Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg11b Structured version   Visualization version   GIF version

Theorem cdlemg11b 40052
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 1209 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → ¬ (𝑅𝐺) (𝑃 𝑄))
2 simpl1 1189 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3 simpl31 1252 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝐺𝑇)
4 simpl2l 1224 . . . . . 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 39573 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐺) = ((𝑃 (𝐺𝑃)) 𝑊))
132, 3, 4, 12syl3anc 1369 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑅𝐺) = ((𝑃 (𝐺𝑃)) 𝑊))
14 eqid 2727 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
15 simpl1l 1222 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝐾 ∈ HL)
1615hllatd 38773 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝐾 ∈ Lat)
17 simp2ll 1238 . . . . . . . . . 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 38698 . . . . . . . . 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 39535 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑃 ∈ (Base‘𝐾)) → (𝐺𝑃) ∈ (Base‘𝐾))
222, 3, 20, 21syl3anc 1369 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑃) ∈ (Base‘𝐾))
2314, 6latjcl 18422 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐺𝑃) ∈ (Base‘𝐾)) → (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾))
2416, 20, 22, 23syl3anc 1369 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾))
25 simpl1r 1223 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑊𝐻)
2614, 9lhpbase 39408 . . . . . . . 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 18423 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐺𝑃)) 𝑊) ∈ (Base‘𝐾))
2916, 24, 27, 28syl3anc 1369 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → ((𝑃 (𝐺𝑃)) 𝑊) ∈ (Base‘𝐾))
30 simpl2r 1225 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑄𝐴)
3114, 8atbase 38698 . . . . . . . 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 18422 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
3416, 20, 32, 33syl3anc 1369 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑃 𝑄) ∈ (Base‘𝐾))
3514, 5, 7latmle1 18447 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐺𝑃)) 𝑊) (𝑃 (𝐺𝑃)))
3616, 24, 27, 35syl3anc 1369 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → ((𝑃 (𝐺𝑃)) 𝑊) (𝑃 (𝐺𝑃)))
3714, 5, 6latlej1 18431 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑃 (𝑃 𝑄))
3816, 20, 32, 37syl3anc 1369 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑃 (𝑃 𝑄))
3914, 9, 10ltrncl 39535 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑄 ∈ (Base‘𝐾)) → (𝐺𝑄) ∈ (Base‘𝐾))
402, 3, 32, 39syl3anc 1369 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑄) ∈ (Base‘𝐾))
4114, 5, 6latlej1 18431 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝐺𝑃) ∈ (Base‘𝐾) ∧ (𝐺𝑄) ∈ (Base‘𝐾)) → (𝐺𝑃) ((𝐺𝑃) (𝐺𝑄)))
4216, 22, 40, 41syl3anc 1369 . . . . . . . 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 18433 . . . . . . . 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 1370 . . . . . . 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 18429 . . . . 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 2949 . 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 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wne 2935   class class class wbr 5142  cfv 6542  (class class class)co 7414  Basecbs 17171  lecple 17231  joincjn 18294  meetcmee 18295  Latclat 18414  Atomscatm 38672  HLchlt 38759  LHypclh 39394  LTrncltrn 39511  trLctrl 39568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8838  df-poset 18296  df-lub 18329  df-glb 18330  df-join 18331  df-meet 18332  df-lat 18415  df-ats 38676  df-atl 38707  df-cvlat 38731  df-hlat 38760  df-lhyp 39398  df-laut 39399  df-ldil 39514  df-ltrn 39515  df-trl 39569
This theorem is referenced by:  cdlemg12b  40054
  Copyright terms: Public domain W3C validator