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Theorem cph2subdi 21391
Description: Distributive law for inner product subtraction. Complex version of ip2subdi 18446. (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 21371 . . . . . 6  |-  ( W  e.  CPreHil  ->  W  e. CMod )
31, 2syl 16 . . . . 5  |-  ( ph  ->  W  e. CMod )
4 eqid 2467 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
54clmadd 21309 . . . . 5  |-  ( W  e. CMod  ->  +  =  ( +g  `  (Scalar `  W ) ) )
63, 5syl 16 . . . 4  |-  ( ph  ->  +  =  ( +g  `  (Scalar `  W )
) )
76oveqd 6299 . . 3  |-  ( ph  ->  ( ( A  .,  C )  +  ( B  .,  D ) )  =  ( ( A  .,  C ) ( +g  `  (Scalar `  W ) ) ( B  .,  D ) ) )
86oveqd 6299 . . 3  |-  ( ph  ->  ( ( A  .,  D )  +  ( B  .,  C ) )  =  ( ( A  .,  D ) ( +g  `  (Scalar `  W ) ) ( B  .,  C ) ) )
97, 8oveq12d 6300 . 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 21353 . . . . . 6  |-  ( W  e.  CPreHil  ->  W  e.  PreHil )
111, 10syl 16 . . . . 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 2467 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
174, 14, 15, 16ipcl 18435 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  (Scalar `  W ) ) )
1811, 12, 13, 17syl3anc 1228 . . . 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 18435 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  D  e.  V )  ->  ( B  .,  D )  e.  ( Base `  (Scalar `  W ) ) )
2211, 19, 20, 21syl3anc 1228 . . . 4  |-  ( ph  ->  ( B  .,  D
)  e.  ( Base `  (Scalar `  W )
) )
234, 16clmacl 21318 . . . 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 1228 . . 3  |-  ( ph  ->  ( ( A  .,  C )  +  ( B  .,  D ) )  e.  ( Base `  (Scalar `  W )
) )
254, 14, 15, 16ipcl 18435 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  D  e.  V )  ->  ( A  .,  D )  e.  ( Base `  (Scalar `  W ) ) )
2611, 12, 20, 25syl3anc 1228 . . . 4  |-  ( ph  ->  ( A  .,  D
)  e.  ( Base `  (Scalar `  W )
) )
274, 14, 15, 16ipcl 18435 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .,  C )  e.  ( Base `  (Scalar `  W ) ) )
2811, 19, 13, 27syl3anc 1228 . . . 4  |-  ( ph  ->  ( B  .,  C
)  e.  ( Base `  (Scalar `  W )
) )
294, 16clmacl 21318 . . . 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 1228 . . 3  |-  ( ph  ->  ( ( A  .,  D )  +  ( B  .,  C ) )  e.  ( Base `  (Scalar `  W )
) )
314, 16clmsub 21315 . . 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 1228 . 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 2467 . . 3  |-  ( -g `  (Scalar `  W )
)  =  ( -g `  (Scalar `  W )
)
35 eqid 2467 . . 3  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
364, 14, 15, 33, 34, 35, 11, 12, 19, 13, 20ip2subdi 18446 . 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 2519 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 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282    + caddc 9491    - cmin 9801   Basecbs 14486   +g cplusg 14551  Scalarcsca 14554   .icip 14556   -gcsg 15726   PreHilcphl 18426  CModcclm 21297   CPreHilccph 21348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-seq 12072  df-exp 12131  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-0g 14693  df-mnd 15728  df-mhm 15777  df-grp 15858  df-minusg 15859  df-sbg 15860  df-subg 15993  df-ghm 16060  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-cring 16989  df-oppr 17056  df-dvdsr 17074  df-unit 17075  df-rnghom 17148  df-drng 17181  df-subrg 17210  df-staf 17277  df-srng 17278  df-lmod 17297  df-lmhm 17451  df-lvec 17532  df-sra 17601  df-rgmod 17602  df-cnfld 18192  df-phl 18428  df-nlm 20842  df-clm 21298  df-cph 21350
This theorem is referenced by:  nmparlem  21417
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