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Theorem dihjatcclem3 39814
Description: Lemma for dihjatcc 39816. (Contributed by NM, 28-Sep-2014.)
Hypotheses
Ref Expression
dihjatcclem.b 𝐵 = (Base‘𝐾)
dihjatcclem.l = (le‘𝐾)
dihjatcclem.h 𝐻 = (LHyp‘𝐾)
dihjatcclem.j = (join‘𝐾)
dihjatcclem.m = (meet‘𝐾)
dihjatcclem.a 𝐴 = (Atoms‘𝐾)
dihjatcclem.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihjatcclem.s = (LSSum‘𝑈)
dihjatcclem.i 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihjatcclem.v 𝑉 = ((𝑃 𝑄) 𝑊)
dihjatcclem.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
dihjatcclem.p (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
dihjatcclem.q (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
dihjatcc.w 𝐶 = ((oc‘𝐾)‘𝑊)
dihjatcc.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihjatcc.r 𝑅 = ((trL‘𝐾)‘𝑊)
dihjatcc.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dihjatcc.g 𝐺 = (𝑑𝑇 (𝑑𝐶) = 𝑃)
dihjatcc.dd 𝐷 = (𝑑𝑇 (𝑑𝐶) = 𝑄)
Assertion
Ref Expression
dihjatcclem3 (𝜑 → (𝑅‘(𝐺𝐷)) = 𝑉)
Distinct variable groups:   ,𝑑   𝐴,𝑑   𝐵,𝑑   𝐶,𝑑   𝐻,𝑑   𝑃,𝑑   𝐾,𝑑   𝑄,𝑑   𝑇,𝑑   𝑊,𝑑
Allowed substitution hints:   𝜑(𝑑)   𝐷(𝑑)   (𝑑)   𝑅(𝑑)   𝑈(𝑑)   𝐸(𝑑)   𝐺(𝑑)   𝐼(𝑑)   (𝑑)   (𝑑)   𝑉(𝑑)

Proof of Theorem dihjatcclem3
StepHypRef Expression
1 dihjatcclem.k . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 dihjatcclem.l . . . . . . 7 = (le‘𝐾)
3 dihjatcclem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
4 dihjatcclem.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
5 dihjatcc.w . . . . . . 7 𝐶 = ((oc‘𝐾)‘𝑊)
62, 3, 4, 5lhpocnel2 38413 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐶𝐴 ∧ ¬ 𝐶 𝑊))
71, 6syl 17 . . . . 5 (𝜑 → (𝐶𝐴 ∧ ¬ 𝐶 𝑊))
8 dihjatcclem.p . . . . 5 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
9 dihjatcc.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
10 dihjatcc.g . . . . . 6 𝐺 = (𝑑𝑇 (𝑑𝐶) = 𝑃)
112, 3, 4, 9, 10ltrniotacl 38973 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐺𝑇)
121, 7, 8, 11syl3anc 1371 . . . 4 (𝜑𝐺𝑇)
13 dihjatcclem.q . . . . . 6 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
14 dihjatcc.dd . . . . . . 7 𝐷 = (𝑑𝑇 (𝑑𝐶) = 𝑄)
152, 3, 4, 9, 14ltrniotacl 38973 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐷𝑇)
161, 7, 13, 15syl3anc 1371 . . . . 5 (𝜑𝐷𝑇)
174, 9ltrncnv 38540 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐷𝑇) → 𝐷𝑇)
181, 16, 17syl2anc 584 . . . 4 (𝜑𝐷𝑇)
194, 9ltrnco 39113 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝐷𝑇) → (𝐺𝐷) ∈ 𝑇)
201, 12, 18, 19syl3anc 1371 . . 3 (𝜑 → (𝐺𝐷) ∈ 𝑇)
21 dihjatcclem.j . . . 4 = (join‘𝐾)
22 dihjatcclem.m . . . 4 = (meet‘𝐾)
23 dihjatcc.r . . . 4 𝑅 = ((trL‘𝐾)‘𝑊)
242, 21, 22, 3, 4, 9, 23trlval2 38557 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺𝐷) ∈ 𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑅‘(𝐺𝐷)) = ((𝑄 ((𝐺𝐷)‘𝑄)) 𝑊))
251, 20, 13, 24syl3anc 1371 . 2 (𝜑 → (𝑅‘(𝐺𝐷)) = ((𝑄 ((𝐺𝐷)‘𝑄)) 𝑊))
2613simpld 495 . . . . . . . 8 (𝜑𝑄𝐴)
272, 3, 4, 9ltrncoval 38539 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺𝑇𝐷𝑇) ∧ 𝑄𝐴) → ((𝐺𝐷)‘𝑄) = (𝐺‘(𝐷𝑄)))
281, 12, 18, 26, 27syl121anc 1375 . . . . . . 7 (𝜑 → ((𝐺𝐷)‘𝑄) = (𝐺‘(𝐷𝑄)))
292, 3, 4, 9, 14ltrniotacnvval 38976 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐷𝑄) = 𝐶)
301, 7, 13, 29syl3anc 1371 . . . . . . . . 9 (𝜑 → (𝐷𝑄) = 𝐶)
3130fveq2d 6843 . . . . . . . 8 (𝜑 → (𝐺‘(𝐷𝑄)) = (𝐺𝐶))
322, 3, 4, 9, 10ltrniotaval 38975 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐺𝐶) = 𝑃)
331, 7, 8, 32syl3anc 1371 . . . . . . . 8 (𝜑 → (𝐺𝐶) = 𝑃)
3431, 33eqtrd 2777 . . . . . . 7 (𝜑 → (𝐺‘(𝐷𝑄)) = 𝑃)
3528, 34eqtrd 2777 . . . . . 6 (𝜑 → ((𝐺𝐷)‘𝑄) = 𝑃)
3635oveq2d 7367 . . . . 5 (𝜑 → (𝑄 ((𝐺𝐷)‘𝑄)) = (𝑄 𝑃))
371simpld 495 . . . . . 6 (𝜑𝐾 ∈ HL)
388simpld 495 . . . . . 6 (𝜑𝑃𝐴)
3921, 3hlatjcom 37761 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) = (𝑄 𝑃))
4037, 38, 26, 39syl3anc 1371 . . . . 5 (𝜑 → (𝑃 𝑄) = (𝑄 𝑃))
4136, 40eqtr4d 2780 . . . 4 (𝜑 → (𝑄 ((𝐺𝐷)‘𝑄)) = (𝑃 𝑄))
4241oveq1d 7366 . . 3 (𝜑 → ((𝑄 ((𝐺𝐷)‘𝑄)) 𝑊) = ((𝑃 𝑄) 𝑊))
43 dihjatcclem.v . . 3 𝑉 = ((𝑃 𝑄) 𝑊)
4442, 43eqtr4di 2795 . 2 (𝜑 → ((𝑄 ((𝐺𝐷)‘𝑄)) 𝑊) = 𝑉)
4525, 44eqtrd 2777 1 (𝜑 → (𝑅‘(𝐺𝐷)) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106   class class class wbr 5103  ccnv 5630  ccom 5635  cfv 6493  crio 7306  (class class class)co 7351  Basecbs 17042  lecple 17099  occoc 17100  joincjn 18159  meetcmee 18160  LSSumclsm 19374  Atomscatm 37656  HLchlt 37743  LHypclh 38378  LTrncltrn 38495  trLctrl 38552  TEndoctendo 39146  DVecHcdvh 39472  DIsoHcdih 39622
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 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-riotaBAD 37346
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-iin 4955  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-undef 8196  df-map 8725  df-proset 18143  df-poset 18161  df-plt 18178  df-lub 18194  df-glb 18195  df-join 18196  df-meet 18197  df-p0 18273  df-p1 18274  df-lat 18280  df-clat 18347  df-oposet 37569  df-ol 37571  df-oml 37572  df-covers 37659  df-ats 37660  df-atl 37691  df-cvlat 37715  df-hlat 37744  df-llines 37892  df-lplanes 37893  df-lvols 37894  df-lines 37895  df-psubsp 37897  df-pmap 37898  df-padd 38190  df-lhyp 38382  df-laut 38383  df-ldil 38498  df-ltrn 38499  df-trl 38553
This theorem is referenced by:  dihjatcclem4  39815
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