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Theorem cdlemd4 34300
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)
Hypotheses
Ref Expression
cdlemd4.l = (le‘𝐾)
cdlemd4.j = (join‘𝐾)
cdlemd4.a 𝐴 = (Atoms‘𝐾)
cdlemd4.h 𝐻 = (LHyp‘𝐾)
cdlemd4.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemd4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹𝑅) = (𝐺𝑅))

Proof of Theorem cdlemd4
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 simp11l 1165 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝐾 ∈ HL)
2 simp11r 1166 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑊𝐻)
3 simp21 1087 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
4 simp22 1088 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
5 simp231 1198 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑃𝑄)
6 cdlemd4.l . . . 4 = (le‘𝐾)
7 cdlemd4.j . . . 4 = (join‘𝐾)
8 cdlemd4.a . . . 4 𝐴 = (Atoms‘𝐾)
9 cdlemd4.h . . . 4 𝐻 = (LHyp‘𝐾)
106, 7, 8, 9cdlemb2 34139 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → ∃𝑠𝐴𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))
111, 2, 3, 4, 5, 10syl221anc 1329 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → ∃𝑠𝐴𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))
12 simpl11 1129 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
13 simpl12 1130 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → (𝐹𝑇𝐺𝑇))
14 simpl13 1131 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → 𝑅𝐴)
15 simpl21 1132 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
16 simprl 790 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → 𝑠𝐴)
17 simprrl 800 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → ¬ 𝑠 𝑊)
1816, 17jca 553 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → (𝑠𝐴 ∧ ¬ 𝑠 𝑊))
19 hllat 33462 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ Lat)
201, 19syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝐾 ∈ Lat)
2120adantr 480 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → 𝐾 ∈ Lat)
22 eqid 2610 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2322, 8atbase 33388 . . . . . 6 (𝑠𝐴𝑠 ∈ (Base‘𝐾))
2423ad2antrl 760 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → 𝑠 ∈ (Base‘𝐾))
25 simp21l 1171 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑃𝐴)
2622, 8atbase 33388 . . . . . . 7 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
2725, 26syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑃 ∈ (Base‘𝐾))
2827adantr 480 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → 𝑃 ∈ (Base‘𝐾))
29 simp22l 1173 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑄𝐴)
3022, 8atbase 33388 . . . . . . 7 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3129, 30syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑄 ∈ (Base‘𝐾))
3231adantr 480 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → 𝑄 ∈ (Base‘𝐾))
33 simprrr 801 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → ¬ 𝑠 (𝑃 𝑄))
3422, 6, 7latnlej1l 16841 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑠 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑠𝑃)
3534necomd 2837 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑠 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑃𝑠)
3621, 24, 28, 32, 33, 35syl131anc 1331 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → 𝑃𝑠)
37 simpl22 1133 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
38 simpl23 1134 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃))
396, 7, 8, 9cdlemd3 34299 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑠𝐴 ∧ ¬ 𝑠 (𝑃 𝑄))) → ¬ 𝑅 (𝑃 𝑠))
4012, 15, 37, 38, 14, 16, 33, 39syl133anc 1341 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → ¬ 𝑅 (𝑃 𝑠))
4136, 40jca 553 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → (𝑃𝑠 ∧ ¬ 𝑅 (𝑃 𝑠)))
42 simpl3l 1109 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → (𝐹𝑃) = (𝐺𝑃))
435adantr 480 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → 𝑃𝑄)
4443, 33jca 553 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → (𝑃𝑄 ∧ ¬ 𝑠 (𝑃 𝑄)))
45 simpl3 1059 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄)))
46 cdlemd4.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
476, 7, 8, 9, 46cdlemd2 34298 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑠𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑠 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹𝑠) = (𝐺𝑠))
4812, 13, 16, 15, 37, 44, 45, 47syl331anc 1343 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → (𝐹𝑠) = (𝐺𝑠))
496, 7, 8, 9, 46cdlemd2 34298 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑃𝑠 ∧ ¬ 𝑅 (𝑃 𝑠))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑠) = (𝐺𝑠))) → (𝐹𝑅) = (𝐺𝑅))
5012, 13, 14, 15, 18, 41, 42, 48, 49syl332anc 1349 . 2 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))) → (𝐹𝑅) = (𝐺𝑅))
5111, 50rexlimddv 3017 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹𝑅) = (𝐺𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wrex 2897   class class class wbr 4578  cfv 5790  (class class class)co 6527  Basecbs 15644  lecple 15724  joincjn 16716  Latclat 16817  Atomscatm 33362  HLchlt 33449  LHypclh 34082  LTrncltrn 34199
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-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-iin 4453  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-mpt2 6532  df-1st 7037  df-2nd 7038  df-map 7724  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-psubsp 33601  df-pmap 33602  df-padd 33894  df-lhyp 34086  df-laut 34087  df-ldil 34202  df-ltrn 34203
This theorem is referenced by:  cdlemd5  34301
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