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Theorem cdlemg2fv2 37782
Description: Value of a translation in terms of an associated atom. TODO: FIX COMMENT. TODO: Is this useful elsewhere e.g. around cdlemeg46fjv 37705 that use more complex proofs? TODO: Use ltrnj 37314 to vastly simplify. (Contributed by NM, 23-Apr-2013.)
Hypotheses
Ref Expression
cdlemg2inv.h 𝐻 = (LHyp‘𝐾)
cdlemg2inv.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg2j.l = (le‘𝐾)
cdlemg2j.j = (join‘𝐾)
cdlemg2j.a 𝐴 = (Atoms‘𝐾)
cdlemg2j.m = (meet‘𝐾)
cdlemg2j.u 𝑈 = ((𝑃 𝑄) 𝑊)
Assertion
Ref Expression
cdlemg2fv2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝐹‘(𝑅 𝑈)) = ((𝐹𝑅) 𝑈))

Proof of Theorem cdlemg2fv2
StepHypRef Expression
1 simp1 1133 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp23 1205 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
3 simp1l 1194 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝐾 ∈ HL)
43hllatd 36546 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝐾 ∈ Lat)
5 simp23l 1291 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑅𝐴)
6 eqid 2821 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
7 cdlemg2j.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
86, 7atbase 36471 . . . . . 6 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
95, 8syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑅 ∈ (Base‘𝐾))
10 simp1r 1195 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑊𝐻)
11 simp21l 1287 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑃𝐴)
12 simp22l 1289 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑄𝐴)
13 cdlemg2j.l . . . . . . 7 = (le‘𝐾)
14 cdlemg2j.j . . . . . . 7 = (join‘𝐾)
15 cdlemg2j.m . . . . . . 7 = (meet‘𝐾)
16 cdlemg2inv.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
17 cdlemg2j.u . . . . . . 7 𝑈 = ((𝑃 𝑄) 𝑊)
1813, 14, 15, 7, 16, 17, 6cdleme0aa 37392 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) → 𝑈 ∈ (Base‘𝐾))
193, 10, 11, 12, 18syl211anc 1373 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑈 ∈ (Base‘𝐾))
206, 14latjcl 17640 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → (𝑅 𝑈) ∈ (Base‘𝐾))
214, 9, 19, 20syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝑅 𝑈) ∈ (Base‘𝐾))
22 simp23r 1292 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ¬ 𝑅 𝑊)
236, 13, 14latlej1 17649 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → 𝑅 (𝑅 𝑈))
244, 9, 19, 23syl3anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑅 (𝑅 𝑈))
256, 16lhpbase 37180 . . . . . . . 8 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2610, 25syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑊 ∈ (Base‘𝐾))
276, 13lattr 17645 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ (𝑅 𝑈) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑅 (𝑅 𝑈) ∧ (𝑅 𝑈) 𝑊) → 𝑅 𝑊))
284, 9, 21, 26, 27syl13anc 1369 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 (𝑅 𝑈) ∧ (𝑅 𝑈) 𝑊) → 𝑅 𝑊))
2924, 28mpand 694 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 𝑈) 𝑊𝑅 𝑊))
3022, 29mtod 201 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ¬ (𝑅 𝑈) 𝑊)
3121, 30jca 515 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 𝑈) ∈ (Base‘𝐾) ∧ ¬ (𝑅 𝑈) 𝑊))
32 simp3 1135 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝐹𝑇)
33 eqid 2821 . . . . . . . 8 (0.‘𝐾) = (0.‘𝐾)
3413, 15, 33, 7, 16lhpmat 37212 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 𝑊) = (0.‘𝐾))
351, 2, 34syl2anc 587 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝑅 𝑊) = (0.‘𝐾))
3635oveq1d 7145 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 𝑊) 𝑈) = ((0.‘𝐾) 𝑈))
376, 14, 7hlatjcl 36549 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
383, 11, 12, 37syl3anc 1368 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝑃 𝑄) ∈ (Base‘𝐾))
396, 13, 15latmle2 17666 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
404, 38, 26, 39syl3anc 1368 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑃 𝑄) 𝑊) 𝑊)
4117, 40eqbrtrid 5074 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑈 𝑊)
426, 13, 14, 15, 7atmod4i2 37049 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑈 𝑊) → ((𝑅 𝑊) 𝑈) = ((𝑅 𝑈) 𝑊))
433, 5, 19, 26, 41, 42syl131anc 1380 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 𝑊) 𝑈) = ((𝑅 𝑈) 𝑊))
44 hlol 36543 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ OL)
453, 44syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝐾 ∈ OL)
466, 14, 33olj02 36408 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑈 ∈ (Base‘𝐾)) → ((0.‘𝐾) 𝑈) = 𝑈)
4745, 19, 46syl2anc 587 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((0.‘𝐾) 𝑈) = 𝑈)
4836, 43, 473eqtr3d 2864 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 𝑈) 𝑊) = 𝑈)
4948oveq2d 7146 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝑅 ((𝑅 𝑈) 𝑊)) = (𝑅 𝑈))
50 cdlemg2inv.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
5116, 50, 13, 14, 7, 15, 6cdlemg2fv 37781 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ ((𝑅 𝑈) ∈ (Base‘𝐾) ∧ ¬ (𝑅 𝑈) 𝑊)) ∧ (𝐹𝑇 ∧ (𝑅 ((𝑅 𝑈) 𝑊)) = (𝑅 𝑈))) → (𝐹‘(𝑅 𝑈)) = ((𝐹𝑅) ((𝑅 𝑈) 𝑊)))
521, 2, 31, 32, 49, 51syl122anc 1376 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝐹‘(𝑅 𝑈)) = ((𝐹𝑅) ((𝑅 𝑈) 𝑊)))
5348oveq2d 7146 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝐹𝑅) ((𝑅 𝑈) 𝑊)) = ((𝐹𝑅) 𝑈))
5452, 53eqtrd 2856 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝐹‘(𝑅 𝑈)) = ((𝐹𝑅) 𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115   class class class wbr 5039  cfv 6328  (class class class)co 7130  Basecbs 16462  lecple 16551  joincjn 17533  meetcmee 17534  0.cp0 17626  Latclat 17634  OLcol 36356  Atomscatm 36445  HLchlt 36532  LHypclh 37166  LTrncltrn 37283
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 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-riotaBAD 36135
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-undef 7914  df-map 8383  df-proset 17517  df-poset 17535  df-plt 17547  df-lub 17563  df-glb 17564  df-join 17565  df-meet 17566  df-p0 17628  df-p1 17629  df-lat 17635  df-clat 17697  df-oposet 36358  df-ol 36360  df-oml 36361  df-covers 36448  df-ats 36449  df-atl 36480  df-cvlat 36504  df-hlat 36533  df-llines 36680  df-lplanes 36681  df-lvols 36682  df-lines 36683  df-psubsp 36685  df-pmap 36686  df-padd 36978  df-lhyp 37170  df-laut 37171  df-ldil 37286  df-ltrn 37287  df-trl 37341
This theorem is referenced by:  cdlemg2l  37785
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