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Theorem cdlemg2fv2 39459
Description: Value of a translation in terms of an associated atom. TODO: FIX COMMENT. TODO: Is this useful elsewhere e.g. around cdlemeg46fjv 39382 that use more complex proofs? TODO: Use ltrnj 38991 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 1136 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp23 1208 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
3 simp1l 1197 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝐾 ∈ HL)
43hllatd 38222 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝐾 ∈ Lat)
5 simp23l 1294 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑅𝐴)
6 eqid 2732 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
7 cdlemg2j.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
86, 7atbase 38147 . . . . . 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 1198 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑊𝐻)
11 simp21l 1290 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑃𝐴)
12 simp22l 1292 . . . . . 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 39069 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) → 𝑈 ∈ (Base‘𝐾))
193, 10, 11, 12, 18syl211anc 1376 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑈 ∈ (Base‘𝐾))
206, 14latjcl 18388 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → (𝑅 𝑈) ∈ (Base‘𝐾))
214, 9, 19, 20syl3anc 1371 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝑅 𝑈) ∈ (Base‘𝐾))
22 simp23r 1295 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ¬ 𝑅 𝑊)
236, 13, 14latlej1 18397 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → 𝑅 (𝑅 𝑈))
244, 9, 19, 23syl3anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑅 (𝑅 𝑈))
256, 16lhpbase 38857 . . . . . . . 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 18393 . . . . . . 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 1372 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 (𝑅 𝑈) ∧ (𝑅 𝑈) 𝑊) → 𝑅 𝑊))
2924, 28mpand 693 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 𝑈) 𝑊𝑅 𝑊))
3022, 29mtod 197 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ¬ (𝑅 𝑈) 𝑊)
3121, 30jca 512 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 𝑈) ∈ (Base‘𝐾) ∧ ¬ (𝑅 𝑈) 𝑊))
32 simp3 1138 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝐹𝑇)
33 eqid 2732 . . . . . . . 8 (0.‘𝐾) = (0.‘𝐾)
3413, 15, 33, 7, 16lhpmat 38889 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 𝑊) = (0.‘𝐾))
351, 2, 34syl2anc 584 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝑅 𝑊) = (0.‘𝐾))
3635oveq1d 7420 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 𝑊) 𝑈) = ((0.‘𝐾) 𝑈))
376, 14, 7hlatjcl 38225 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
383, 11, 12, 37syl3anc 1371 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝑃 𝑄) ∈ (Base‘𝐾))
396, 13, 15latmle2 18414 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
404, 38, 26, 39syl3anc 1371 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑃 𝑄) 𝑊) 𝑊)
4117, 40eqbrtrid 5182 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑈 𝑊)
426, 13, 14, 15, 7atmod4i2 38726 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑈 𝑊) → ((𝑅 𝑊) 𝑈) = ((𝑅 𝑈) 𝑊))
433, 5, 19, 26, 41, 42syl131anc 1383 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 𝑊) 𝑈) = ((𝑅 𝑈) 𝑊))
44 hlol 38219 . . . . . . 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 38084 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑈 ∈ (Base‘𝐾)) → ((0.‘𝐾) 𝑈) = 𝑈)
4745, 19, 46syl2anc 584 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((0.‘𝐾) 𝑈) = 𝑈)
4836, 43, 473eqtr3d 2780 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 𝑈) 𝑊) = 𝑈)
4948oveq2d 7421 . . 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 39458 . . 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 1379 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝐹‘(𝑅 𝑈)) = ((𝐹𝑅) ((𝑅 𝑈) 𝑊)))
5348oveq2d 7421 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝐹𝑅) ((𝑅 𝑈) 𝑊)) = ((𝐹𝑅) 𝑈))
5452, 53eqtrd 2772 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 396  w3a 1087   = wceq 1541  wcel 2106   class class class wbr 5147  cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  joincjn 18260  meetcmee 18261  0.cp0 18372  Latclat 18380  OLcol 38032  Atomscatm 38121  HLchlt 38208  LHypclh 38843  LTrncltrn 38960
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-riotaBAD 37811
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-undef 8254  df-map 8818  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-p1 18375  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209  df-llines 38357  df-lplanes 38358  df-lvols 38359  df-lines 38360  df-psubsp 38362  df-pmap 38363  df-padd 38655  df-lhyp 38847  df-laut 38848  df-ldil 38963  df-ltrn 38964  df-trl 39018
This theorem is referenced by:  cdlemg2l  39462
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