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Theorem ip2di 16561
Description: Distributive law for inner product. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
phlsrng.f  |-  F  =  (Scalar `  W )
phllmhm.h  |-  .,  =  ( .i `  W )
phllmhm.v  |-  V  =  ( Base `  W
)
ipdir.g  |-  .+  =  ( +g  `  W )
ipdir.p  |-  .+^  =  ( +g  `  F )
ip2di.1  |-  ( ph  ->  W  e.  PreHil )
ip2di.2  |-  ( ph  ->  A  e.  V )
ip2di.3  |-  ( ph  ->  B  e.  V )
ip2di.4  |-  ( ph  ->  C  e.  V )
ip2di.5  |-  ( ph  ->  D  e.  V )
Assertion
Ref Expression
ip2di  |-  ( ph  ->  ( ( A  .+  B )  .,  ( C  .+  D ) )  =  ( ( ( A  .,  C ) 
.+^  ( B  .,  D ) )  .+^  ( ( A  .,  D )  .+^  ( B 
.,  C ) ) ) )

Proof of Theorem ip2di
StepHypRef Expression
1 ip2di.1 . . 3  |-  ( ph  ->  W  e.  PreHil )
2 ip2di.2 . . 3  |-  ( ph  ->  A  e.  V )
3 ip2di.3 . . 3  |-  ( ph  ->  B  e.  V )
4 phllmod 16550 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
51, 4syl 15 . . . 4  |-  ( ph  ->  W  e.  LMod )
6 ip2di.4 . . . 4  |-  ( ph  ->  C  e.  V )
7 ip2di.5 . . . 4  |-  ( ph  ->  D  e.  V )
8 phllmhm.v . . . . 5  |-  V  =  ( Base `  W
)
9 ipdir.g . . . . 5  |-  .+  =  ( +g  `  W )
108, 9lmodvacl 15657 . . . 4  |-  ( ( W  e.  LMod  /\  C  e.  V  /\  D  e.  V )  ->  ( C  .+  D )  e.  V )
115, 6, 7, 10syl3anc 1182 . . 3  |-  ( ph  ->  ( C  .+  D
)  e.  V )
12 phlsrng.f . . . 4  |-  F  =  (Scalar `  W )
13 phllmhm.h . . . 4  |-  .,  =  ( .i `  W )
14 ipdir.p . . . 4  |-  .+^  =  ( +g  `  F )
1512, 13, 8, 9, 14ipdir 16559 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  ( C  .+  D )  e.  V ) )  ->  ( ( A 
.+  B )  .,  ( C  .+  D ) )  =  ( ( A  .,  ( C 
.+  D ) ) 
.+^  ( B  .,  ( C  .+  D ) ) ) )
161, 2, 3, 11, 15syl13anc 1184 . 2  |-  ( ph  ->  ( ( A  .+  B )  .,  ( C  .+  D ) )  =  ( ( A 
.,  ( C  .+  D ) )  .+^  ( B  .,  ( C 
.+  D ) ) ) )
1712, 13, 8, 9, 14ipdi 16560 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  C  e.  V  /\  D  e.  V )
)  ->  ( A  .,  ( C  .+  D
) )  =  ( ( A  .,  C
)  .+^  ( A  .,  D ) ) )
181, 2, 6, 7, 17syl13anc 1184 . . 3  |-  ( ph  ->  ( A  .,  ( C  .+  D ) )  =  ( ( A 
.,  C )  .+^  ( A  .,  D ) ) )
1912, 13, 8, 9, 14ipdi 16560 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( B  e.  V  /\  C  e.  V  /\  D  e.  V )
)  ->  ( B  .,  ( C  .+  D
) )  =  ( ( B  .,  C
)  .+^  ( B  .,  D ) ) )
201, 3, 6, 7, 19syl13anc 1184 . . . 4  |-  ( ph  ->  ( B  .,  ( C  .+  D ) )  =  ( ( B 
.,  C )  .+^  ( B  .,  D ) ) )
2112phlsrng 16551 . . . . . . 7  |-  ( W  e.  PreHil  ->  F  e.  *Ring )
221, 21syl 15 . . . . . 6  |-  ( ph  ->  F  e.  *Ring )
23 srngrng 15633 . . . . . 6  |-  ( F  e.  *Ring  ->  F  e.  Ring )
24 rngcmn 15387 . . . . . 6  |-  ( F  e.  Ring  ->  F  e. CMnd
)
2522, 23, 243syl 18 . . . . 5  |-  ( ph  ->  F  e. CMnd )
26 eqid 2296 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  F )
2712, 13, 8, 26ipcl 16553 . . . . . 6  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .,  C )  e.  ( Base `  F
) )
281, 3, 6, 27syl3anc 1182 . . . . 5  |-  ( ph  ->  ( B  .,  C
)  e.  ( Base `  F ) )
2912, 13, 8, 26ipcl 16553 . . . . . 6  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  D  e.  V )  ->  ( B  .,  D )  e.  ( Base `  F
) )
301, 3, 7, 29syl3anc 1182 . . . . 5  |-  ( ph  ->  ( B  .,  D
)  e.  ( Base `  F ) )
3126, 14cmncom 15121 . . . . 5  |-  ( ( F  e. CMnd  /\  ( B  .,  C )  e.  ( Base `  F
)  /\  ( B  .,  D )  e.  (
Base `  F )
)  ->  ( ( B  .,  C )  .+^  ( B  .,  D ) )  =  ( ( B  .,  D ) 
.+^  ( B  .,  C ) ) )
3225, 28, 30, 31syl3anc 1182 . . . 4  |-  ( ph  ->  ( ( B  .,  C )  .+^  ( B 
.,  D ) )  =  ( ( B 
.,  D )  .+^  ( B  .,  C ) ) )
3320, 32eqtrd 2328 . . 3  |-  ( ph  ->  ( B  .,  ( C  .+  D ) )  =  ( ( B 
.,  D )  .+^  ( B  .,  C ) ) )
3418, 33oveq12d 5892 . 2  |-  ( ph  ->  ( ( A  .,  ( C  .+  D ) )  .+^  ( B  .,  ( C  .+  D
) ) )  =  ( ( ( A 
.,  C )  .+^  ( A  .,  D ) )  .+^  ( ( B  .,  D )  .+^  ( B  .,  C ) ) ) )
3512, 13, 8, 26ipcl 16553 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  F
) )
361, 2, 6, 35syl3anc 1182 . . 3  |-  ( ph  ->  ( A  .,  C
)  e.  ( Base `  F ) )
3712, 13, 8, 26ipcl 16553 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  D  e.  V )  ->  ( A  .,  D )  e.  ( Base `  F
) )
381, 2, 7, 37syl3anc 1182 . . 3  |-  ( ph  ->  ( A  .,  D
)  e.  ( Base `  F ) )
3926, 14cmn4 15124 . . 3  |-  ( ( F  e. CMnd  /\  (
( A  .,  C
)  e.  ( Base `  F )  /\  ( A  .,  D )  e.  ( Base `  F
) )  /\  (
( B  .,  D
)  e.  ( Base `  F )  /\  ( B  .,  C )  e.  ( Base `  F
) ) )  -> 
( ( ( A 
.,  C )  .+^  ( A  .,  D ) )  .+^  ( ( B  .,  D )  .+^  ( B  .,  C ) ) )  =  ( ( ( A  .,  C )  .+^  ( B 
.,  D ) ) 
.+^  ( ( A 
.,  D )  .+^  ( B  .,  C ) ) ) )
4025, 36, 38, 30, 28, 39syl122anc 1191 . 2  |-  ( ph  ->  ( ( ( A 
.,  C )  .+^  ( A  .,  D ) )  .+^  ( ( B  .,  D )  .+^  ( B  .,  C ) ) )  =  ( ( ( A  .,  C )  .+^  ( B 
.,  D ) ) 
.+^  ( ( A 
.,  D )  .+^  ( B  .,  C ) ) ) )
4116, 34, 403eqtrd 2332 1  |-  ( ph  ->  ( ( A  .+  B )  .,  ( C  .+  D ) )  =  ( ( ( A  .,  C ) 
.+^  ( B  .,  D ) )  .+^  ( ( A  .,  D )  .+^  ( B 
.,  C ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   .icip 13229  CMndccmn 15105   Ringcrg 15353   *Ringcsr 15625   LModclmod 15643   PreHilcphl 16544
This theorem is referenced by:  cph2di  18658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-mnd 14383  df-mhm 14431  df-grp 14505  df-minusg 14506  df-ghm 14697  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-rnghom 15512  df-staf 15626  df-srng 15627  df-lmod 15645  df-lmhm 15795  df-lvec 15872  df-sra 15941  df-rgmod 15942  df-phl 16546
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