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Theorem trlval3 40189
Description: The value of the trace of a lattice translation in terms of 2 atoms. TODO: Try to shorten proof. (Contributed by NM, 3-May-2013.)
Hypotheses
Ref Expression
trlval3.l = (le‘𝐾)
trlval3.j = (join‘𝐾)
trlval3.m = (meet‘𝐾)
trlval3.a 𝐴 = (Atoms‘𝐾)
trlval3.h 𝐻 = (LHyp‘𝐾)
trlval3.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
trlval3.r 𝑅 = ((trL‘𝐾)‘𝑊)
Assertion
Ref Expression
trlval3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))

Proof of Theorem trlval3
StepHypRef Expression
1 simpl1 1192 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl31 1255 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3 simpl2 1193 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝐹𝑇)
4 simpr 484 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝐹𝑃) = 𝑃)
5 trlval3.l . . . . 5 = (le‘𝐾)
6 eqid 2737 . . . . 5 (0.‘𝐾) = (0.‘𝐾)
7 trlval3.a . . . . 5 𝐴 = (Atoms‘𝐾)
8 trlval3.h . . . . 5 𝐻 = (LHyp‘𝐾)
9 trlval3.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
10 trlval3.r . . . . 5 𝑅 = ((trL‘𝐾)‘𝑊)
115, 6, 7, 8, 9, 10trl0 40172 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑅𝐹) = (0.‘𝐾))
121, 2, 3, 4, 11syl112anc 1376 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑅𝐹) = (0.‘𝐾))
13 simpl33 1257 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))
14 simpl1l 1225 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝐾 ∈ HL)
15 hlatl 39361 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
1614, 15syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝐾 ∈ AtLat)
174oveq2d 7447 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) = (𝑃 𝑃))
18 simp31l 1297 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → 𝑃𝐴)
1918adantr 480 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝑃𝐴)
20 trlval3.j . . . . . . . . 9 = (join‘𝐾)
2120, 7hlatjidm 39370 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑃 𝑃) = 𝑃)
2214, 19, 21syl2anc 584 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 𝑃) = 𝑃)
2317, 22eqtrd 2777 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) = 𝑃)
2423, 19eqeltrd 2841 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) ∈ 𝐴)
25 simp1 1137 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
26 simp2 1138 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → 𝐹𝑇)
27 simp31 1210 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
28 simp32 1211 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
295, 7, 8, 9ltrn2ateq 40182 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝐹𝑃) = 𝑃 ↔ (𝐹𝑄) = 𝑄))
3025, 26, 27, 28, 29syl13anc 1374 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → ((𝐹𝑃) = 𝑃 ↔ (𝐹𝑄) = 𝑄))
3130biimpa 476 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝐹𝑄) = 𝑄)
3231oveq2d 7447 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (𝐹𝑄)) = (𝑄 𝑄))
33 simp32l 1299 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → 𝑄𝐴)
3433adantr 480 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝑄𝐴)
3520, 7hlatjidm 39370 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑄 𝑄) = 𝑄)
3614, 34, 35syl2anc 584 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑄 𝑄) = 𝑄)
3732, 36eqtrd 2777 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (𝐹𝑄)) = 𝑄)
3837, 34eqeltrd 2841 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (𝐹𝑄)) ∈ 𝐴)
39 trlval3.m . . . . . 6 = (meet‘𝐾)
4039, 6, 7atnem0 39319 . . . . 5 ((𝐾 ∈ AtLat ∧ (𝑃 (𝐹𝑃)) ∈ 𝐴 ∧ (𝑄 (𝐹𝑄)) ∈ 𝐴) → ((𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)) ↔ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾)))
4116, 24, 38, 40syl3anc 1373 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → ((𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)) ↔ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾)))
4213, 41mpbid 232 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾))
4312, 42eqtr4d 2780 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))
44 simpl1 1192 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐾 ∈ HL ∧ 𝑊𝐻))
45 simpl2 1193 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐹𝑇)
46 simpl31 1255 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
475, 20, 39, 7, 8, 9, 10trlval2 40165 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
4844, 45, 46, 47syl3anc 1373 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
49 simpl1l 1225 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐾 ∈ HL)
5049hllatd 39365 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐾 ∈ Lat)
5118adantr 480 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑃𝐴)
525, 7, 8, 9ltrnat 40142 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃𝐴) → (𝐹𝑃) ∈ 𝐴)
5344, 45, 51, 52syl3anc 1373 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐹𝑃) ∈ 𝐴)
54 eqid 2737 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
5554, 20, 7hlatjcl 39368 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
5649, 51, 53, 55syl3anc 1373 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
57 simpl1r 1226 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑊𝐻)
5854, 8lhpbase 40000 . . . . . . 7 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
5957, 58syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑊 ∈ (Base‘𝐾))
6054, 5, 39latmle1 18509 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃)) 𝑊) (𝑃 (𝐹𝑃)))
6150, 56, 59, 60syl3anc 1373 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) 𝑊) (𝑃 (𝐹𝑃)))
6248, 61eqbrtrd 5165 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) (𝑃 (𝐹𝑃)))
63 simpl32 1256 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
645, 20, 39, 7, 8, 9, 10trlval2 40165 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑅𝐹) = ((𝑄 (𝐹𝑄)) 𝑊))
6544, 45, 63, 64syl3anc 1373 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) = ((𝑄 (𝐹𝑄)) 𝑊))
6633adantr 480 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑄𝐴)
675, 7, 8, 9ltrnat 40142 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑄𝐴) → (𝐹𝑄) ∈ 𝐴)
6844, 45, 66, 67syl3anc 1373 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐹𝑄) ∈ 𝐴)
6954, 20, 7hlatjcl 39368 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑄𝐴 ∧ (𝐹𝑄) ∈ 𝐴) → (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾))
7049, 66, 68, 69syl3anc 1373 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾))
7154, 5, 39latmle1 18509 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑄 (𝐹𝑄)) 𝑊) (𝑄 (𝐹𝑄)))
7250, 70, 59, 71syl3anc 1373 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑄 (𝐹𝑄)) 𝑊) (𝑄 (𝐹𝑄)))
7365, 72eqbrtrd 5165 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) (𝑄 (𝐹𝑄)))
7454, 8, 9, 10trlcl 40166 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) ∈ (Base‘𝐾))
7544, 45, 74syl2anc 584 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) ∈ (Base‘𝐾))
7654, 5, 39latlem12 18511 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑅𝐹) ∈ (Base‘𝐾) ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾))) → (((𝑅𝐹) (𝑃 (𝐹𝑃)) ∧ (𝑅𝐹) (𝑄 (𝐹𝑄))) ↔ (𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄)))))
7750, 75, 56, 70, 76syl13anc 1374 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (((𝑅𝐹) (𝑃 (𝐹𝑃)) ∧ (𝑅𝐹) (𝑄 (𝐹𝑄))) ↔ (𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄)))))
7862, 73, 77mpbi2and 712 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))
7949, 15syl 17 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐾 ∈ AtLat)
80 simpr 484 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐹𝑃) ≠ 𝑃)
815, 7, 8, 9, 10trlat 40171 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
8244, 46, 45, 80, 81syl112anc 1376 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) ∈ 𝐴)
8354, 39latmcl 18485 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ (Base‘𝐾))
8450, 56, 70, 83syl3anc 1373 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ (Base‘𝐾))
8554, 5, 6, 7atlen0 39311 . . . . . . 7 (((𝐾 ∈ AtLat ∧ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ (Base‘𝐾) ∧ (𝑅𝐹) ∈ 𝐴) ∧ (𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄)))) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ≠ (0.‘𝐾))
8679, 84, 82, 78, 85syl31anc 1375 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ≠ (0.‘𝐾))
8786neneqd 2945 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ¬ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾))
88 simpl33 1257 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))
8920, 39, 6, 72atmat0 39528 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) ∧ (𝑄𝐴 ∧ (𝐹𝑄) ∈ 𝐴 ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴 ∨ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾)))
9049, 51, 53, 66, 68, 88, 89syl33anc 1387 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴 ∨ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾)))
9190ord 865 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (¬ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴 → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾)))
9287, 91mt3d 148 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴)
935, 7atcmp 39312 . . . 4 ((𝐾 ∈ AtLat ∧ (𝑅𝐹) ∈ 𝐴 ∧ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴) → ((𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ↔ (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄)))))
9479, 82, 92, 93syl3anc 1373 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ↔ (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄)))))
9578, 94mpbid 232 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))
9643, 95pm2.61dane 3029 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  meetcmee 18358  0.cp0 18468  Latclat 18476  Atomscatm 39264  AtLatcal 39265  HLchlt 39351  LHypclh 39986  LTrncltrn 40103  trLctrl 40160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-p1 18471  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lhyp 39990  df-laut 39991  df-ldil 40106  df-ltrn 40107  df-trl 40161
This theorem is referenced by:  trlval4  40190
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