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Theorem hlhilphllem 35619
Description: Lemma for hlhil 20942. (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 35596 . 2  |-  ( ph  ->  V  =  ( Base `  U ) )
7 hlhilphllem.a . . 3  |-  .+  =  ( +g  `  L )
81, 2, 3, 4, 7hlhilplus 35597 . 2  |-  ( ph  ->  .+  =  ( +g  `  U ) )
9 hlhilphllem.s . . 3  |-  .x.  =  ( .s `  L )
101, 4, 9, 2, 3hlhilvsca 35607 . 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 35615 . 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 35602 . 2  |-  ( ph  ->  B  =  ( Base `  F ) )
20 hlhilphllem.p . . 3  |-  .+^  =  ( +g  `  R )
211, 4, 17, 2, 15, 3, 20hlhilsplus2 35603 . 2  |-  ( ph  -> 
.+^  =  ( +g  `  F ) )
22 hlhilphllem.t . . 3  |-  .X.  =  ( .r `  R )
231, 4, 17, 2, 15, 3, 22hlhilsmul2 35604 . 2  |-  ( ph  ->  .X.  =  ( .r
`  F ) )
24 hlhilphllem.g . . 3  |-  G  =  ( (HGMap `  K
) `  W )
251, 2, 15, 24, 3hlhilnvl 35610 . 2  |-  ( ph  ->  G  =  ( *r `  F ) )
26 hlhilphllem.q . . 3  |-  Q  =  ( 0g `  R
)
271, 4, 17, 2, 15, 3, 26hlhils0 35605 . 2  |-  ( ph  ->  Q  =  ( 0g
`  F ) )
281, 2, 3hlhillvec 35611 . 2  |-  ( ph  ->  U  e.  LVec )
291, 2, 3, 15hlhilsrng 35614 . 2  |-  ( ph  ->  F  e.  *Ring )
30 hlhilphllem.j . . . 4  |-  J  =  ( (HDMap `  K
) `  W )
3133ad2ant1 1009 . . . 4  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( K  e.  HL  /\  W  e.  H ) )
32 simp2 989 . . . 4  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  a  e.  V )
33 simp3 990 . . . 4  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  b  e.  V )
341, 4, 5, 30, 2, 31, 11, 32, 33hlhilipval 35609 . . 3  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( a  .,  b )  =  ( ( J `  b
) `  a )
)
351, 4, 5, 17, 18, 30, 31, 32, 33hdmapipcl 35565 . . 3  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( ( J `  b ) `  a )  e.  B
)
3634, 35eqeltrd 2517 . 2  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( a  .,  b )  e.  B
)
3733ad2ant1 1009 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
38 simp31 1024 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
a  e.  V )
39 simp32 1025 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
b  e.  V )
40 simp33 1026 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
c  e.  V )
41 simp2 989 . . . 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 35566 . . 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 34767 . . . . . 6  |-  ( ph  ->  L  e.  LMod )
44433ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  ->  L  e.  LMod )
455, 17, 9, 18lmodvscl 16977 . . . . . 6  |-  ( ( L  e.  LMod  /\  d  e.  B  /\  a  e.  V )  ->  (
d  .x.  a )  e.  V )
4644, 41, 38, 45syl3anc 1218 . . . . 5  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( d  .x.  a
)  e.  V )
475, 7lmodvacl 16974 . . . . 5  |-  ( ( L  e.  LMod  /\  (
d  .x.  a )  e.  V  /\  b  e.  V )  ->  (
( d  .x.  a
)  .+  b )  e.  V )
4844, 46, 39, 47syl3anc 1218 . . . 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 35609 . . 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 35609 . . . . 5  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( a  .,  c
)  =  ( ( J `  c ) `
 a ) )
5150oveq2d 6119 . . . 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 35609 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( b  .,  c
)  =  ( ( J `  c ) `
 b ) )
5351, 52oveq12d 6121 . . 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 2485 . 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 465 . . . . . 6  |-  ( (
ph  /\  a  e.  V )  ->  ( K  e.  HL  /\  W  e.  H ) )
56 simpr 461 . . . . . 6  |-  ( (
ph  /\  a  e.  V )  ->  a  e.  V )
571, 4, 5, 30, 2, 55, 11, 56, 56hlhilipval 35609 . . . . 5  |-  ( (
ph  /\  a  e.  V )  ->  (
a  .,  a )  =  ( ( J `
 a ) `  a ) )
5857eqeq1d 2451 . . . 4  |-  ( (
ph  /\  a  e.  V )  ->  (
( a  .,  a
)  =  Q  <->  ( ( J `  a ) `  a )  =  Q ) )
591, 4, 5, 13, 17, 26, 30, 55, 56hdmapip0 35575 . . . 4  |-  ( (
ph  /\  a  e.  V )  ->  (
( ( J `  a ) `  a
)  =  Q  <->  a  =  .0.  ) )
6058, 59bitrd 253 . . 3  |-  ( (
ph  /\  a  e.  V )  ->  (
( a  .,  a
)  =  Q  <->  a  =  .0.  ) )
6160biimp3a 1318 . 2  |-  ( (
ph  /\  a  e.  V  /\  ( a  .,  a )  =  Q )  ->  a  =  .0.  )
621, 4, 5, 30, 24, 31, 32, 33hdmapg 35590 . . 3  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( G `  ( ( J `  b ) `  a
) )  =  ( ( J `  a
) `  b )
)
6334fveq2d 5707 . . 3  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( G `  ( a  .,  b
) )  =  ( G `  ( ( J `  b ) `
 a ) ) )
641, 4, 5, 30, 2, 31, 11, 33, 32hlhilipval 35609 . . 3  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( b  .,  a )  =  ( ( J `  a
) `  b )
)
6562, 63, 643eqtr4d 2485 . 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 18095 1  |-  ( ph  ->  U  e.  PreHil )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105   Basecbs 14186   +g cplusg 14250   .rcmulr 14251  Scalarcsca 14253   .scvsca 14254   .icip 14255   0gc0g 14390   LModclmod 16960   PreHilcphl 18065   HLchlt 33007   LHypclh 33640   DVecHcdvh 34735  HDMapchdma 35450  HGMapchg 35543  HLHilchlh 35592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-riotaBAD 32616
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-ot 3898  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-tpos 6757  df-undef 6804  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-starv 14265  df-sca 14266  df-vsca 14267  df-ip 14268  df-0g 14392  df-mre 14536  df-mrc 14537  df-acs 14539  df-poset 15128  df-plt 15140  df-lub 15156  df-glb 15157  df-join 15158  df-meet 15159  df-p0 15221  df-p1 15222  df-lat 15228  df-clat 15290  df-mnd 15427  df-mhm 15476  df-submnd 15477  df-grp 15557  df-minusg 15558  df-sbg 15559  df-subg 15690  df-ghm 15757  df-cntz 15847  df-oppg 15873  df-lsm 16147  df-cmn 16291  df-abl 16292  df-mgp 16604  df-ur 16616  df-rng 16659  df-oppr 16727  df-dvdsr 16745  df-unit 16746  df-invr 16776  df-dvr 16787  df-rnghom 16818  df-drng 16846  df-subrg 16875  df-staf 16942  df-srng 16943  df-lmod 16962  df-lss 17026  df-lsp 17065  df-lmhm 17115  df-lvec 17196  df-sra 17265  df-rgmod 17266  df-phl 18067  df-lsatoms 32633  df-lshyp 32634  df-lcv 32676  df-lfl 32715  df-lkr 32743  df-ldual 32781  df-oposet 32833  df-ol 32835  df-oml 32836  df-covers 32923  df-ats 32924  df-atl 32955  df-cvlat 32979  df-hlat 33008  df-llines 33154  df-lplanes 33155  df-lvols 33156  df-lines 33157  df-psubsp 33159  df-pmap 33160  df-padd 33452  df-lhyp 33644  df-laut 33645  df-ldil 33760  df-ltrn 33761  df-trl 33815  df-tgrp 34399  df-tendo 34411  df-edring 34413  df-dveca 34659  df-disoa 34686  df-dvech 34736  df-dib 34796  df-dic 34830  df-dih 34886  df-doch 35005  df-djh 35052  df-lcdual 35244  df-mapd 35282  df-hvmap 35414  df-hdmap1 35451  df-hdmap 35452  df-hgmap 35544  df-hlhil 35593
This theorem is referenced by:  hlhilhillem  35620
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