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Theorem cph2subdi 21946
Description: Distributive law for inner product subtraction. Complex version of ip2subdi 18975. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
cphipcj.h  |-  .,  =  ( .i `  W )
cphipcj.v  |-  V  =  ( Base `  W
)
cphsubdir.m  |-  .-  =  ( -g `  W )
cph2subdi.1  |-  ( ph  ->  W  e.  CPreHil )
cph2subdi.2  |-  ( ph  ->  A  e.  V )
cph2subdi.3  |-  ( ph  ->  B  e.  V )
cph2subdi.4  |-  ( ph  ->  C  e.  V )
cph2subdi.5  |-  ( ph  ->  D  e.  V )
Assertion
Ref Expression
cph2subdi  |-  ( ph  ->  ( ( A  .-  B )  .,  ( C  .-  D ) )  =  ( ( ( A  .,  C )  +  ( B  .,  D ) )  -  ( ( A  .,  D )  +  ( B  .,  C ) ) ) )

Proof of Theorem cph2subdi
StepHypRef Expression
1 cph2subdi.1 . . . . . 6  |-  ( ph  ->  W  e.  CPreHil )
2 cphclm 21926 . . . . . 6  |-  ( W  e.  CPreHil  ->  W  e. CMod )
31, 2syl 17 . . . . 5  |-  ( ph  ->  W  e. CMod )
4 eqid 2402 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
54clmadd 21864 . . . . 5  |-  ( W  e. CMod  ->  +  =  ( +g  `  (Scalar `  W ) ) )
63, 5syl 17 . . . 4  |-  ( ph  ->  +  =  ( +g  `  (Scalar `  W )
) )
76oveqd 6294 . . 3  |-  ( ph  ->  ( ( A  .,  C )  +  ( B  .,  D ) )  =  ( ( A  .,  C ) ( +g  `  (Scalar `  W ) ) ( B  .,  D ) ) )
86oveqd 6294 . . 3  |-  ( ph  ->  ( ( A  .,  D )  +  ( B  .,  C ) )  =  ( ( A  .,  D ) ( +g  `  (Scalar `  W ) ) ( B  .,  C ) ) )
97, 8oveq12d 6295 . 2  |-  ( ph  ->  ( ( ( A 
.,  C )  +  ( B  .,  D
) ) ( -g `  (Scalar `  W )
) ( ( A 
.,  D )  +  ( B  .,  C
) ) )  =  ( ( ( A 
.,  C ) ( +g  `  (Scalar `  W ) ) ( B  .,  D ) ) ( -g `  (Scalar `  W ) ) ( ( A  .,  D
) ( +g  `  (Scalar `  W ) ) ( B  .,  C ) ) ) )
10 cphphl 21908 . . . . . 6  |-  ( W  e.  CPreHil  ->  W  e.  PreHil )
111, 10syl 17 . . . . 5  |-  ( ph  ->  W  e.  PreHil )
12 cph2subdi.2 . . . . 5  |-  ( ph  ->  A  e.  V )
13 cph2subdi.4 . . . . 5  |-  ( ph  ->  C  e.  V )
14 cphipcj.h . . . . . 6  |-  .,  =  ( .i `  W )
15 cphipcj.v . . . . . 6  |-  V  =  ( Base `  W
)
16 eqid 2402 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
174, 14, 15, 16ipcl 18964 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  (Scalar `  W ) ) )
1811, 12, 13, 17syl3anc 1230 . . . 4  |-  ( ph  ->  ( A  .,  C
)  e.  ( Base `  (Scalar `  W )
) )
19 cph2subdi.3 . . . . 5  |-  ( ph  ->  B  e.  V )
20 cph2subdi.5 . . . . 5  |-  ( ph  ->  D  e.  V )
214, 14, 15, 16ipcl 18964 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  D  e.  V )  ->  ( B  .,  D )  e.  ( Base `  (Scalar `  W ) ) )
2211, 19, 20, 21syl3anc 1230 . . . 4  |-  ( ph  ->  ( B  .,  D
)  e.  ( Base `  (Scalar `  W )
) )
234, 16clmacl 21873 . . . 4  |-  ( ( W  e. CMod  /\  ( A  .,  C )  e.  ( Base `  (Scalar `  W ) )  /\  ( B  .,  D )  e.  ( Base `  (Scalar `  W ) ) )  ->  ( ( A 
.,  C )  +  ( B  .,  D
) )  e.  (
Base `  (Scalar `  W
) ) )
243, 18, 22, 23syl3anc 1230 . . 3  |-  ( ph  ->  ( ( A  .,  C )  +  ( B  .,  D ) )  e.  ( Base `  (Scalar `  W )
) )
254, 14, 15, 16ipcl 18964 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  D  e.  V )  ->  ( A  .,  D )  e.  ( Base `  (Scalar `  W ) ) )
2611, 12, 20, 25syl3anc 1230 . . . 4  |-  ( ph  ->  ( A  .,  D
)  e.  ( Base `  (Scalar `  W )
) )
274, 14, 15, 16ipcl 18964 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .,  C )  e.  ( Base `  (Scalar `  W ) ) )
2811, 19, 13, 27syl3anc 1230 . . . 4  |-  ( ph  ->  ( B  .,  C
)  e.  ( Base `  (Scalar `  W )
) )
294, 16clmacl 21873 . . . 4  |-  ( ( W  e. CMod  /\  ( A  .,  D )  e.  ( Base `  (Scalar `  W ) )  /\  ( B  .,  C )  e.  ( Base `  (Scalar `  W ) ) )  ->  ( ( A 
.,  D )  +  ( B  .,  C
) )  e.  (
Base `  (Scalar `  W
) ) )
303, 26, 28, 29syl3anc 1230 . . 3  |-  ( ph  ->  ( ( A  .,  D )  +  ( B  .,  C ) )  e.  ( Base `  (Scalar `  W )
) )
314, 16clmsub 21870 . . 3  |-  ( ( W  e. CMod  /\  (
( A  .,  C
)  +  ( B 
.,  D ) )  e.  ( Base `  (Scalar `  W ) )  /\  ( ( A  .,  D )  +  ( B  .,  C ) )  e.  ( Base `  (Scalar `  W )
) )  ->  (
( ( A  .,  C )  +  ( B  .,  D ) )  -  ( ( A  .,  D )  +  ( B  .,  C ) ) )  =  ( ( ( A  .,  C )  +  ( B  .,  D ) ) (
-g `  (Scalar `  W
) ) ( ( A  .,  D )  +  ( B  .,  C ) ) ) )
323, 24, 30, 31syl3anc 1230 . 2  |-  ( ph  ->  ( ( ( A 
.,  C )  +  ( B  .,  D
) )  -  (
( A  .,  D
)  +  ( B 
.,  C ) ) )  =  ( ( ( A  .,  C
)  +  ( B 
.,  D ) ) ( -g `  (Scalar `  W ) ) ( ( A  .,  D
)  +  ( B 
.,  C ) ) ) )
33 cphsubdir.m . . 3  |-  .-  =  ( -g `  W )
34 eqid 2402 . . 3  |-  ( -g `  (Scalar `  W )
)  =  ( -g `  (Scalar `  W )
)
35 eqid 2402 . . 3  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
364, 14, 15, 33, 34, 35, 11, 12, 19, 13, 20ip2subdi 18975 . 2  |-  ( ph  ->  ( ( A  .-  B )  .,  ( C  .-  D ) )  =  ( ( ( A  .,  C ) ( +g  `  (Scalar `  W ) ) ( B  .,  D ) ) ( -g `  (Scalar `  W ) ) ( ( A  .,  D
) ( +g  `  (Scalar `  W ) ) ( B  .,  C ) ) ) )
379, 32, 363eqtr4rd 2454 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 1405    e. wcel 1842   ` cfv 5568  (class class class)co 6277    + caddc 9524    - cmin 9840   Basecbs 14839   +g cplusg 14907  Scalarcsca 14910   .icip 14912   -gcsg 16377   PreHilcphl 18955  CModcclm 21852   CPreHilccph 21903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-addf 9600  ax-mulf 9601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-tpos 6957  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-fz 11725  df-seq 12150  df-exp 12209  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-starv 14922  df-sca 14923  df-vsca 14924  df-ip 14925  df-tset 14926  df-ple 14927  df-ds 14929  df-unif 14930  df-0g 15054  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-mhm 16288  df-grp 16379  df-minusg 16380  df-sbg 16381  df-subg 16520  df-ghm 16587  df-cmn 17122  df-abl 17123  df-mgp 17460  df-ur 17472  df-ring 17518  df-cring 17519  df-oppr 17590  df-dvdsr 17608  df-unit 17609  df-rnghom 17682  df-drng 17716  df-subrg 17745  df-staf 17812  df-srng 17813  df-lmod 17832  df-lmhm 17986  df-lvec 18067  df-sra 18136  df-rgmod 18137  df-cnfld 18739  df-phl 18957  df-nlm 21397  df-clm 21853  df-cph 21905
This theorem is referenced by:  nmparlem  21972
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