝐴,𝑟   ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑅,𝑟   𝑈,𝑟
Allowed substitution hints:   𝐻(𝑟)   𝐾(𝑟)   (𝑟)   (𝑟)   𝑊(𝑟)

Proof of Theorem cdleme0moN
StepHypRef Expression
1 simp23r 1176 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ 𝑅 𝑊)
2 neanior 2874 . . 3 ((𝑅𝑃𝑅𝑄) ↔ ¬ (𝑅 = 𝑃𝑅 = 𝑄))
3 simpl33 1137 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
4 simp23l 1175 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑅𝐴)
54adantr 480 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → 𝑅𝐴)
6 simprl 790 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → 𝑅𝑃)
7 simprr 792 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → 𝑅𝑄)
8 simpl32 1136 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → 𝑅 (𝑃 𝑄))
9 simpl1l 1105 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → 𝐾 ∈ HL)
10 hlcvl 33458 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ CvLat)
119, 10syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → 𝐾 ∈ CvLat)
12 simp21l 1171 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑃𝐴)
1312adantr 480 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → 𝑃𝐴)
14 simp22l 1173 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑄𝐴)
1514adantr 480 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → 𝑄𝐴)
16 simpl31 1135 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → 𝑃𝑄)
17 cdleme0.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
18 cdleme0.l . . . . . . . . 9 = (le‘𝐾)
19 cdleme0.j . . . . . . . . 9 = (join‘𝐾)
2017, 18, 19cvlsupr2 33442 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
2111, 13, 15, 5, 16, 20syl131anc 1331 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
226, 7, 8, 21mpbir3and 1238 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → (𝑃 𝑅) = (𝑄 𝑅))
23 simp1l 1078 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐾 ∈ HL)
24 simp1r 1079 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑊𝐻)
25 simp21r 1172 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ 𝑃 𝑊)
26 simp31 1090 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑃𝑄)
27 cdleme0.m . . . . . . . . 9 = (meet‘𝐾)
28 cdleme0.h . . . . . . . . 9 𝐻 = (LHyp‘𝐾)
29 cdleme0.u . . . . . . . . 9 𝑈 = ((𝑃 𝑄) 𝑊)
3018, 19, 27, 17, 28, 29lhpat2 34143 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → 𝑈𝐴)
3123, 24, 12, 25, 14, 26, 30syl222anc 1334 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑈𝐴)
3231adantr 480 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → 𝑈𝐴)
33 simpl1 1057 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
34 simpl21 1132 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
35 simpl22 1133 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
3618, 19, 27, 17, 28, 29cdleme02N 34321 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → ((𝑃 𝑈) = (𝑄 𝑈) ∧ 𝑈 𝑊))
3736simpld 474 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → (𝑃 𝑈) = (𝑄 𝑈))
3833, 34, 35, 16, 37syl121anc 1323 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → (𝑃 𝑈) = (𝑄 𝑈))
39 df-rmo 2904 . . . . . . 7 (∃*𝑟𝐴 (𝑃 𝑟) = (𝑄 𝑟) ↔ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
40 oveq2 6535 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑃 𝑟) = (𝑃 𝑅))
41 oveq2 6535 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
4240, 41eqeq12d 2625 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑃 𝑟) = (𝑄 𝑟) ↔ (𝑃 𝑅) = (𝑄 𝑅)))
43 oveq2 6535 . . . . . . . . 9 (𝑟 = 𝑈 → (𝑃 𝑟) = (𝑃 𝑈))
44 oveq2 6535 . . . . . . . . 9 (𝑟 = 𝑈 → (𝑄 𝑟) = (𝑄 𝑈))
4543, 44eqeq12d 2625 . . . . . . . 8 (𝑟 = 𝑈 → ((𝑃 𝑟) = (𝑄 𝑟) ↔ (𝑃 𝑈) = (𝑄 𝑈)))
4642, 45rmoi 3496 . . . . . . 7 ((∃*𝑟𝐴 (𝑃 𝑟) = (𝑄 𝑟) ∧ (𝑅𝐴 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑈𝐴 ∧ (𝑃 𝑈) = (𝑄 𝑈))) → 𝑅 = 𝑈)
4739, 46syl3an1br 1357 . . . . . 6 ((∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ∧ (𝑅𝐴 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑈𝐴 ∧ (𝑃 𝑈) = (𝑄 𝑈))) → 𝑅 = 𝑈)
483, 5, 22, 32, 38, 47syl122anc 1327 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → 𝑅 = 𝑈)
4936simprd 478 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → 𝑈 𝑊)
5033, 34, 35, 16, 49syl121anc 1323 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → 𝑈 𝑊)
5148, 50eqbrtrd 4600 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ (𝑅𝑃𝑅𝑄)) → 𝑅 𝑊)
5251ex 449 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑅𝑃𝑅𝑄) → 𝑅 𝑊))
532, 52syl5bir 232 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (¬ (𝑅 = 𝑃𝑅 = 𝑄) → 𝑅 𝑊))
541, 53mt3d 139 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑅 = 𝑃𝑅 = 𝑄))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383  w3a 1031   = wceq 1475  wcel 1977  ∃*wmo 2459  wne 2780  ∃*wrmo 2899   class class class wbr 4578  cfv 5790  (class class class)co 6527  lecple 15724  joincjn 16716  meetcmee 16717  Atomscatm 33362  CvLatclc 33364  HLchlt 33449  LHypclh 34082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-preset 16700  df-poset 16718  df-plt 16730  df-lub 16746  df-glb 16747  df-join 16748  df-meet 16749  df-p0 16811  df-p1 16812  df-lat 16818  df-clat 16880  df-oposet 33275  df-ol 33277  df-oml 33278  df-covers 33365  df-ats 33366  df-atl 33397  df-cvlat 33421  df-hlat 33450  df-lhyp 34086
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator