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Theorem hlhilphllem 35455
Description: Lemma for hlhil 22389. (Contributed by NM, 23-Jun-2015.)
Hypotheses
Ref Expression
hlhilphl.h  |-  H  =  ( LHyp `  K
)
hlhilphllem.u  |-  U  =  ( (HLHil `  K
) `  W )
hlhilphl.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hlhilphllem.f  |-  F  =  (Scalar `  U )
hlhilphllem.l  |-  L  =  ( ( DVecH `  K
) `  W )
hlhilphllem.v  |-  V  =  ( Base `  L
)
hlhilphllem.a  |-  .+  =  ( +g  `  L )
hlhilphllem.s  |-  .x.  =  ( .s `  L )
hlhilphllem.r  |-  R  =  (Scalar `  L )
hlhilphllem.b  |-  B  =  ( Base `  R
)
hlhilphllem.p  |-  .+^  =  ( +g  `  R )
hlhilphllem.t  |-  .X.  =  ( .r `  R )
hlhilphllem.q  |-  Q  =  ( 0g `  R
)
hlhilphllem.z  |-  .0.  =  ( 0g `  L )
hlhilphllem.i  |-  .,  =  ( .i `  U )
hlhilphllem.j  |-  J  =  ( (HDMap `  K
) `  W )
hlhilphllem.g  |-  G  =  ( (HGMap `  K
) `  W )
hlhilphllem.e  |-  E  =  ( x  e.  V ,  y  e.  V  |->  ( ( J `  y ) `  x
) )
Assertion
Ref Expression
hlhilphllem  |-  ( ph  ->  U  e.  PreHil )
Distinct variable groups:    x, y, K    x, U    x, W, y    ph, x    x, J, y    x, V, y
Allowed substitution hints:    ph( y)    B( x, y)    .+ ( x, y)    .+^ (
x, y)    Q( x, y)    R( x, y)    .x. ( x, y)   
.X. ( x, y)    U( y)    E( x, y)    F( x, y)    G( x, y)    H( x, y)    ., ( x, y)    L( x, y)    .0. ( x, y)

Proof of Theorem hlhilphllem
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hlhilphl.h . . 3  |-  H  =  ( LHyp `  K
)
2 hlhilphllem.u . . 3  |-  U  =  ( (HLHil `  K
) `  W )
3 hlhilphl.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
4 hlhilphllem.l . . 3  |-  L  =  ( ( DVecH `  K
) `  W )
5 hlhilphllem.v . . 3  |-  V  =  ( Base `  L
)
61, 2, 3, 4, 5hlhilbase 35432 . 2  |-  ( ph  ->  V  =  ( Base `  U ) )
7 hlhilphllem.a . . 3  |-  .+  =  ( +g  `  L )
81, 2, 3, 4, 7hlhilplus 35433 . 2  |-  ( ph  ->  .+  =  ( +g  `  U ) )
9 hlhilphllem.s . . 3  |-  .x.  =  ( .s `  L )
101, 4, 9, 2, 3hlhilvsca 35443 . 2  |-  ( ph  ->  .x.  =  ( .s
`  U ) )
11 hlhilphllem.i . . 3  |-  .,  =  ( .i `  U )
1211a1i 11 . 2  |-  ( ph  ->  .,  =  ( .i
`  U ) )
13 hlhilphllem.z . . 3  |-  .0.  =  ( 0g `  L )
141, 4, 2, 3, 13hlhil0 35451 . 2  |-  ( ph  ->  .0.  =  ( 0g
`  U ) )
15 hlhilphllem.f . . 3  |-  F  =  (Scalar `  U )
1615a1i 11 . 2  |-  ( ph  ->  F  =  (Scalar `  U ) )
17 hlhilphllem.r . . 3  |-  R  =  (Scalar `  L )
18 hlhilphllem.b . . 3  |-  B  =  ( Base `  R
)
191, 4, 17, 2, 15, 3, 18hlhilsbase2 35438 . 2  |-  ( ph  ->  B  =  ( Base `  F ) )
20 hlhilphllem.p . . 3  |-  .+^  =  ( +g  `  R )
211, 4, 17, 2, 15, 3, 20hlhilsplus2 35439 . 2  |-  ( ph  -> 
.+^  =  ( +g  `  F ) )
22 hlhilphllem.t . . 3  |-  .X.  =  ( .r `  R )
231, 4, 17, 2, 15, 3, 22hlhilsmul2 35440 . 2  |-  ( ph  ->  .X.  =  ( .r
`  F ) )
24 hlhilphllem.g . . 3  |-  G  =  ( (HGMap `  K
) `  W )
251, 2, 15, 24, 3hlhilnvl 35446 . 2  |-  ( ph  ->  G  =  ( *r `  F ) )
26 hlhilphllem.q . . 3  |-  Q  =  ( 0g `  R
)
271, 4, 17, 2, 15, 3, 26hlhils0 35441 . 2  |-  ( ph  ->  Q  =  ( 0g
`  F ) )
281, 2, 3hlhillvec 35447 . 2  |-  ( ph  ->  U  e.  LVec )
291, 2, 3, 15hlhilsrng 35450 . 2  |-  ( ph  ->  F  e.  *Ring )
30 hlhilphllem.j . . . 4  |-  J  =  ( (HDMap `  K
) `  W )
3133ad2ant1 1027 . . . 4  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( K  e.  HL  /\  W  e.  H ) )
32 simp2 1007 . . . 4  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  a  e.  V )
33 simp3 1008 . . . 4  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  b  e.  V )
341, 4, 5, 30, 2, 31, 11, 32, 33hlhilipval 35445 . . 3  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( a  .,  b )  =  ( ( J `  b
) `  a )
)
351, 4, 5, 17, 18, 30, 31, 32, 33hdmapipcl 35401 . . 3  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( ( J `  b ) `  a )  e.  B
)
3634, 35eqeltrd 2511 . 2  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( a  .,  b )  e.  B
)
3733ad2ant1 1027 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
38 simp31 1042 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
a  e.  V )
39 simp32 1043 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
b  e.  V )
40 simp33 1044 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
c  e.  V )
41 simp2 1007 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
d  e.  B )
421, 4, 5, 7, 9, 17, 18, 20, 22, 30, 37, 38, 39, 40, 41hdmapln1 35402 . . 3  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( ( J `  c ) `  (
( d  .x.  a
)  .+  b )
)  =  ( ( d  .X.  ( ( J `  c ) `  a ) )  .+^  ( ( J `  c ) `  b
) ) )
431, 4, 3dvhlmod 34603 . . . . . 6  |-  ( ph  ->  L  e.  LMod )
44433ad2ant1 1027 . . . . 5  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  ->  L  e.  LMod )
455, 17, 9, 18lmodvscl 18101 . . . . . 6  |-  ( ( L  e.  LMod  /\  d  e.  B  /\  a  e.  V )  ->  (
d  .x.  a )  e.  V )
4644, 41, 38, 45syl3anc 1265 . . . . 5  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( d  .x.  a
)  e.  V )
475, 7lmodvacl 18098 . . . . 5  |-  ( ( L  e.  LMod  /\  (
d  .x.  a )  e.  V  /\  b  e.  V )  ->  (
( d  .x.  a
)  .+  b )  e.  V )
4844, 46, 39, 47syl3anc 1265 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( ( d  .x.  a )  .+  b
)  e.  V )
491, 4, 5, 30, 2, 37, 11, 48, 40hlhilipval 35445 . . 3  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( ( ( d 
.x.  a )  .+  b )  .,  c
)  =  ( ( J `  c ) `
 ( ( d 
.x.  a )  .+  b ) ) )
501, 4, 5, 30, 2, 37, 11, 38, 40hlhilipval 35445 . . . . 5  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( a  .,  c
)  =  ( ( J `  c ) `
 a ) )
5150oveq2d 6319 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( d  .X.  (
a  .,  c )
)  =  ( d 
.X.  ( ( J `
 c ) `  a ) ) )
521, 4, 5, 30, 2, 37, 11, 39, 40hlhilipval 35445 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( b  .,  c
)  =  ( ( J `  c ) `
 b ) )
5351, 52oveq12d 6321 . . 3  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( ( d  .X.  ( a  .,  c
) )  .+^  ( b 
.,  c ) )  =  ( ( d 
.X.  ( ( J `
 c ) `  a ) )  .+^  ( ( J `  c ) `  b
) ) )
5442, 49, 533eqtr4d 2474 . 2  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( ( ( d 
.x.  a )  .+  b )  .,  c
)  =  ( ( d  .X.  ( a  .,  c ) )  .+^  ( b  .,  c
) ) )
553adantr 467 . . . . . 6  |-  ( (
ph  /\  a  e.  V )  ->  ( K  e.  HL  /\  W  e.  H ) )
56 simpr 463 . . . . . 6  |-  ( (
ph  /\  a  e.  V )  ->  a  e.  V )
571, 4, 5, 30, 2, 55, 11, 56, 56hlhilipval 35445 . . . . 5  |-  ( (
ph  /\  a  e.  V )  ->  (
a  .,  a )  =  ( ( J `
 a ) `  a ) )
5857eqeq1d 2425 . . . 4  |-  ( (
ph  /\  a  e.  V )  ->  (
( a  .,  a
)  =  Q  <->  ( ( J `  a ) `  a )  =  Q ) )
591, 4, 5, 13, 17, 26, 30, 55, 56hdmapip0 35411 . . . 4  |-  ( (
ph  /\  a  e.  V )  ->  (
( ( J `  a ) `  a
)  =  Q  <->  a  =  .0.  ) )
6058, 59bitrd 257 . . 3  |-  ( (
ph  /\  a  e.  V )  ->  (
( a  .,  a
)  =  Q  <->  a  =  .0.  ) )
6160biimp3a 1365 . 2  |-  ( (
ph  /\  a  e.  V  /\  ( a  .,  a )  =  Q )  ->  a  =  .0.  )
621, 4, 5, 30, 24, 31, 32, 33hdmapg 35426 . . 3  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( G `  ( ( J `  b ) `  a
) )  =  ( ( J `  a
) `  b )
)
6334fveq2d 5883 . . 3  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( G `  ( a  .,  b
) )  =  ( G `  ( ( J `  b ) `
 a ) ) )
641, 4, 5, 30, 2, 31, 11, 33, 32hlhilipval 35445 . . 3  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( b  .,  a )  =  ( ( J `  a
) `  b )
)
6562, 63, 643eqtr4d 2474 . 2  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( G `  ( a  .,  b
) )  =  ( b  .,  a ) )
666, 8, 10, 12, 14, 16, 19, 21, 23, 25, 27, 28, 29, 36, 54, 61, 65isphld 19213 1  |-  ( ph  ->  U  e.  PreHil )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   ` cfv 5599  (class class class)co 6303    |-> cmpt2 6305   Basecbs 15114   +g cplusg 15183   .rcmulr 15184  Scalarcsca 15186   .scvsca 15187   .icip 15188   0gc0g 15331   LModclmod 18084   PreHilcphl 19183   HLchlt 32841   LHypclh 33474   DVecHcdvh 34571  HDMapchdma 35286  HGMapchg 35379  HLHilchlh 35428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-riotaBAD 32450
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-ot 4006  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-tpos 6979  df-undef 7026  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-map 7480  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-n0 10872  df-z 10940  df-uz 11162  df-fz 11787  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-0g 15333  df-mre 15485  df-mrc 15486  df-acs 15488  df-preset 16166  df-poset 16184  df-plt 16197  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-p0 16278  df-p1 16279  df-lat 16285  df-clat 16347  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-mhm 16575  df-submnd 16576  df-grp 16666  df-minusg 16667  df-sbg 16668  df-subg 16807  df-ghm 16874  df-cntz 16964  df-oppg 16990  df-lsm 17281  df-cmn 17425  df-abl 17426  df-mgp 17717  df-ur 17729  df-ring 17775  df-oppr 17844  df-dvdsr 17862  df-unit 17863  df-invr 17893  df-dvr 17904  df-rnghom 17936  df-drng 17970  df-subrg 17999  df-staf 18066  df-srng 18067  df-lmod 18086  df-lss 18149  df-lsp 18188  df-lmhm 18238  df-lvec 18319  df-sra 18388  df-rgmod 18389  df-phl 19185  df-lsatoms 32467  df-lshyp 32468  df-lcv 32510  df-lfl 32549  df-lkr 32577  df-ldual 32615  df-oposet 32667  df-ol 32669  df-oml 32670  df-covers 32757  df-ats 32758  df-atl 32789  df-cvlat 32813  df-hlat 32842  df-llines 32988  df-lplanes 32989  df-lvols 32990  df-lines 32991  df-psubsp 32993  df-pmap 32994  df-padd 33286  df-lhyp 33478  df-laut 33479  df-ldil 33594  df-ltrn 33595  df-trl 33650  df-tgrp 34235  df-tendo 34247  df-edring 34249  df-dveca 34495  df-disoa 34522  df-dvech 34572  df-dib 34632  df-dic 34666  df-dih 34722  df-doch 34841  df-djh 34888  df-lcdual 35080  df-mapd 35118  df-hvmap 35250  df-hdmap1 35287  df-hdmap 35288  df-hgmap 35380  df-hlhil 35429
This theorem is referenced by:  hlhilhillem  35456
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