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Theorem cdlemg11b 40625
Description: TODO: FIX COMMENT. (Contributed by NM, 5-May-2013.)
Hypotheses
Ref Expression
cdlemg8.l = (le‘𝐾)
cdlemg8.j = (join‘𝐾)
cdlemg8.m = (meet‘𝐾)
cdlemg8.a 𝐴 = (Atoms‘𝐾)
cdlemg8.h 𝐻 = (LHyp‘𝐾)
cdlemg8.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg10.r 𝑅 = ((trL‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemg11b (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → (𝑃 𝑄) ≠ ((𝐺𝑃) (𝐺𝑄)))

Proof of Theorem cdlemg11b
StepHypRef Expression
1 simp33 1210 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → ¬ (𝑅𝐺) (𝑃 𝑄))
2 simpl1 1190 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3 simpl31 1253 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝐺𝑇)
4 simpl2l 1225 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
5 cdlemg8.l . . . . . . 7 = (le‘𝐾)
6 cdlemg8.j . . . . . . 7 = (join‘𝐾)
7 cdlemg8.m . . . . . . 7 = (meet‘𝐾)
8 cdlemg8.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
9 cdlemg8.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
10 cdlemg8.t . . . . . . 7 𝑇 = ((LTrn‘𝐾)‘𝑊)
11 cdlemg10.r . . . . . . 7 𝑅 = ((trL‘𝐾)‘𝑊)
125, 6, 7, 8, 9, 10, 11trlval2 40146 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐺) = ((𝑃 (𝐺𝑃)) 𝑊))
132, 3, 4, 12syl3anc 1370 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑅𝐺) = ((𝑃 (𝐺𝑃)) 𝑊))
14 eqid 2735 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
15 simpl1l 1223 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝐾 ∈ HL)
1615hllatd 39346 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝐾 ∈ Lat)
17 simp2ll 1239 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → 𝑃𝐴)
1817adantr 480 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑃𝐴)
1914, 8atbase 39271 . . . . . . . . 9 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
2018, 19syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑃 ∈ (Base‘𝐾))
2114, 9, 10ltrncl 40108 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑃 ∈ (Base‘𝐾)) → (𝐺𝑃) ∈ (Base‘𝐾))
222, 3, 20, 21syl3anc 1370 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑃) ∈ (Base‘𝐾))
2314, 6latjcl 18497 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐺𝑃) ∈ (Base‘𝐾)) → (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾))
2416, 20, 22, 23syl3anc 1370 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾))
25 simpl1r 1224 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑊𝐻)
2614, 9lhpbase 39981 . . . . . . . 8 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2725, 26syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑊 ∈ (Base‘𝐾))
2814, 7latmcl 18498 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐺𝑃)) 𝑊) ∈ (Base‘𝐾))
2916, 24, 27, 28syl3anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → ((𝑃 (𝐺𝑃)) 𝑊) ∈ (Base‘𝐾))
30 simpl2r 1226 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑄𝐴)
3114, 8atbase 39271 . . . . . . . 8 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3230, 31syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑄 ∈ (Base‘𝐾))
3314, 6latjcl 18497 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
3416, 20, 32, 33syl3anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑃 𝑄) ∈ (Base‘𝐾))
3514, 5, 7latmle1 18522 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐺𝑃)) 𝑊) (𝑃 (𝐺𝑃)))
3616, 24, 27, 35syl3anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → ((𝑃 (𝐺𝑃)) 𝑊) (𝑃 (𝐺𝑃)))
3714, 5, 6latlej1 18506 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑃 (𝑃 𝑄))
3816, 20, 32, 37syl3anc 1370 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑃 (𝑃 𝑄))
3914, 9, 10ltrncl 40108 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑄 ∈ (Base‘𝐾)) → (𝐺𝑄) ∈ (Base‘𝐾))
402, 3, 32, 39syl3anc 1370 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑄) ∈ (Base‘𝐾))
4114, 5, 6latlej1 18506 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝐺𝑃) ∈ (Base‘𝐾) ∧ (𝐺𝑄) ∈ (Base‘𝐾)) → (𝐺𝑃) ((𝐺𝑃) (𝐺𝑄)))
4216, 22, 40, 41syl3anc 1370 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑃) ((𝐺𝑃) (𝐺𝑄)))
43 simpr 484 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄)))
4442, 43breqtrrd 5176 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑃) (𝑃 𝑄))
4514, 5, 6latjle12 18508 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ (𝐺𝑃) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑃 (𝑃 𝑄) ∧ (𝐺𝑃) (𝑃 𝑄)) ↔ (𝑃 (𝐺𝑃)) (𝑃 𝑄)))
4616, 20, 22, 34, 45syl13anc 1371 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → ((𝑃 (𝑃 𝑄) ∧ (𝐺𝑃) (𝑃 𝑄)) ↔ (𝑃 (𝐺𝑃)) (𝑃 𝑄)))
4738, 44, 46mpbi2and 712 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑃 (𝐺𝑃)) (𝑃 𝑄))
4814, 5, 16, 29, 24, 34, 36, 47lattrd 18504 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → ((𝑃 (𝐺𝑃)) 𝑊) (𝑃 𝑄))
4913, 48eqbrtrd 5170 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑅𝐺) (𝑃 𝑄))
5049ex 412 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → ((𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄)) → (𝑅𝐺) (𝑃 𝑄)))
5150necon3bd 2952 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → (¬ (𝑅𝐺) (𝑃 𝑄) → (𝑃 𝑄) ≠ ((𝐺𝑃) (𝐺𝑄))))
521, 51mpd 15 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → (𝑃 𝑄) ≠ ((𝐺𝑃) (𝐺𝑄)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  Latclat 18489  Atomscatm 39245  HLchlt 39332  LHypclh 39967  LTrncltrn 40084  trLctrl 40141
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-mpo 7436  df-map 8867  df-poset 18371  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-lat 18490  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142
This theorem is referenced by:  cdlemg12b  40627
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