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Theorem hlhilphllem 32152
Description: Lemma for hlhil 18807. (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 32129 . 2  |-  ( ph  ->  V  =  ( Base `  U ) )
7 hlhilphllem.a . . 3  |-  .+  =  ( +g  `  L )
81, 2, 3, 4, 7hlhilplus 32130 . 2  |-  ( ph  ->  .+  =  ( +g  `  U ) )
9 hlhilphllem.s . . 3  |-  .x.  =  ( .s `  L )
101, 4, 9, 2, 3hlhilvsca 32140 . 2  |-  ( ph  ->  .x.  =  ( .s
`  U ) )
11 hlhilphllem.i . . 3  |-  .,  =  ( .i `  U )
1211a1i 10 . 2  |-  ( ph  ->  .,  =  ( .i
`  U ) )
13 hlhilphllem.z . . 3  |-  .0.  =  ( 0g `  L )
141, 4, 2, 3, 13hlhil0 32148 . 2  |-  ( ph  ->  .0.  =  ( 0g
`  U ) )
15 hlhilphllem.f . . 3  |-  F  =  (Scalar `  U )
1615a1i 10 . 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 32135 . 2  |-  ( ph  ->  B  =  ( Base `  F ) )
20 hlhilphllem.p . . 3  |-  .+^  =  ( +g  `  R )
211, 4, 17, 2, 15, 3, 20hlhilsplus2 32136 . 2  |-  ( ph  -> 
.+^  =  ( +g  `  F ) )
22 hlhilphllem.t . . 3  |-  .X.  =  ( .r `  R )
231, 4, 17, 2, 15, 3, 22hlhilsmul2 32137 . 2  |-  ( ph  ->  .X.  =  ( .r
`  F ) )
24 hlhilphllem.g . . 3  |-  G  =  ( (HGMap `  K
) `  W )
251, 2, 15, 24, 3hlhilnvl 32143 . 2  |-  ( ph  ->  G  =  ( * r `  F ) )
26 hlhilphllem.q . . 3  |-  Q  =  ( 0g `  R
)
271, 4, 17, 2, 15, 3, 26hlhils0 32138 . 2  |-  ( ph  ->  Q  =  ( 0g
`  F ) )
281, 2, 3hlhillvec 32144 . 2  |-  ( ph  ->  U  e.  LVec )
291, 2, 3, 15hlhilsrng 32147 . 2  |-  ( ph  ->  F  e.  *Ring )
30 hlhilphllem.j . . . 4  |-  J  =  ( (HDMap `  K
) `  W )
3133ad2ant1 976 . . . 4  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( K  e.  HL  /\  W  e.  H ) )
32 simp2 956 . . . 4  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  a  e.  V )
33 simp3 957 . . . 4  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  b  e.  V )
341, 4, 5, 30, 2, 31, 11, 32, 33hlhilipval 32142 . . 3  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( a  .,  b )  =  ( ( J `  b
) `  a )
)
351, 4, 5, 17, 18, 30, 31, 32, 33hdmapipcl 32098 . . 3  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( ( J `  b ) `  a )  e.  B
)
3634, 35eqeltrd 2357 . 2  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( a  .,  b )  e.  B
)
3733ad2ant1 976 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
38 simp31 991 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
a  e.  V )
39 simp32 992 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
b  e.  V )
40 simp33 993 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
c  e.  V )
41 simp2 956 . . . 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 32099 . . 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 31300 . . . . . 6  |-  ( ph  ->  L  e.  LMod )
44433ad2ant1 976 . . . . 5  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  ->  L  e.  LMod )
455, 17, 9, 18lmodvscl 15644 . . . . . 6  |-  ( ( L  e.  LMod  /\  d  e.  B  /\  a  e.  V )  ->  (
d  .x.  a )  e.  V )
4644, 41, 38, 45syl3anc 1182 . . . . 5  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( d  .x.  a
)  e.  V )
475, 7lmodvacl 15641 . . . . 5  |-  ( ( L  e.  LMod  /\  (
d  .x.  a )  e.  V  /\  b  e.  V )  ->  (
( d  .x.  a
)  .+  b )  e.  V )
4844, 46, 39, 47syl3anc 1182 . . . 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 32142 . . 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 32142 . . . . 5  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( a  .,  c
)  =  ( ( J `  c ) `
 a ) )
5150oveq2d 5874 . . . 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 32142 . . . 4  |-  ( (
ph  /\  d  e.  B  /\  ( a  e.  V  /\  b  e.  V  /\  c  e.  V ) )  -> 
( b  .,  c
)  =  ( ( J `  c ) `
 b ) )
5351, 52oveq12d 5876 . . 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 2325 . 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 451 . . . . . 6  |-  ( (
ph  /\  a  e.  V )  ->  ( K  e.  HL  /\  W  e.  H ) )
56 simpr 447 . . . . . 6  |-  ( (
ph  /\  a  e.  V )  ->  a  e.  V )
571, 4, 5, 30, 2, 55, 11, 56, 56hlhilipval 32142 . . . . 5  |-  ( (
ph  /\  a  e.  V )  ->  (
a  .,  a )  =  ( ( J `
 a ) `  a ) )
5857eqeq1d 2291 . . . 4  |-  ( (
ph  /\  a  e.  V )  ->  (
( a  .,  a
)  =  Q  <->  ( ( J `  a ) `  a )  =  Q ) )
591, 4, 5, 13, 17, 26, 30, 55, 56hdmapip0 32108 . . . 4  |-  ( (
ph  /\  a  e.  V )  ->  (
( ( J `  a ) `  a
)  =  Q  <->  a  =  .0.  ) )
6058, 59bitrd 244 . . 3  |-  ( (
ph  /\  a  e.  V )  ->  (
( a  .,  a
)  =  Q  <->  a  =  .0.  ) )
6160biimp3a 1281 . 2  |-  ( (
ph  /\  a  e.  V  /\  ( a  .,  a )  =  Q )  ->  a  =  .0.  )
621, 4, 5, 30, 24, 31, 32, 33hdmapg 32123 . . 3  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( G `  ( ( J `  b ) `  a
) )  =  ( ( J `  a
) `  b )
)
6334fveq2d 5529 . . 3  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( G `  ( a  .,  b
) )  =  ( G `  ( ( J `  b ) `
 a ) ) )
641, 4, 5, 30, 2, 31, 11, 33, 32hlhilipval 32142 . . 3  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( b  .,  a )  =  ( ( J `  a
) `  b )
)
6562, 63, 643eqtr4d 2325 . 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 16558 1  |-  ( ph  ->  U  e.  PreHil )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   .icip 13213   0gc0g 13400   LModclmod 15627   PreHilcphl 16528   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268  HDMapchdma 31983  HGMapchg 32076  HLHilchlh 32125
This theorem is referenced by:  hlhilhillem  32153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-ip 13226  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-ghm 14681  df-cntz 14793  df-oppg 14819  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-rnghom 15496  df-drng 15514  df-subrg 15543  df-staf 15610  df-srng 15611  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lmhm 15779  df-lvec 15856  df-sra 15925  df-rgmod 15926  df-phl 16530  df-lsatoms 29166  df-lshyp 29167  df-lcv 29209  df-lfl 29248  df-lkr 29276  df-ldual 29314  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tgrp 30932  df-tendo 30944  df-edring 30946  df-dveca 31192  df-disoa 31219  df-dvech 31269  df-dib 31329  df-dic 31363  df-dih 31419  df-doch 31538  df-djh 31585  df-lcdual 31777  df-mapd 31815  df-hvmap 31947  df-hdmap1 31984  df-hdmap 31985  df-hgmap 32077  df-hlhil 32126
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