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Theorem ip2di 18767
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 18756 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
51, 4syl 16 . . . 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 17639 . . . 4  |-  ( ( W  e.  LMod  /\  C  e.  V  /\  D  e.  V )  ->  ( C  .+  D )  e.  V )
115, 6, 7, 10syl3anc 1226 . . 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 18765 . . 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 1228 . 2  |-  ( ph  ->  ( ( A  .+  B )  .,  ( C  .+  D ) )  =  ( ( A 
.,  ( C  .+  D ) )  .+^  ( B  .,  ( C 
.+  D ) ) ) )
1712, 13, 8, 9, 14ipdi 18766 . . . 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 1228 . . 3  |-  ( ph  ->  ( A  .,  ( C  .+  D ) )  =  ( ( A 
.,  C )  .+^  ( A  .,  D ) ) )
1912, 13, 8, 9, 14ipdi 18766 . . . . 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 1228 . . . 4  |-  ( ph  ->  ( B  .,  ( C  .+  D ) )  =  ( ( B 
.,  C )  .+^  ( B  .,  D ) ) )
2112phlsrng 18757 . . . . . 6  |-  ( W  e.  PreHil  ->  F  e.  *Ring )
22 srngring 17614 . . . . . 6  |-  ( F  e.  *Ring  ->  F  e.  Ring )
23 ringcmn 17342 . . . . . 6  |-  ( F  e.  Ring  ->  F  e. CMnd
)
241, 21, 22, 234syl 21 . . . . 5  |-  ( ph  ->  F  e. CMnd )
25 eqid 2382 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  F )
2612, 13, 8, 25ipcl 18759 . . . . . 6  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .,  C )  e.  ( Base `  F
) )
271, 3, 6, 26syl3anc 1226 . . . . 5  |-  ( ph  ->  ( B  .,  C
)  e.  ( Base `  F ) )
2812, 13, 8, 25ipcl 18759 . . . . . 6  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  D  e.  V )  ->  ( B  .,  D )  e.  ( Base `  F
) )
291, 3, 7, 28syl3anc 1226 . . . . 5  |-  ( ph  ->  ( B  .,  D
)  e.  ( Base `  F ) )
3025, 14cmncom 16931 . . . . 5  |-  ( ( F  e. CMnd  /\  ( B  .,  C )  e.  ( Base `  F
)  /\  ( B  .,  D )  e.  (
Base `  F )
)  ->  ( ( B  .,  C )  .+^  ( B  .,  D ) )  =  ( ( B  .,  D ) 
.+^  ( B  .,  C ) ) )
3124, 27, 29, 30syl3anc 1226 . . . 4  |-  ( ph  ->  ( ( B  .,  C )  .+^  ( B 
.,  D ) )  =  ( ( B 
.,  D )  .+^  ( B  .,  C ) ) )
3220, 31eqtrd 2423 . . 3  |-  ( ph  ->  ( B  .,  ( C  .+  D ) )  =  ( ( B 
.,  D )  .+^  ( B  .,  C ) ) )
3318, 32oveq12d 6214 . 2  |-  ( ph  ->  ( ( A  .,  ( C  .+  D ) )  .+^  ( B  .,  ( C  .+  D
) ) )  =  ( ( ( A 
.,  C )  .+^  ( A  .,  D ) )  .+^  ( ( B  .,  D )  .+^  ( B  .,  C ) ) ) )
3412, 13, 8, 25ipcl 18759 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  F
) )
351, 2, 6, 34syl3anc 1226 . . 3  |-  ( ph  ->  ( A  .,  C
)  e.  ( Base `  F ) )
3612, 13, 8, 25ipcl 18759 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  D  e.  V )  ->  ( A  .,  D )  e.  ( Base `  F
) )
371, 2, 7, 36syl3anc 1226 . . 3  |-  ( ph  ->  ( A  .,  D
)  e.  ( Base `  F ) )
3825, 14cmn4 16934 . . 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 ) ) ) )
3924, 35, 37, 29, 27, 38syl122anc 1235 . 2  |-  ( ph  ->  ( ( ( A 
.,  C )  .+^  ( A  .,  D ) )  .+^  ( ( B  .,  D )  .+^  ( B  .,  C ) ) )  =  ( ( ( A  .,  C )  .+^  ( B 
.,  D ) ) 
.+^  ( ( A 
.,  D )  .+^  ( B  .,  C ) ) ) )
4016, 33, 393eqtrd 2427 1  |-  ( ph  ->  ( ( A  .+  B )  .,  ( C  .+  D ) )  =  ( ( ( A  .,  C ) 
.+^  ( B  .,  D ) )  .+^  ( ( A  .,  D )  .+^  ( B 
.,  C ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1826   ` cfv 5496  (class class class)co 6196   Basecbs 14634   +g cplusg 14702  Scalarcsca 14705   .icip 14707  CMndccmn 16915   Ringcrg 17311   *Ringcsr 17606   LModclmod 17625   PreHilcphl 18750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-tpos 6873  df-recs 6960  df-rdg 6994  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-plusg 14715  df-mulr 14716  df-sca 14718  df-vsca 14719  df-ip 14720  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-mhm 16083  df-grp 16174  df-minusg 16175  df-ghm 16382  df-cmn 16917  df-abl 16918  df-mgp 17255  df-ur 17267  df-ring 17313  df-oppr 17385  df-rnghom 17477  df-staf 17607  df-srng 17608  df-lmod 17627  df-lmhm 17781  df-lvec 17862  df-sra 17931  df-rgmod 17932  df-phl 18752
This theorem is referenced by:  cph2di  21738
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