Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihjatcclem3 Structured version   Visualization version   GIF version

Theorem dihjatcclem3 40291
Description: Lemma for dihjatcc 40293. (Contributed by NM, 28-Sep-2014.)
Hypotheses
Ref Expression
dihjatcclem.b 𝐡 = (Baseβ€˜πΎ)
dihjatcclem.l ≀ = (leβ€˜πΎ)
dihjatcclem.h 𝐻 = (LHypβ€˜πΎ)
dihjatcclem.j ∨ = (joinβ€˜πΎ)
dihjatcclem.m ∧ = (meetβ€˜πΎ)
dihjatcclem.a 𝐴 = (Atomsβ€˜πΎ)
dihjatcclem.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
dihjatcclem.s βŠ• = (LSSumβ€˜π‘ˆ)
dihjatcclem.i 𝐼 = ((DIsoHβ€˜πΎ)β€˜π‘Š)
dihjatcclem.v 𝑉 = ((𝑃 ∨ 𝑄) ∧ π‘Š)
dihjatcclem.k (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
dihjatcclem.p (πœ‘ β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
dihjatcclem.q (πœ‘ β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
dihjatcc.w 𝐢 = ((ocβ€˜πΎ)β€˜π‘Š)
dihjatcc.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dihjatcc.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
dihjatcc.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
dihjatcc.g 𝐺 = (℩𝑑 ∈ 𝑇 (π‘‘β€˜πΆ) = 𝑃)
dihjatcc.dd 𝐷 = (℩𝑑 ∈ 𝑇 (π‘‘β€˜πΆ) = 𝑄)
Assertion
Ref Expression
dihjatcclem3 (πœ‘ β†’ (π‘…β€˜(𝐺 ∘ ◑𝐷)) = 𝑉)
Distinct variable groups:   ≀ ,𝑑   𝐴,𝑑   𝐡,𝑑   𝐢,𝑑   𝐻,𝑑   𝑃,𝑑   𝐾,𝑑   𝑄,𝑑   𝑇,𝑑   π‘Š,𝑑
Allowed substitution hints:   πœ‘(𝑑)   𝐷(𝑑)   βŠ• (𝑑)   𝑅(𝑑)   π‘ˆ(𝑑)   𝐸(𝑑)   𝐺(𝑑)   𝐼(𝑑)   ∨ (𝑑)   ∧ (𝑑)   𝑉(𝑑)

Proof of Theorem dihjatcclem3
StepHypRef Expression
1 dihjatcclem.k . . 3 (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 dihjatcclem.l . . . . . . 7 ≀ = (leβ€˜πΎ)
3 dihjatcclem.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
4 dihjatcclem.h . . . . . . 7 𝐻 = (LHypβ€˜πΎ)
5 dihjatcc.w . . . . . . 7 𝐢 = ((ocβ€˜πΎ)β€˜π‘Š)
62, 3, 4, 5lhpocnel2 38890 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝐢 ∈ 𝐴 ∧ Β¬ 𝐢 ≀ π‘Š))
71, 6syl 17 . . . . 5 (πœ‘ β†’ (𝐢 ∈ 𝐴 ∧ Β¬ 𝐢 ≀ π‘Š))
8 dihjatcclem.p . . . . 5 (πœ‘ β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
9 dihjatcc.t . . . . . 6 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
10 dihjatcc.g . . . . . 6 𝐺 = (℩𝑑 ∈ 𝑇 (π‘‘β€˜πΆ) = 𝑃)
112, 3, 4, 9, 10ltrniotacl 39450 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐢 ∈ 𝐴 ∧ Β¬ 𝐢 ≀ π‘Š) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ 𝐺 ∈ 𝑇)
121, 7, 8, 11syl3anc 1372 . . . 4 (πœ‘ β†’ 𝐺 ∈ 𝑇)
13 dihjatcclem.q . . . . . 6 (πœ‘ β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
14 dihjatcc.dd . . . . . . 7 𝐷 = (℩𝑑 ∈ 𝑇 (π‘‘β€˜πΆ) = 𝑄)
152, 3, 4, 9, 14ltrniotacl 39450 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐢 ∈ 𝐴 ∧ Β¬ 𝐢 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ 𝐷 ∈ 𝑇)
161, 7, 13, 15syl3anc 1372 . . . . 5 (πœ‘ β†’ 𝐷 ∈ 𝑇)
174, 9ltrncnv 39017 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) β†’ ◑𝐷 ∈ 𝑇)
181, 16, 17syl2anc 585 . . . 4 (πœ‘ β†’ ◑𝐷 ∈ 𝑇)
194, 9ltrnco 39590 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ◑𝐷 ∈ 𝑇) β†’ (𝐺 ∘ ◑𝐷) ∈ 𝑇)
201, 12, 18, 19syl3anc 1372 . . 3 (πœ‘ β†’ (𝐺 ∘ ◑𝐷) ∈ 𝑇)
21 dihjatcclem.j . . . 4 ∨ = (joinβ€˜πΎ)
22 dihjatcclem.m . . . 4 ∧ = (meetβ€˜πΎ)
23 dihjatcc.r . . . 4 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
242, 21, 22, 3, 4, 9, 23trlval2 39034 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐺 ∘ ◑𝐷) ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (π‘…β€˜(𝐺 ∘ ◑𝐷)) = ((𝑄 ∨ ((𝐺 ∘ ◑𝐷)β€˜π‘„)) ∧ π‘Š))
251, 20, 13, 24syl3anc 1372 . 2 (πœ‘ β†’ (π‘…β€˜(𝐺 ∘ ◑𝐷)) = ((𝑄 ∨ ((𝐺 ∘ ◑𝐷)β€˜π‘„)) ∧ π‘Š))
2613simpld 496 . . . . . . . 8 (πœ‘ β†’ 𝑄 ∈ 𝐴)
272, 3, 4, 9ltrncoval 39016 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ ◑𝐷 ∈ 𝑇) ∧ 𝑄 ∈ 𝐴) β†’ ((𝐺 ∘ ◑𝐷)β€˜π‘„) = (πΊβ€˜(β—‘π·β€˜π‘„)))
281, 12, 18, 26, 27syl121anc 1376 . . . . . . 7 (πœ‘ β†’ ((𝐺 ∘ ◑𝐷)β€˜π‘„) = (πΊβ€˜(β—‘π·β€˜π‘„)))
292, 3, 4, 9, 14ltrniotacnvval 39453 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐢 ∈ 𝐴 ∧ Β¬ 𝐢 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (β—‘π·β€˜π‘„) = 𝐢)
301, 7, 13, 29syl3anc 1372 . . . . . . . . 9 (πœ‘ β†’ (β—‘π·β€˜π‘„) = 𝐢)
3130fveq2d 6896 . . . . . . . 8 (πœ‘ β†’ (πΊβ€˜(β—‘π·β€˜π‘„)) = (πΊβ€˜πΆ))
322, 3, 4, 9, 10ltrniotaval 39452 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐢 ∈ 𝐴 ∧ Β¬ 𝐢 ≀ π‘Š) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (πΊβ€˜πΆ) = 𝑃)
331, 7, 8, 32syl3anc 1372 . . . . . . . 8 (πœ‘ β†’ (πΊβ€˜πΆ) = 𝑃)
3431, 33eqtrd 2773 . . . . . . 7 (πœ‘ β†’ (πΊβ€˜(β—‘π·β€˜π‘„)) = 𝑃)
3528, 34eqtrd 2773 . . . . . 6 (πœ‘ β†’ ((𝐺 ∘ ◑𝐷)β€˜π‘„) = 𝑃)
3635oveq2d 7425 . . . . 5 (πœ‘ β†’ (𝑄 ∨ ((𝐺 ∘ ◑𝐷)β€˜π‘„)) = (𝑄 ∨ 𝑃))
371simpld 496 . . . . . 6 (πœ‘ β†’ 𝐾 ∈ HL)
388simpld 496 . . . . . 6 (πœ‘ β†’ 𝑃 ∈ 𝐴)
3921, 3hlatjcom 38238 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
4037, 38, 26, 39syl3anc 1372 . . . . 5 (πœ‘ β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
4136, 40eqtr4d 2776 . . . 4 (πœ‘ β†’ (𝑄 ∨ ((𝐺 ∘ ◑𝐷)β€˜π‘„)) = (𝑃 ∨ 𝑄))
4241oveq1d 7424 . . 3 (πœ‘ β†’ ((𝑄 ∨ ((𝐺 ∘ ◑𝐷)β€˜π‘„)) ∧ π‘Š) = ((𝑃 ∨ 𝑄) ∧ π‘Š))
43 dihjatcclem.v . . 3 𝑉 = ((𝑃 ∨ 𝑄) ∧ π‘Š)
4442, 43eqtr4di 2791 . 2 (πœ‘ β†’ ((𝑄 ∨ ((𝐺 ∘ ◑𝐷)β€˜π‘„)) ∧ π‘Š) = 𝑉)
4525, 44eqtrd 2773 1 (πœ‘ β†’ (π‘…β€˜(𝐺 ∘ ◑𝐷)) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   class class class wbr 5149  β—‘ccnv 5676   ∘ ccom 5681  β€˜cfv 6544  β„©crio 7364  (class class class)co 7409  Basecbs 17144  lecple 17204  occoc 17205  joincjn 18264  meetcmee 18265  LSSumclsm 19502  Atomscatm 38133  HLchlt 38220  LHypclh 38855  LTrncltrn 38972  trLctrl 39029  TEndoctendo 39623  DVecHcdvh 39949  DIsoHcdih 40099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-riotaBAD 37823
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-undef 8258  df-map 8822  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18385  df-clat 18452  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-llines 38369  df-lplanes 38370  df-lvols 38371  df-lines 38372  df-psubsp 38374  df-pmap 38375  df-padd 38667  df-lhyp 38859  df-laut 38860  df-ldil 38975  df-ltrn 38976  df-trl 39030
This theorem is referenced by:  dihjatcclem4  40292
  Copyright terms: Public domain W3C validator