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Theorem dihjatcclem3 39883
Description: Lemma for dihjatcc 39885. (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 38482 . . . . . 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 39042 . . . . 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 39042 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐷𝑇)
161, 7, 13, 15syl3anc 1371 . . . . 5 (𝜑𝐷𝑇)
174, 9ltrncnv 38609 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐷𝑇) → 𝐷𝑇)
181, 16, 17syl2anc 584 . . . 4 (𝜑𝐷𝑇)
194, 9ltrnco 39182 . . . 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 38626 . . 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 38608 . . . . . . . 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 39045 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐷𝑄) = 𝐶)
301, 7, 13, 29syl3anc 1371 . . . . . . . . 9 (𝜑 → (𝐷𝑄) = 𝐶)
3130fveq2d 6846 . . . . . . . 8 (𝜑 → (𝐺‘(𝐷𝑄)) = (𝐺𝐶))
322, 3, 4, 9, 10ltrniotaval 39044 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐺𝐶) = 𝑃)
331, 7, 8, 32syl3anc 1371 . . . . . . . 8 (𝜑 → (𝐺𝐶) = 𝑃)
3431, 33eqtrd 2776 . . . . . . 7 (𝜑 → (𝐺‘(𝐷𝑄)) = 𝑃)
3528, 34eqtrd 2776 . . . . . 6 (𝜑 → ((𝐺𝐷)‘𝑄) = 𝑃)
3635oveq2d 7373 . . . . 5 (𝜑 → (𝑄 ((𝐺𝐷)‘𝑄)) = (𝑄 𝑃))
371simpld 495 . . . . . 6 (𝜑𝐾 ∈ HL)
388simpld 495 . . . . . 6 (𝜑𝑃𝐴)
3921, 3hlatjcom 37830 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) = (𝑄 𝑃))
4037, 38, 26, 39syl3anc 1371 . . . . 5 (𝜑 → (𝑃 𝑄) = (𝑄 𝑃))
4136, 40eqtr4d 2779 . . . 4 (𝜑 → (𝑄 ((𝐺𝐷)‘𝑄)) = (𝑃 𝑄))
4241oveq1d 7372 . . 3 (𝜑 → ((𝑄 ((𝐺𝐷)‘𝑄)) 𝑊) = ((𝑃 𝑄) 𝑊))
43 dihjatcclem.v . . 3 𝑉 = ((𝑃 𝑄) 𝑊)
4442, 43eqtr4di 2794 . 2 (𝜑 → ((𝑄 ((𝐺𝐷)‘𝑄)) 𝑊) = 𝑉)
4525, 44eqtrd 2776 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 5105  ccnv 5632  ccom 5637  cfv 6496  crio 7312  (class class class)co 7357  Basecbs 17083  lecple 17140  occoc 17141  joincjn 18200  meetcmee 18201  LSSumclsm 19416  Atomscatm 37725  HLchlt 37812  LHypclh 38447  LTrncltrn 38564  trLctrl 38621  TEndoctendo 39215  DVecHcdvh 39541  DIsoHcdih 39691
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-riotaBAD 37415
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-undef 8204  df-map 8767  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-p1 18315  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962  df-lvols 37963  df-lines 37964  df-psubsp 37966  df-pmap 37967  df-padd 38259  df-lhyp 38451  df-laut 38452  df-ldil 38567  df-ltrn 38568  df-trl 38622
This theorem is referenced by:  dihjatcclem4  39884
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