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Theorem cdleme0moN 35830
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 1203 . 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 2915 . . 3 ((𝑅𝑃𝑅𝑄) ↔ ¬ (𝑅 = 𝑃𝑅 = 𝑄))
3 simpl33 1164 . . . . . 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 1202 . . . . . . 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 809 . . . . . . 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 811 . . . . . . 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 1163 . . . . . . 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 1132 . . . . . . . . 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 34964 . . . . . . . . 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 1198 . . . . . . . . 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 1200 . . . . . . . . 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 1162 . . . . . . . 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 34948 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
2111, 13, 15, 5, 16, 20syl131anc 1379 . . . . . . 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 1264 . . . . . 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 1105 . . . . . . . 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 1106 . . . . . . . 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 1199 . . . . . . . 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 1117 . . . . . . . 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 35649 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → 𝑈𝐴)
3123, 24, 12, 25, 14, 26, 30syl222anc 1382 . . . . . . 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 1084 . . . . . . 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 1159 . . . . . . 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 1160 . . . . . . 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 35827 . . . . . . . 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 1371 . . . . . 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 2949 . . . . . . 7 (∃*𝑟𝐴 (𝑃 𝑟) = (𝑄 𝑟) ↔ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
40 oveq2 6698 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑃 𝑟) = (𝑃 𝑅))
41 oveq2 6698 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
4240, 41eqeq12d 2666 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑃 𝑟) = (𝑄 𝑟) ↔ (𝑃 𝑅) = (𝑄 𝑅)))
43 oveq2 6698 . . . . . . . . 9 (𝑟 = 𝑈 → (𝑃 𝑟) = (𝑃 𝑈))
44 oveq2 6698 . . . . . . . . 9 (𝑟 = 𝑈 → (𝑄 𝑟) = (𝑄 𝑈))
4543, 44eqeq12d 2666 . . . . . . . 8 (𝑟 = 𝑈 → ((𝑃 𝑟) = (𝑄 𝑟) ↔ (𝑃 𝑈) = (𝑄 𝑈)))
4642, 45rmoi 3563 . . . . . . 7 ((∃*𝑟𝐴 (𝑃 𝑟) = (𝑄 𝑟) ∧ (𝑅𝐴 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑈𝐴 ∧ (𝑃 𝑈) = (𝑄 𝑈))) → 𝑅 = 𝑈)
4739, 46syl3an1br 1405 . . . . . 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 1375 . . . . 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 1371 . . . . 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 4707 . . . 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 233 . 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 140 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 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  ∃*wmo 2499  wne 2823  ∃*wrmo 2944   class class class wbr 4685  cfv 5926  (class class class)co 6690  lecple 15995  joincjn 16991  meetcmee 16992  Atomscatm 34868  CvLatclc 34870  HLchlt 34955  LHypclh 35588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-lhyp 35592
This theorem is referenced by: (None)
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