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Theorem dihjatcclem3 39171
Description: Lemma for dihjatcc 39173. (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 37770 . . . . . 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 38330 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐺𝑇)
121, 7, 8, 11syl3anc 1373 . . . 4 (𝜑𝐺𝑇)
13 dihjatcclem.q . . . . . 6 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
14 dihjatcc.dd . . . . . . 7 𝐷 = (𝑑𝑇 (𝑑𝐶) = 𝑄)
152, 3, 4, 9, 14ltrniotacl 38330 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐷𝑇)
161, 7, 13, 15syl3anc 1373 . . . . 5 (𝜑𝐷𝑇)
174, 9ltrncnv 37897 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐷𝑇) → 𝐷𝑇)
181, 16, 17syl2anc 587 . . . 4 (𝜑𝐷𝑇)
194, 9ltrnco 38470 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝐷𝑇) → (𝐺𝐷) ∈ 𝑇)
201, 12, 18, 19syl3anc 1373 . . 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 37914 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺𝐷) ∈ 𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑅‘(𝐺𝐷)) = ((𝑄 ((𝐺𝐷)‘𝑄)) 𝑊))
251, 20, 13, 24syl3anc 1373 . 2 (𝜑 → (𝑅‘(𝐺𝐷)) = ((𝑄 ((𝐺𝐷)‘𝑄)) 𝑊))
2613simpld 498 . . . . . . . 8 (𝜑𝑄𝐴)
272, 3, 4, 9ltrncoval 37896 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺𝑇𝐷𝑇) ∧ 𝑄𝐴) → ((𝐺𝐷)‘𝑄) = (𝐺‘(𝐷𝑄)))
281, 12, 18, 26, 27syl121anc 1377 . . . . . . 7 (𝜑 → ((𝐺𝐷)‘𝑄) = (𝐺‘(𝐷𝑄)))
292, 3, 4, 9, 14ltrniotacnvval 38333 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐷𝑄) = 𝐶)
301, 7, 13, 29syl3anc 1373 . . . . . . . . 9 (𝜑 → (𝐷𝑄) = 𝐶)
3130fveq2d 6721 . . . . . . . 8 (𝜑 → (𝐺‘(𝐷𝑄)) = (𝐺𝐶))
322, 3, 4, 9, 10ltrniotaval 38332 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐺𝐶) = 𝑃)
331, 7, 8, 32syl3anc 1373 . . . . . . . 8 (𝜑 → (𝐺𝐶) = 𝑃)
3431, 33eqtrd 2777 . . . . . . 7 (𝜑 → (𝐺‘(𝐷𝑄)) = 𝑃)
3528, 34eqtrd 2777 . . . . . 6 (𝜑 → ((𝐺𝐷)‘𝑄) = 𝑃)
3635oveq2d 7229 . . . . 5 (𝜑 → (𝑄 ((𝐺𝐷)‘𝑄)) = (𝑄 𝑃))
371simpld 498 . . . . . 6 (𝜑𝐾 ∈ HL)
388simpld 498 . . . . . 6 (𝜑𝑃𝐴)
3921, 3hlatjcom 37119 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) = (𝑄 𝑃))
4037, 38, 26, 39syl3anc 1373 . . . . 5 (𝜑 → (𝑃 𝑄) = (𝑄 𝑃))
4136, 40eqtr4d 2780 . . . 4 (𝜑 → (𝑄 ((𝐺𝐷)‘𝑄)) = (𝑃 𝑄))
4241oveq1d 7228 . . 3 (𝜑 → ((𝑄 ((𝐺𝐷)‘𝑄)) 𝑊) = ((𝑃 𝑄) 𝑊))
43 dihjatcclem.v . . 3 𝑉 = ((𝑃 𝑄) 𝑊)
4442, 43eqtr4di 2796 . 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 399   = wceq 1543  wcel 2110   class class class wbr 5053  ccnv 5550  ccom 5555  cfv 6380  crio 7169  (class class class)co 7213  Basecbs 16760  lecple 16809  occoc 16810  joincjn 17818  meetcmee 17819  LSSumclsm 19023  Atomscatm 37014  HLchlt 37101  LHypclh 37735  LTrncltrn 37852  trLctrl 37909  TEndoctendo 38503  DVecHcdvh 38829  DIsoHcdih 38979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-riotaBAD 36704
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-undef 8015  df-map 8510  df-proset 17802  df-poset 17820  df-plt 17836  df-lub 17852  df-glb 17853  df-join 17854  df-meet 17855  df-p0 17931  df-p1 17932  df-lat 17938  df-clat 18005  df-oposet 36927  df-ol 36929  df-oml 36930  df-covers 37017  df-ats 37018  df-atl 37049  df-cvlat 37073  df-hlat 37102  df-llines 37249  df-lplanes 37250  df-lvols 37251  df-lines 37252  df-psubsp 37254  df-pmap 37255  df-padd 37547  df-lhyp 37739  df-laut 37740  df-ldil 37855  df-ltrn 37856  df-trl 37910
This theorem is referenced by:  dihjatcclem4  39172
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