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Theorem cdlemg11b 40645
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 1211 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → ¬ (𝑅𝐺) (𝑃 𝑄))
2 simpl1 1191 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3 simpl31 1254 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝐺𝑇)
4 simpl2l 1226 . . . . . 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 40166 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐺) = ((𝑃 (𝐺𝑃)) 𝑊))
132, 3, 4, 12syl3anc 1372 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑅𝐺) = ((𝑃 (𝐺𝑃)) 𝑊))
14 eqid 2736 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
15 simpl1l 1224 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝐾 ∈ HL)
1615hllatd 39366 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝐾 ∈ Lat)
17 simp2ll 1240 . . . . . . . . . 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 39291 . . . . . . . . 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 40128 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑃 ∈ (Base‘𝐾)) → (𝐺𝑃) ∈ (Base‘𝐾))
222, 3, 20, 21syl3anc 1372 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑃) ∈ (Base‘𝐾))
2314, 6latjcl 18485 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐺𝑃) ∈ (Base‘𝐾)) → (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾))
2416, 20, 22, 23syl3anc 1372 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾))
25 simpl1r 1225 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑊𝐻)
2614, 9lhpbase 40001 . . . . . . . 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 18486 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐺𝑃)) 𝑊) ∈ (Base‘𝐾))
2916, 24, 27, 28syl3anc 1372 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → ((𝑃 (𝐺𝑃)) 𝑊) ∈ (Base‘𝐾))
30 simpl2r 1227 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑄𝐴)
3114, 8atbase 39291 . . . . . . . 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 18485 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
3416, 20, 32, 33syl3anc 1372 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑃 𝑄) ∈ (Base‘𝐾))
3514, 5, 7latmle1 18510 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐺𝑃)) 𝑊) (𝑃 (𝐺𝑃)))
3616, 24, 27, 35syl3anc 1372 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → ((𝑃 (𝐺𝑃)) 𝑊) (𝑃 (𝐺𝑃)))
3714, 5, 6latlej1 18494 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑃 (𝑃 𝑄))
3816, 20, 32, 37syl3anc 1372 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑃 (𝑃 𝑄))
3914, 9, 10ltrncl 40128 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑄 ∈ (Base‘𝐾)) → (𝐺𝑄) ∈ (Base‘𝐾))
402, 3, 32, 39syl3anc 1372 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑄) ∈ (Base‘𝐾))
4114, 5, 6latlej1 18494 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝐺𝑃) ∈ (Base‘𝐾) ∧ (𝐺𝑄) ∈ (Base‘𝐾)) → (𝐺𝑃) ((𝐺𝑃) (𝐺𝑄)))
4216, 22, 40, 41syl3anc 1372 . . . . . . . 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 5170 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑃) (𝑃 𝑄))
4514, 5, 6latjle12 18496 . . . . . . . 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 1373 . . . . . . 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 18492 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → ((𝑃 (𝐺𝑃)) 𝑊) (𝑃 𝑄))
4913, 48eqbrtrd 5164 . . . 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 2953 . 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 1539  wcel 2107  wne 2939   class class class wbr 5142  cfv 6560  (class class class)co 7432  Basecbs 17248  lecple 17305  joincjn 18358  meetcmee 18359  Latclat 18477  Atomscatm 39265  HLchlt 39352  LHypclh 39987  LTrncltrn 40104  trLctrl 40161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-poset 18360  df-lub 18392  df-glb 18393  df-join 18394  df-meet 18395  df-lat 18478  df-ats 39269  df-atl 39300  df-cvlat 39324  df-hlat 39353  df-lhyp 39991  df-laut 39992  df-ldil 40107  df-ltrn 40108  df-trl 40162
This theorem is referenced by:  cdlemg12b  40647
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