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Theorem dia2dimlem6 34346
Description: Lemma for dia2dim 34354. Eliminate auxiliary translations  G and  D. (Contributed by NM, 8-Sep-2014.)
Hypotheses
Ref Expression
dia2dimlem6.l  |-  .<_  =  ( le `  K )
dia2dimlem6.j  |-  .\/  =  ( join `  K )
dia2dimlem6.m  |-  ./\  =  ( meet `  K )
dia2dimlem6.a  |-  A  =  ( Atoms `  K )
dia2dimlem6.h  |-  H  =  ( LHyp `  K
)
dia2dimlem6.t  |-  T  =  ( ( LTrn `  K
) `  W )
dia2dimlem6.r  |-  R  =  ( ( trL `  K
) `  W )
dia2dimlem6.y  |-  Y  =  ( ( DVecA `  K
) `  W )
dia2dimlem6.s  |-  S  =  ( LSubSp `  Y )
dia2dimlem6.pl  |-  .(+)  =  (
LSSum `  Y )
dia2dimlem6.n  |-  N  =  ( LSpan `  Y )
dia2dimlem6.i  |-  I  =  ( ( DIsoA `  K
) `  W )
dia2dimlem6.q  |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )
dia2dimlem6.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
dia2dimlem6.u  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
dia2dimlem6.v  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
dia2dimlem6.p  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
dia2dimlem6.f  |-  ( ph  ->  ( F  e.  T  /\  ( F `  P
)  =/=  P ) )
dia2dimlem6.rf  |-  ( ph  ->  ( R `  F
)  .<_  ( U  .\/  V ) )
dia2dimlem6.uv  |-  ( ph  ->  U  =/=  V )
dia2dimlem6.ru  |-  ( ph  ->  ( R `  F
)  =/=  U )
dia2dimlem6.rv  |-  ( ph  ->  ( R `  F
)  =/=  V )
Assertion
Ref Expression
dia2dimlem6  |-  ( ph  ->  F  e.  ( ( I `  U ) 
.(+)  ( I `  V ) ) )

Proof of Theorem dia2dimlem6
Dummy variables  g 
d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dia2dimlem6.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
2 dia2dimlem6.l . . . 4  |-  .<_  =  ( le `  K )
3 dia2dimlem6.j . . . 4  |-  .\/  =  ( join `  K )
4 dia2dimlem6.m . . . 4  |-  ./\  =  ( meet `  K )
5 dia2dimlem6.a . . . 4  |-  A  =  ( Atoms `  K )
6 dia2dimlem6.h . . . 4  |-  H  =  ( LHyp `  K
)
7 dia2dimlem6.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
8 dia2dimlem6.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
9 dia2dimlem6.q . . . 4  |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )
10 dia2dimlem6.u . . . 4  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
11 dia2dimlem6.v . . . 4  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
12 dia2dimlem6.p . . . 4  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
13 dia2dimlem6.f . . . 4  |-  ( ph  ->  ( F  e.  T  /\  ( F `  P
)  =/=  P ) )
14 dia2dimlem6.rf . . . 4  |-  ( ph  ->  ( R `  F
)  .<_  ( U  .\/  V ) )
15 dia2dimlem6.uv . . . 4  |-  ( ph  ->  U  =/=  V )
16 dia2dimlem6.ru . . . 4  |-  ( ph  ->  ( R `  F
)  =/=  U )
172, 3, 4, 5, 6, 7, 8, 9, 1, 10, 11, 12, 13, 14, 15, 16dia2dimlem1 34341 . . 3  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
1813simpld 460 . . . 4  |-  ( ph  ->  F  e.  T )
192, 5, 6, 7ltrnel 33413 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
201, 18, 12, 19syl3anc 1264 . . 3  |-  ( ph  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
212, 5, 6, 7cdleme50ex 33835 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
( F `  P
)  e.  A  /\  -.  ( F `  P
)  .<_  W ) )  ->  E. d  e.  T  ( d `  Q
)  =  ( F `
 P ) )
221, 17, 20, 21syl3anc 1264 . 2  |-  ( ph  ->  E. d  e.  T  ( d `  Q
)  =  ( F `
 P ) )
232, 5, 6, 7cdleme50ex 33835 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  E. g  e.  T  ( g `  P )  =  Q )
241, 12, 17, 23syl3anc 1264 . . . 4  |-  ( ph  ->  E. g  e.  T  ( g `  P
)  =  Q )
25 dia2dimlem6.y . . . . . . . 8  |-  Y  =  ( ( DVecA `  K
) `  W )
26 dia2dimlem6.s . . . . . . . 8  |-  S  =  ( LSubSp `  Y )
27 dia2dimlem6.pl . . . . . . . 8  |-  .(+)  =  (
LSSum `  Y )
28 dia2dimlem6.n . . . . . . . 8  |-  N  =  ( LSpan `  Y )
29 dia2dimlem6.i . . . . . . . 8  |-  I  =  ( ( DIsoA `  K
) `  W )
3013ad2ant1 1026 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  T  /\  (
g `  P )  =  Q  /\  d  e.  T )  /\  (
d `  Q )  =  ( F `  P ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
31103ad2ant1 1026 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  T  /\  (
g `  P )  =  Q  /\  d  e.  T )  /\  (
d `  Q )  =  ( F `  P ) )  -> 
( U  e.  A  /\  U  .<_  W ) )
32113ad2ant1 1026 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  T  /\  (
g `  P )  =  Q  /\  d  e.  T )  /\  (
d `  Q )  =  ( F `  P ) )  -> 
( V  e.  A  /\  V  .<_  W ) )
33123ad2ant1 1026 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  T  /\  (
g `  P )  =  Q  /\  d  e.  T )  /\  (
d `  Q )  =  ( F `  P ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
34133ad2ant1 1026 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  T  /\  (
g `  P )  =  Q  /\  d  e.  T )  /\  (
d `  Q )  =  ( F `  P ) )  -> 
( F  e.  T  /\  ( F `  P
)  =/=  P ) )
35143ad2ant1 1026 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  T  /\  (
g `  P )  =  Q  /\  d  e.  T )  /\  (
d `  Q )  =  ( F `  P ) )  -> 
( R `  F
)  .<_  ( U  .\/  V ) )
36153ad2ant1 1026 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  T  /\  (
g `  P )  =  Q  /\  d  e.  T )  /\  (
d `  Q )  =  ( F `  P ) )  ->  U  =/=  V )
37163ad2ant1 1026 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  T  /\  (
g `  P )  =  Q  /\  d  e.  T )  /\  (
d `  Q )  =  ( F `  P ) )  -> 
( R `  F
)  =/=  U )
38 dia2dimlem6.rv . . . . . . . . 9  |-  ( ph  ->  ( R `  F
)  =/=  V )
39383ad2ant1 1026 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  T  /\  (
g `  P )  =  Q  /\  d  e.  T )  /\  (
d `  Q )  =  ( F `  P ) )  -> 
( R `  F
)  =/=  V )
40 simp21 1038 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  T  /\  (
g `  P )  =  Q  /\  d  e.  T )  /\  (
d `  Q )  =  ( F `  P ) )  -> 
g  e.  T )
41 simp22 1039 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  T  /\  (
g `  P )  =  Q  /\  d  e.  T )  /\  (
d `  Q )  =  ( F `  P ) )  -> 
( g `  P
)  =  Q )
42 simp23 1040 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  T  /\  (
g `  P )  =  Q  /\  d  e.  T )  /\  (
d `  Q )  =  ( F `  P ) )  -> 
d  e.  T )
43 simp3 1007 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  T  /\  (
g `  P )  =  Q  /\  d  e.  T )  /\  (
d `  Q )  =  ( F `  P ) )  -> 
( d `  Q
)  =  ( F `
 P ) )
442, 3, 4, 5, 6, 7, 8, 25, 26, 27, 28, 29, 9, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43dia2dimlem5 34345 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  T  /\  (
g `  P )  =  Q  /\  d  e.  T )  /\  (
d `  Q )  =  ( F `  P ) )  ->  F  e.  ( (
I `  U )  .(+)  ( I `  V
) ) )
45443exp 1204 . . . . . 6  |-  ( ph  ->  ( ( g  e.  T  /\  ( g `
 P )  =  Q  /\  d  e.  T )  ->  (
( d `  Q
)  =  ( F `
 P )  ->  F  e.  ( (
I `  U )  .(+)  ( I `  V
) ) ) ) )
46453expd 1222 . . . . 5  |-  ( ph  ->  ( g  e.  T  ->  ( ( g `  P )  =  Q  ->  ( d  e.  T  ->  ( (
d `  Q )  =  ( F `  P )  ->  F  e.  ( ( I `  U )  .(+)  ( I `
 V ) ) ) ) ) ) )
4746rexlimdv 2922 . . . 4  |-  ( ph  ->  ( E. g  e.  T  ( g `  P )  =  Q  ->  ( d  e.  T  ->  ( (
d `  Q )  =  ( F `  P )  ->  F  e.  ( ( I `  U )  .(+)  ( I `
 V ) ) ) ) ) )
4824, 47mpd 15 . . 3  |-  ( ph  ->  ( d  e.  T  ->  ( ( d `  Q )  =  ( F `  P )  ->  F  e.  ( ( I `  U
)  .(+)  ( I `  V ) ) ) ) )
4948rexlimdv 2922 . 2  |-  ( ph  ->  ( E. d  e.  T  ( d `  Q )  =  ( F `  P )  ->  F  e.  ( ( I `  U
)  .(+)  ( I `  V ) ) ) )
5022, 49mpd 15 1  |-  ( ph  ->  F  e.  ( ( I `  U ) 
.(+)  ( I `  V ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   lecple 15159   joincjn 16140   meetcmee 16141   LSSumclsm 17221   LSubSpclss 18090   LSpanclspn 18129   Atomscatm 32538   HLchlt 32625   LHypclh 33258   LTrncltrn 33375   trLctrl 33433   DVecAcdveca 34278   DIsoAcdia 34305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-riotaBAD 32234
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-undef 7028  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-0g 15299  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-sbg 16626  df-subg 16765  df-lsm 17223  df-cmn 17367  df-abl 17368  df-mgp 17659  df-ur 17671  df-ring 17717  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-dvr 17846  df-drng 17912  df-lmod 18028  df-lss 18091  df-lsp 18130  df-lvec 18261  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-llines 32772  df-lplanes 32773  df-lvols 32774  df-lines 32775  df-psubsp 32777  df-pmap 32778  df-padd 33070  df-lhyp 33262  df-laut 33263  df-ldil 33378  df-ltrn 33379  df-trl 33434  df-tgrp 34019  df-tendo 34031  df-edring 34033  df-dveca 34279  df-disoa 34306
This theorem is referenced by:  dia2dimlem7  34347
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