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Theorem cdlemg11b 37887
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 1208 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → ¬ (𝑅𝐺) (𝑃 𝑄))
2 simpl1 1188 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3 simpl31 1251 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝐺𝑇)
4 simpl2l 1223 . . . . . 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 37408 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐺) = ((𝑃 (𝐺𝑃)) 𝑊))
132, 3, 4, 12syl3anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑅𝐺) = ((𝑃 (𝐺𝑃)) 𝑊))
14 eqid 2824 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
15 simpl1l 1221 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝐾 ∈ HL)
1615hllatd 36609 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝐾 ∈ Lat)
17 simp2ll 1237 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → 𝑃𝐴)
1817adantr 484 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑃𝐴)
1914, 8atbase 36534 . . . . . . . . 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 37370 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑃 ∈ (Base‘𝐾)) → (𝐺𝑃) ∈ (Base‘𝐾))
222, 3, 20, 21syl3anc 1368 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑃) ∈ (Base‘𝐾))
2314, 6latjcl 17661 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐺𝑃) ∈ (Base‘𝐾)) → (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾))
2416, 20, 22, 23syl3anc 1368 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾))
25 simpl1r 1222 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑊𝐻)
2614, 9lhpbase 37243 . . . . . . . 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 17662 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐺𝑃)) 𝑊) ∈ (Base‘𝐾))
2916, 24, 27, 28syl3anc 1368 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → ((𝑃 (𝐺𝑃)) 𝑊) ∈ (Base‘𝐾))
30 simpl2r 1224 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑄𝐴)
3114, 8atbase 36534 . . . . . . . 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 17661 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
3416, 20, 32, 33syl3anc 1368 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑃 𝑄) ∈ (Base‘𝐾))
3514, 5, 7latmle1 17686 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐺𝑃)) 𝑊) (𝑃 (𝐺𝑃)))
3616, 24, 27, 35syl3anc 1368 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → ((𝑃 (𝐺𝑃)) 𝑊) (𝑃 (𝐺𝑃)))
3714, 5, 6latlej1 17670 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑃 (𝑃 𝑄))
3816, 20, 32, 37syl3anc 1368 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑃 (𝑃 𝑄))
3914, 9, 10ltrncl 37370 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑄 ∈ (Base‘𝐾)) → (𝐺𝑄) ∈ (Base‘𝐾))
402, 3, 32, 39syl3anc 1368 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑄) ∈ (Base‘𝐾))
4114, 5, 6latlej1 17670 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝐺𝑃) ∈ (Base‘𝐾) ∧ (𝐺𝑄) ∈ (Base‘𝐾)) → (𝐺𝑃) ((𝐺𝑃) (𝐺𝑄)))
4216, 22, 40, 41syl3anc 1368 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑃) ((𝐺𝑃) (𝐺𝑄)))
43 simpr 488 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄)))
4442, 43breqtrrd 5080 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑃) (𝑃 𝑄))
4514, 5, 6latjle12 17672 . . . . . . . 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 1369 . . . . . . 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 711 . . . . . 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 17668 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → ((𝑃 (𝐺𝑃)) 𝑊) (𝑃 𝑄))
4913, 48eqbrtrd 5074 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑅𝐺) (𝑃 𝑄))
5049ex 416 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → ((𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄)) → (𝑅𝐺) (𝑃 𝑄)))
5150necon3bd 3028 . 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014   class class class wbr 5052  cfv 6343  (class class class)co 7149  Basecbs 16483  lecple 16572  joincjn 17554  meetcmee 17555  Latclat 17655  Atomscatm 36508  HLchlt 36595  LHypclh 37229  LTrncltrn 37346  trLctrl 37403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404  df-poset 17556  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-lat 17656  df-ats 36512  df-atl 36543  df-cvlat 36567  df-hlat 36596  df-lhyp 37233  df-laut 37234  df-ldil 37349  df-ltrn 37350  df-trl 37404
This theorem is referenced by:  cdlemg12b  37889
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