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Theorem cdlemc6 37199
Description: Lemma for cdlemc 37200. (Contributed by NM, 26-May-2012.)
Hypotheses
Ref Expression
cdlemc3.l = (le‘𝐾)
cdlemc3.j = (join‘𝐾)
cdlemc3.m = (meet‘𝐾)
cdlemc3.a 𝐴 = (Atoms‘𝐾)
cdlemc3.h 𝐻 = (LHyp‘𝐾)
cdlemc3.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemc3.r 𝑅 = ((trL‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemc6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝐹𝑄) = ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))))

Proof of Theorem cdlemc6
StepHypRef Expression
1 simp1l 1191 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → 𝐾 ∈ HL)
2 simp22l 1286 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → 𝑃𝐴)
3 simp23l 1288 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → 𝑄𝐴)
4 cdlemc3.j . . . . . 6 = (join‘𝐾)
5 cdlemc3.a . . . . . 6 𝐴 = (Atoms‘𝐾)
64, 5hlatjcom 36371 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) = (𝑄 𝑃))
71, 2, 3, 6syl3anc 1365 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝑃 𝑄) = (𝑄 𝑃))
87oveq2d 7167 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (𝑃 𝑄)) = (𝑄 (𝑄 𝑃)))
91hllatd 36367 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → 𝐾 ∈ Lat)
10 eqid 2825 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
1110, 5atbase 36292 . . . . 5 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
123, 11syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → 𝑄 ∈ (Base‘𝐾))
1310, 5atbase 36292 . . . . 5 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
142, 13syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → 𝑃 ∈ (Base‘𝐾))
15 cdlemc3.m . . . . 5 = (meet‘𝐾)
1610, 4, 15latabs2 17690 . . . 4 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑄 (𝑄 𝑃)) = 𝑄)
179, 12, 14, 16syl3anc 1365 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (𝑄 𝑃)) = 𝑄)
188, 17eqtrd 2860 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (𝑃 𝑄)) = 𝑄)
19 simp1 1130 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝐾 ∈ HL ∧ 𝑊𝐻))
20 simp22 1201 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
21 simp21 1200 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → 𝐹𝑇)
22 simp3 1132 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝐹𝑃) = 𝑃)
23 cdlemc3.l . . . . . . 7 = (le‘𝐾)
24 eqid 2825 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
25 cdlemc3.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
26 cdlemc3.t . . . . . . 7 𝑇 = ((LTrn‘𝐾)‘𝑊)
27 cdlemc3.r . . . . . . 7 𝑅 = ((trL‘𝐾)‘𝑊)
2823, 24, 5, 25, 26, 27trl0 37173 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑅𝐹) = (0.‘𝐾))
2919, 20, 21, 22, 28syl112anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝑅𝐹) = (0.‘𝐾))
3029oveq2d 7167 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (𝑅𝐹)) = (𝑄 (0.‘𝐾)))
31 hlol 36364 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ OL)
321, 31syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → 𝐾 ∈ OL)
3310, 4, 24olj01 36228 . . . . 5 ((𝐾 ∈ OL ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑄 (0.‘𝐾)) = 𝑄)
3432, 12, 33syl2anc 584 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (0.‘𝐾)) = 𝑄)
3530, 34eqtrd 2860 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (𝑅𝐹)) = 𝑄)
3622oveq1d 7166 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) = (𝑃 ((𝑃 𝑄) 𝑊)))
3710, 4, 5hlatjcl 36370 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
381, 2, 3, 37syl3anc 1365 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝑃 𝑄) ∈ (Base‘𝐾))
39 simp1r 1192 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → 𝑊𝐻)
4010, 25lhpbase 37001 . . . . . . 7 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
4139, 40syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → 𝑊 ∈ (Base‘𝐾))
4210, 15latmcl 17654 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
439, 38, 41, 42syl3anc 1365 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
4410, 4latjcom 17661 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾)) → (𝑃 ((𝑃 𝑄) 𝑊)) = (((𝑃 𝑄) 𝑊) 𝑃))
459, 14, 43, 44syl3anc 1365 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝑃 ((𝑃 𝑄) 𝑊)) = (((𝑃 𝑄) 𝑊) 𝑃))
4623, 4, 5hlatlej1 36378 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑃 (𝑃 𝑄))
471, 2, 3, 46syl3anc 1365 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → 𝑃 (𝑃 𝑄))
4810, 23, 4, 15, 5atmod2i1 36864 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴 ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑃 (𝑃 𝑄)) → (((𝑃 𝑄) 𝑊) 𝑃) = ((𝑃 𝑄) (𝑊 𝑃)))
491, 2, 38, 41, 47, 48syl131anc 1377 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (((𝑃 𝑄) 𝑊) 𝑃) = ((𝑃 𝑄) (𝑊 𝑃)))
50 eqid 2825 . . . . . . . 8 (1.‘𝐾) = (1.‘𝐾)
5123, 4, 50, 5, 25lhpjat1 37023 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑊 𝑃) = (1.‘𝐾))
521, 39, 20, 51syl21anc 835 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝑊 𝑃) = (1.‘𝐾))
5352oveq2d 7167 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → ((𝑃 𝑄) (𝑊 𝑃)) = ((𝑃 𝑄) (1.‘𝐾)))
5410, 15, 50olm11 36230 . . . . . 6 ((𝐾 ∈ OL ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (1.‘𝐾)) = (𝑃 𝑄))
5532, 38, 54syl2anc 584 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → ((𝑃 𝑄) (1.‘𝐾)) = (𝑃 𝑄))
5649, 53, 553eqtrd 2864 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (((𝑃 𝑄) 𝑊) 𝑃) = (𝑃 𝑄))
5736, 45, 563eqtrd 2864 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) = (𝑃 𝑄))
5835, 57oveq12d 7169 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) = (𝑄 (𝑃 𝑄)))
5923, 5, 25, 26ltrnateq 37184 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝐹𝑄) = 𝑄)
6018, 58, 593eqtr4rd 2871 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝐹𝑄) = ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107   class class class wbr 5062  cfv 6351  (class class class)co 7151  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  0.cp0 17639  1.cp1 17640  Latclat 17647  OLcol 36177  Atomscatm 36266  HLchlt 36353  LHypclh 36987  LTrncltrn 37104  trLctrl 37161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-map 8401  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36179  df-ol 36181  df-oml 36182  df-covers 36269  df-ats 36270  df-atl 36301  df-cvlat 36325  df-hlat 36354  df-psubsp 36506  df-pmap 36507  df-padd 36799  df-lhyp 36991  df-laut 36992  df-ldil 37107  df-ltrn 37108  df-trl 37162
This theorem is referenced by:  cdlemc  37200
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