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Theorem cdleme0moN 40208
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 1294 . 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 3033 . . 3 ((𝑅𝑃𝑅𝑄) ↔ ¬ (𝑅 = 𝑃𝑅 = 𝑄))
3 simpl33 1255 . . . . . 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 1293 . . . . . . 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 771 . . . . . . 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 773 . . . . . . 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 1254 . . . . . . 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 1223 . . . . . . . . 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 39341 . . . . . . . . 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 1289 . . . . . . . . 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 1291 . . . . . . . . 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 1253 . . . . . . . 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 39325 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
2111, 13, 15, 5, 16, 20syl131anc 1382 . . . . . . 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 1341 . . . . . 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 1196 . . . . . . . 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 1197 . . . . . . . 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 1290 . . . . . . . 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 1208 . . . . . . . 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 40028 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → 𝑈𝐴)
3123, 24, 12, 25, 14, 26, 30syl222anc 1385 . . . . . . 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 1190 . . . . . . 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 1250 . . . . . . 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 1251 . . . . . . 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 40205 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → ((𝑃 𝑈) = (𝑄 𝑈) ∧ 𝑈 𝑊))
3736simpld 494 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → (𝑃 𝑈) = (𝑄 𝑈))
3833, 34, 35, 16, 37syl121anc 1374 . . . . . 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 3378 . . . . . . 7 (∃*𝑟𝐴 (𝑃 𝑟) = (𝑄 𝑟) ↔ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
40 oveq2 7439 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑃 𝑟) = (𝑃 𝑅))
41 oveq2 7439 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
4240, 41eqeq12d 2751 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑃 𝑟) = (𝑄 𝑟) ↔ (𝑃 𝑅) = (𝑄 𝑅)))
43 oveq2 7439 . . . . . . . . 9 (𝑟 = 𝑈 → (𝑃 𝑟) = (𝑃 𝑈))
44 oveq2 7439 . . . . . . . . 9 (𝑟 = 𝑈 → (𝑄 𝑟) = (𝑄 𝑈))
4543, 44eqeq12d 2751 . . . . . . . 8 (𝑟 = 𝑈 → ((𝑃 𝑟) = (𝑄 𝑟) ↔ (𝑃 𝑈) = (𝑄 𝑈)))
4642, 45rmoi 3900 . . . . . . 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 1378 . . . . 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 495 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → 𝑈 𝑊)
5033, 34, 35, 16, 49syl121anc 1374 . . . . 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 5170 . . . 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 412 . . 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, 52biimtrrid 243 . 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 148 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 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  ∃*wmo 2536  wne 2938  ∃*wrmo 3377   class class class wbr 5148  cfv 6563  (class class class)co 7431  lecple 17305  joincjn 18369  meetcmee 18370  Atomscatm 39245  CvLatclc 39247  HLchlt 39332  LHypclh 39967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-lhyp 39971
This theorem is referenced by: (None)
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