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Theorem cdlemg11b 35449
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 1097 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → ¬ (𝑅𝐺) (𝑃 𝑄))
2 simpl1 1062 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3 simpl31 1140 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝐺𝑇)
4 simpl2l 1112 . . . . . 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 34969 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐺) = ((𝑃 (𝐺𝑃)) 𝑊))
132, 3, 4, 12syl3anc 1323 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑅𝐺) = ((𝑃 (𝐺𝑃)) 𝑊))
14 eqid 2621 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
15 simpl1l 1110 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝐾 ∈ HL)
16 hllat 34169 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1715, 16syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝐾 ∈ Lat)
18 simp2ll 1126 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → 𝑃𝐴)
1918adantr 481 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑃𝐴)
2014, 8atbase 34095 . . . . . . . . 9 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
2119, 20syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑃 ∈ (Base‘𝐾))
2214, 9, 10ltrncl 34930 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑃 ∈ (Base‘𝐾)) → (𝐺𝑃) ∈ (Base‘𝐾))
232, 3, 21, 22syl3anc 1323 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑃) ∈ (Base‘𝐾))
2414, 6latjcl 16991 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐺𝑃) ∈ (Base‘𝐾)) → (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾))
2517, 21, 23, 24syl3anc 1323 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾))
26 simpl1r 1111 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑊𝐻)
2714, 9lhpbase 34803 . . . . . . . 8 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2826, 27syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑊 ∈ (Base‘𝐾))
2914, 7latmcl 16992 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐺𝑃)) 𝑊) ∈ (Base‘𝐾))
3017, 25, 28, 29syl3anc 1323 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → ((𝑃 (𝐺𝑃)) 𝑊) ∈ (Base‘𝐾))
31 simpl2r 1113 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑄𝐴)
3214, 8atbase 34095 . . . . . . . 8 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3331, 32syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑄 ∈ (Base‘𝐾))
3414, 6latjcl 16991 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
3517, 21, 33, 34syl3anc 1323 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑃 𝑄) ∈ (Base‘𝐾))
3614, 5, 7latmle1 17016 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 (𝐺𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐺𝑃)) 𝑊) (𝑃 (𝐺𝑃)))
3717, 25, 28, 36syl3anc 1323 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → ((𝑃 (𝐺𝑃)) 𝑊) (𝑃 (𝐺𝑃)))
3814, 5, 6latlej1 17000 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑃 (𝑃 𝑄))
3917, 21, 33, 38syl3anc 1323 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → 𝑃 (𝑃 𝑄))
4014, 9, 10ltrncl 34930 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑄 ∈ (Base‘𝐾)) → (𝐺𝑄) ∈ (Base‘𝐾))
412, 3, 33, 40syl3anc 1323 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑄) ∈ (Base‘𝐾))
4214, 5, 6latlej1 17000 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝐺𝑃) ∈ (Base‘𝐾) ∧ (𝐺𝑄) ∈ (Base‘𝐾)) → (𝐺𝑃) ((𝐺𝑃) (𝐺𝑄)))
4317, 23, 41, 42syl3anc 1323 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑃) ((𝐺𝑃) (𝐺𝑄)))
44 simpr 477 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄)))
4543, 44breqtrrd 4651 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝐺𝑃) (𝑃 𝑄))
4614, 5, 6latjle12 17002 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ (𝐺𝑃) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑃 (𝑃 𝑄) ∧ (𝐺𝑃) (𝑃 𝑄)) ↔ (𝑃 (𝐺𝑃)) (𝑃 𝑄)))
4717, 21, 23, 35, 46syl13anc 1325 . . . . . . 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)))
4839, 45, 47mpbi2and 955 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑃 (𝐺𝑃)) (𝑃 𝑄))
4914, 5, 17, 30, 25, 35, 37, 48lattrd 16998 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → ((𝑃 (𝐺𝑃)) 𝑊) (𝑃 𝑄))
5013, 49eqbrtrd 4645 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) ∧ (𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄))) → (𝑅𝐺) (𝑃 𝑄))
5150ex 450 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → ((𝑃 𝑄) = ((𝐺𝑃) (𝐺𝑄)) → (𝑅𝐺) (𝑃 𝑄)))
5251necon3bd 2804 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → (¬ (𝑅𝐺) (𝑃 𝑄) → (𝑃 𝑄) ≠ ((𝐺𝑃) (𝐺𝑄))))
531, 52mpd 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4623  cfv 5857  (class class class)co 6615  Basecbs 15800  lecple 15888  joincjn 16884  meetcmee 16885  Latclat 16985  Atomscatm 34069  HLchlt 34156  LHypclh 34789  LTrncltrn 34906  trLctrl 34964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-poset 16886  df-lub 16914  df-glb 16915  df-join 16916  df-meet 16917  df-lat 16986  df-ats 34073  df-atl 34104  df-cvlat 34128  df-hlat 34157  df-lhyp 34793  df-laut 34794  df-ldil 34909  df-ltrn 34910  df-trl 34965
This theorem is referenced by:  cdlemg12b  35451
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