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Theorem cdlemg2fv2 37616
Description: Value of a translation in terms of an associated atom. TODO: FIX COMMENT. TODO: Is this useful elsewhere e.g. around cdlemeg46fjv 37539 that use more complex proofs? TODO: Use ltrnj 37148 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 1128 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp23 1200 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
3 simp1l 1189 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝐾 ∈ HL)
43hllatd 36380 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝐾 ∈ Lat)
5 simp23l 1286 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑅𝐴)
6 eqid 2818 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
7 cdlemg2j.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
86, 7atbase 36305 . . . . . 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 1190 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑊𝐻)
11 simp21l 1282 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑃𝐴)
12 simp22l 1284 . . . . . 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 37226 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) → 𝑈 ∈ (Base‘𝐾))
193, 10, 11, 12, 18syl211anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑈 ∈ (Base‘𝐾))
206, 14latjcl 17649 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → (𝑅 𝑈) ∈ (Base‘𝐾))
214, 9, 19, 20syl3anc 1363 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝑅 𝑈) ∈ (Base‘𝐾))
22 simp23r 1287 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ¬ 𝑅 𝑊)
236, 13, 14latlej1 17658 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → 𝑅 (𝑅 𝑈))
244, 9, 19, 23syl3anc 1363 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑅 (𝑅 𝑈))
256, 16lhpbase 37014 . . . . . . . 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 17654 . . . . . . 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 1364 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 (𝑅 𝑈) ∧ (𝑅 𝑈) 𝑊) → 𝑅 𝑊))
2924, 28mpand 691 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 𝑈) 𝑊𝑅 𝑊))
3022, 29mtod 199 . . . 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 1130 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝐹𝑇)
33 eqid 2818 . . . . . . . 8 (0.‘𝐾) = (0.‘𝐾)
3413, 15, 33, 7, 16lhpmat 37046 . . . . . . 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 7160 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 𝑊) 𝑈) = ((0.‘𝐾) 𝑈))
376, 14, 7hlatjcl 36383 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
383, 11, 12, 37syl3anc 1363 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝑃 𝑄) ∈ (Base‘𝐾))
396, 13, 15latmle2 17675 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
404, 38, 26, 39syl3anc 1363 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑃 𝑄) 𝑊) 𝑊)
4117, 40eqbrtrid 5092 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → 𝑈 𝑊)
426, 13, 14, 15, 7atmod4i2 36883 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑈 𝑊) → ((𝑅 𝑊) 𝑈) = ((𝑅 𝑈) 𝑊))
433, 5, 19, 26, 41, 42syl131anc 1375 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 𝑊) 𝑈) = ((𝑅 𝑈) 𝑊))
44 hlol 36377 . . . . . . 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 36242 . . . . . 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 2861 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝑅 𝑈) 𝑊) = 𝑈)
4948oveq2d 7161 . . 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 37615 . . 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 1371 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝐹‘(𝑅 𝑈)) = ((𝐹𝑅) ((𝑅 𝑈) 𝑊)))
5348oveq2d 7161 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → ((𝐹𝑅) ((𝑅 𝑈) 𝑊)) = ((𝐹𝑅) 𝑈))
5452, 53eqtrd 2853 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 1079   = wceq 1528  wcel 2105   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  joincjn 17542  meetcmee 17543  0.cp0 17635  Latclat 17643  OLcol 36190  Atomscatm 36279  HLchlt 36366  LHypclh 37000  LTrncltrn 37117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-riotaBAD 35969
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-undef 7928  df-map 8397  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-p1 17638  df-lat 17644  df-clat 17706  df-oposet 36192  df-ol 36194  df-oml 36195  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367  df-llines 36514  df-lplanes 36515  df-lvols 36516  df-lines 36517  df-psubsp 36519  df-pmap 36520  df-padd 36812  df-lhyp 37004  df-laut 37005  df-ldil 37120  df-ltrn 37121  df-trl 37175
This theorem is referenced by:  cdlemg2l  37619
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