Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemj2 Structured version   Visualization version   Unicode version
Theorem cdlemj2 36110
Description: Part of proof of Lemma J of [Crawley] p. 118. Eliminate p. (Contributed by NM, 20-Jun-2013.)
Hypotheses
Ref Expression
cdlemj.b |- B = ( Base ` K )
cdlemj.h |- H = ( LHyp ` K )
cdlemj.t |- T = ( ( LTrn ` K ) ` W )
cdlemj.r |- R = ( ( trL ` K ) ` W )
cdlemj.e |- E = ( ( TEndo ` K ) ` W )
Assertion
Ref Expression
cdlemj2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) -> ( U ` h ) = ( V ` h ) )
Proof of Theorem cdlemj2
Dummy variable p is distinct from all other variables.
StepHypRef Expression
1 simpl1 1064 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) /\ ( p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) ) -> ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) )
2 simpl2 1065 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) /\ ( p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) ) -> ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) )
3 simpl3l 1116 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) /\ ( p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) ) -> ( R ` F ) =/= ( R ` g ) )
4 simpl3r 1117 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) /\ ( p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) ) -> ( R ` g ) =/= ( R ` h ) )
5 simpr 477 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) /\ ( p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) ) -> ( p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) )
6 cdlemj.b . . . . . 6 |- B = ( Base ` K )
7 cdlemj.h . . . . . 6 |- H = ( LHyp ` K )
8 cdlemj.t . . . . . 6 |- T = ( ( LTrn ` K ) ` W )
9 cdlemj.r . . . . . 6 |- R = ( ( trL ` K ) ` W )
10 cdlemj.e . . . . . 6 |- E = ( ( TEndo ` K ) ` W )
11 eqid 2622 . . . . . 6 |- ( le ` K ) = ( le ` K )
12 eqid 2622 . . . . . 6 |- ( Atoms ` K ) = ( Atoms ` K )
136, 7, 8, 9, 10, 11, 12cdlemj1 36109 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) /\ ( p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) ) ) -> ( ( U ` h ) ` p ) = ( ( V ` h ) ` p ) )
141, 2, 3, 4, 5, 13syl113anc 1338 . . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) /\ ( p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) ) -> ( ( U ` h ) ` p ) = ( ( V ` h ) ` p ) )
1514exp32 631 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) -> ( p e. ( Atoms ` K ) -> ( -. p ( le ` K ) W -> ( ( U ` h ) ` p ) = ( ( V ` h ) ` p ) ) ) )
1615ralrimiv 2965 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) -> A. p e. ( Atoms ` K ) ( -. p ( le ` K ) W -> ( ( U ` h ) ` p ) = ( ( V ` h ) ` p ) ) )
17 simp11 1091 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) -> ( K e. HL /\ W e. H ) )
18 simp121 1193 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) -> U e. E )
19 simp133 1198 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) -> h e. T )
207, 8, 10tendocl 36055 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ h e. T ) -> ( U ` h ) e. T )
2117, 18, 19, 20syl3anc 1326 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) -> ( U ` h ) e. T )
22 simp122 1194 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) -> V e. E )
237, 8, 10tendocl 36055 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ V e. E /\ h e. T ) -> ( V ` h ) e. T )
2417, 22, 19, 23syl3anc 1326 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) -> ( V ` h ) e. T )
2511, 12, 7, 8ltrneq 35435 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U ` h ) e. T /\ ( V ` h ) e. T ) -> ( A. p e. ( Atoms ` K ) ( -. p ( le ` K ) W -> ( ( U ` h ) ` p ) = ( ( V ` h ) ` p ) ) <-> ( U ` h ) = ( V ` h ) ) )
2617, 21, 24, 25syl3anc 1326 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) -> ( A. p e. ( Atoms ` K ) ( -. p ( le ` K ) W -> ( ( U ` h ) ` p ) = ( ( V ` h ) ` p ) ) <-> ( U ` h ) = ( V ` h ) ) )
2716, 26mpbid 222 1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) -> ( U ` h ) = ( V ` h ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   class class class wbr 4653   _I cid 5023   |` cres 5116   ` cfv 5888   Basecbs 15857   lecple 15948   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445   TEndoctendo 36040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tendo 36043
This theorem is referenced by:  cdlemj3  36111
  Copyright terms: Public domain W3C validator