Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg6 Structured version   Visualization version   GIF version
Theorem cdlemg6 40128
Description: TODO: FIX COMMENT. (Contributed by NM, 27-Apr-2013.)
Hypotheses
Ref Expression
cdlemg6.l ≀ = (leβ€˜πΎ)
cdlemg6.a 𝐴 = (Atomsβ€˜πΎ)
cdlemg6.h 𝐻 = (LHypβ€˜πΎ)
cdlemg6.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
cdlemg6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄)
Proof of Theorem cdlemg6
StepHypRef Expression
1 simpl1 1188 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 simpl2l 1223 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
3 simpl2r 1224 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
4 simpl31 1251 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ 𝐹 ∈ 𝑇)
5 simpl32 1252 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ 𝐺 ∈ 𝑇)
6 simpr 483 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ)))
7 simpl33 1253 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)
8 cdlemg6.l . . . 4 ≀ = (leβ€˜πΎ)
9 cdlemg6.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
10 cdlemg6.h . . . 4 𝐻 = (LHypβ€˜πΎ)
11 cdlemg6.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
12 eqid 2728 . . . 4 ((trLβ€˜πΎ)β€˜π‘Š) = ((trLβ€˜πΎ)β€˜π‘Š)
13 eqid 2728 . . . 4 (joinβ€˜πΎ) = (joinβ€˜πΎ)
14 eqid 2728 . . . 4 (((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ) = (((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ)
158, 9, 10, 11, 12, 13, 14cdlemg6e 40127 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ)) ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄)
161, 2, 3, 4, 5, 6, 7, 15syl133anc 1390 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄)
17 simpl1 1188 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
18 simpl2l 1223 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
19 simpl2r 1224 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
20 simpl31 1251 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ 𝐹 ∈ 𝑇)
21 simpl32 1252 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ 𝐺 ∈ 𝑇)
22 simpr 483 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ)))
23 simpl33 1253 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)
248, 9, 10, 11, 12, 13, 14cdlemg4 40122 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ)) ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄)
2517, 18, 19, 20, 21, 22, 23, 24syl133anc 1390 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄)
2616, 25pm2.61dan 811 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  lecple 17247  joincjn 18310  Atomscatm 38767  HLchlt 38854  LHypclh 39489  LTrncltrn 39606  trLctrl 39663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-riotaBAD 38457
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-undef 8285  df-map 8853  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-p1 18425  df-lat 18431  df-clat 18498  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-llines 39003  df-lplanes 39004  df-lvols 39005  df-lines 39006  df-psubsp 39008  df-pmap 39009  df-padd 39301  df-lhyp 39493  df-laut 39494  df-ldil 39609  df-ltrn 39610  df-trl 39664
This theorem is referenced by:  cdlemg7aN  40130  cdlemg8a  40132  cdlemg8c  40134  cdlemg11a  40142
  Copyright terms: Public domain W3C validator