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Theorem dihjatcclem3 37228
Description: Lemma for dihjatcc 37230. (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 35826 . . . . . 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 36387 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐺𝑇)
121, 7, 8, 11syl3anc 1476 . . . 4 (𝜑𝐺𝑇)
13 dihjatcclem.q . . . . . 6 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
14 dihjatcc.dd . . . . . . 7 𝐷 = (𝑑𝑇 (𝑑𝐶) = 𝑄)
152, 3, 4, 9, 14ltrniotacl 36387 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐷𝑇)
161, 7, 13, 15syl3anc 1476 . . . . 5 (𝜑𝐷𝑇)
174, 9ltrncnv 35953 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐷𝑇) → 𝐷𝑇)
181, 16, 17syl2anc 573 . . . 4 (𝜑𝐷𝑇)
194, 9ltrnco 36527 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝐷𝑇) → (𝐺𝐷) ∈ 𝑇)
201, 12, 18, 19syl3anc 1476 . . 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 35971 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺𝐷) ∈ 𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑅‘(𝐺𝐷)) = ((𝑄 ((𝐺𝐷)‘𝑄)) 𝑊))
251, 20, 13, 24syl3anc 1476 . 2 (𝜑 → (𝑅‘(𝐺𝐷)) = ((𝑄 ((𝐺𝐷)‘𝑄)) 𝑊))
2613simpld 482 . . . . . . . 8 (𝜑𝑄𝐴)
272, 3, 4, 9ltrncoval 35952 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺𝑇𝐷𝑇) ∧ 𝑄𝐴) → ((𝐺𝐷)‘𝑄) = (𝐺‘(𝐷𝑄)))
281, 12, 18, 26, 27syl121anc 1481 . . . . . . 7 (𝜑 → ((𝐺𝐷)‘𝑄) = (𝐺‘(𝐷𝑄)))
292, 3, 4, 9, 14ltrniotacnvval 36390 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐷𝑄) = 𝐶)
301, 7, 13, 29syl3anc 1476 . . . . . . . . 9 (𝜑 → (𝐷𝑄) = 𝐶)
3130fveq2d 6337 . . . . . . . 8 (𝜑 → (𝐺‘(𝐷𝑄)) = (𝐺𝐶))
322, 3, 4, 9, 10ltrniotaval 36389 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐺𝐶) = 𝑃)
331, 7, 8, 32syl3anc 1476 . . . . . . . 8 (𝜑 → (𝐺𝐶) = 𝑃)
3431, 33eqtrd 2805 . . . . . . 7 (𝜑 → (𝐺‘(𝐷𝑄)) = 𝑃)
3528, 34eqtrd 2805 . . . . . 6 (𝜑 → ((𝐺𝐷)‘𝑄) = 𝑃)
3635oveq2d 6812 . . . . 5 (𝜑 → (𝑄 ((𝐺𝐷)‘𝑄)) = (𝑄 𝑃))
371simpld 482 . . . . . 6 (𝜑𝐾 ∈ HL)
388simpld 482 . . . . . 6 (𝜑𝑃𝐴)
3921, 3hlatjcom 35175 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) = (𝑄 𝑃))
4037, 38, 26, 39syl3anc 1476 . . . . 5 (𝜑 → (𝑃 𝑄) = (𝑄 𝑃))
4136, 40eqtr4d 2808 . . . 4 (𝜑 → (𝑄 ((𝐺𝐷)‘𝑄)) = (𝑃 𝑄))
4241oveq1d 6811 . . 3 (𝜑 → ((𝑄 ((𝐺𝐷)‘𝑄)) 𝑊) = ((𝑃 𝑄) 𝑊))
43 dihjatcclem.v . . 3 𝑉 = ((𝑃 𝑄) 𝑊)
4442, 43syl6eqr 2823 . 2 (𝜑 → ((𝑄 ((𝐺𝐷)‘𝑄)) 𝑊) = 𝑉)
4525, 44eqtrd 2805 1 (𝜑 → (𝑅‘(𝐺𝐷)) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wcel 2145   class class class wbr 4787  ccnv 5249  ccom 5254  cfv 6030  crio 6756  (class class class)co 6796  Basecbs 16064  lecple 16156  occoc 16157  joincjn 17152  meetcmee 17153  LSSumclsm 18256  Atomscatm 35070  HLchlt 35157  LHypclh 35791  LTrncltrn 35908  trLctrl 35966  TEndoctendo 36560  DVecHcdvh 36886  DIsoHcdih 37036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-riotaBAD 34759
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-undef 7555  df-map 8015  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-p1 17248  df-lat 17254  df-clat 17316  df-oposet 34983  df-ol 34985  df-oml 34986  df-covers 35073  df-ats 35074  df-atl 35105  df-cvlat 35129  df-hlat 35158  df-llines 35305  df-lplanes 35306  df-lvols 35307  df-lines 35308  df-psubsp 35310  df-pmap 35311  df-padd 35603  df-lhyp 35795  df-laut 35796  df-ldil 35911  df-ltrn 35912  df-trl 35967
This theorem is referenced by:  dihjatcclem4  37229
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