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Theorem ip2subdi 18552
Description: Distributive law for inner product subtraction. (Contributed by Mario Carneiro, 8-Oct-2015.)
Hypotheses
Ref Expression
phlsrng.f  |-  F  =  (Scalar `  W )
phllmhm.h  |-  .,  =  ( .i `  W )
phllmhm.v  |-  V  =  ( Base `  W
)
ipsubdir.m  |-  .-  =  ( -g `  W )
ipsubdir.s  |-  S  =  ( -g `  F
)
ip2subdi.p  |-  .+  =  ( +g  `  F )
ip2subdi.1  |-  ( ph  ->  W  e.  PreHil )
ip2subdi.2  |-  ( ph  ->  A  e.  V )
ip2subdi.3  |-  ( ph  ->  B  e.  V )
ip2subdi.4  |-  ( ph  ->  C  e.  V )
ip2subdi.5  |-  ( ph  ->  D  e.  V )
Assertion
Ref Expression
ip2subdi  |-  ( ph  ->  ( ( A  .-  B )  .,  ( C  .-  D ) )  =  ( ( ( A  .,  C ) 
.+  ( B  .,  D ) ) S ( ( A  .,  D )  .+  ( B  .,  C ) ) ) )

Proof of Theorem ip2subdi
StepHypRef Expression
1 eqid 2443 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
2 ip2subdi.p . . . 4  |-  .+  =  ( +g  `  F )
3 ipsubdir.s . . . 4  |-  S  =  ( -g `  F
)
4 ip2subdi.1 . . . . . . 7  |-  ( ph  ->  W  e.  PreHil )
5 phllmod 18538 . . . . . . 7  |-  ( W  e.  PreHil  ->  W  e.  LMod )
64, 5syl 16 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
7 phlsrng.f . . . . . . 7  |-  F  =  (Scalar `  W )
87lmodring 17394 . . . . . 6  |-  ( W  e.  LMod  ->  F  e. 
Ring )
96, 8syl 16 . . . . 5  |-  ( ph  ->  F  e.  Ring )
10 ringabl 17102 . . . . 5  |-  ( F  e.  Ring  ->  F  e. 
Abel )
119, 10syl 16 . . . 4  |-  ( ph  ->  F  e.  Abel )
12 ip2subdi.2 . . . . 5  |-  ( ph  ->  A  e.  V )
13 ip2subdi.4 . . . . 5  |-  ( ph  ->  C  e.  V )
14 phllmhm.h . . . . . 6  |-  .,  =  ( .i `  W )
15 phllmhm.v . . . . . 6  |-  V  =  ( Base `  W
)
167, 14, 15, 1ipcl 18541 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  F
) )
174, 12, 13, 16syl3anc 1229 . . . 4  |-  ( ph  ->  ( A  .,  C
)  e.  ( Base `  F ) )
18 ip2subdi.5 . . . . 5  |-  ( ph  ->  D  e.  V )
197, 14, 15, 1ipcl 18541 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  D  e.  V )  ->  ( A  .,  D )  e.  ( Base `  F
) )
204, 12, 18, 19syl3anc 1229 . . . 4  |-  ( ph  ->  ( A  .,  D
)  e.  ( Base `  F ) )
21 ip2subdi.3 . . . . 5  |-  ( ph  ->  B  e.  V )
227, 14, 15, 1ipcl 18541 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .,  C )  e.  ( Base `  F
) )
234, 21, 13, 22syl3anc 1229 . . . 4  |-  ( ph  ->  ( B  .,  C
)  e.  ( Base `  F ) )
241, 2, 3, 11, 17, 20, 23ablsubsub4 16703 . . 3  |-  ( ph  ->  ( ( ( A 
.,  C ) S ( A  .,  D
) ) S ( B  .,  C ) )  =  ( ( A  .,  C ) S ( ( A 
.,  D )  .+  ( B  .,  C ) ) ) )
2524oveq1d 6296 . 2  |-  ( ph  ->  ( ( ( ( A  .,  C ) S ( A  .,  D ) ) S ( B  .,  C
) )  .+  ( B  .,  D ) )  =  ( ( ( A  .,  C ) S ( ( A 
.,  D )  .+  ( B  .,  C ) ) )  .+  ( B  .,  D ) ) )
26 ipsubdir.m . . . . . 6  |-  .-  =  ( -g `  W )
2715, 26lmodvsubcl 17429 . . . . 5  |-  ( ( W  e.  LMod  /\  C  e.  V  /\  D  e.  V )  ->  ( C  .-  D )  e.  V )
286, 13, 18, 27syl3anc 1229 . . . 4  |-  ( ph  ->  ( C  .-  D
)  e.  V )
297, 14, 15, 26, 3ipsubdir 18550 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  ( C  .-  D )  e.  V ) )  ->  ( ( A 
.-  B )  .,  ( C  .-  D ) )  =  ( ( A  .,  ( C 
.-  D ) ) S ( B  .,  ( C  .-  D ) ) ) )
304, 12, 21, 28, 29syl13anc 1231 . . 3  |-  ( ph  ->  ( ( A  .-  B )  .,  ( C  .-  D ) )  =  ( ( A 
.,  ( C  .-  D ) ) S ( B  .,  ( C  .-  D ) ) ) )
317, 14, 15, 26, 3ipsubdi 18551 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  C  e.  V  /\  D  e.  V )
)  ->  ( A  .,  ( C  .-  D
) )  =  ( ( A  .,  C
) S ( A 
.,  D ) ) )
324, 12, 13, 18, 31syl13anc 1231 . . . 4  |-  ( ph  ->  ( A  .,  ( C  .-  D ) )  =  ( ( A 
.,  C ) S ( A  .,  D
) ) )
337, 14, 15, 26, 3ipsubdi 18551 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( B  e.  V  /\  C  e.  V  /\  D  e.  V )
)  ->  ( B  .,  ( C  .-  D
) )  =  ( ( B  .,  C
) S ( B 
.,  D ) ) )
344, 21, 13, 18, 33syl13anc 1231 . . . 4  |-  ( ph  ->  ( B  .,  ( C  .-  D ) )  =  ( ( B 
.,  C ) S ( B  .,  D
) ) )
3532, 34oveq12d 6299 . . 3  |-  ( ph  ->  ( ( A  .,  ( C  .-  D ) ) S ( B 
.,  ( C  .-  D ) ) )  =  ( ( ( A  .,  C ) S ( A  .,  D ) ) S ( ( B  .,  C ) S ( B  .,  D ) ) ) )
36 ringgrp 17077 . . . . . 6  |-  ( F  e.  Ring  ->  F  e. 
Grp )
379, 36syl 16 . . . . 5  |-  ( ph  ->  F  e.  Grp )
381, 3grpsubcl 15992 . . . . 5  |-  ( ( F  e.  Grp  /\  ( A  .,  C )  e.  ( Base `  F
)  /\  ( A  .,  D )  e.  (
Base `  F )
)  ->  ( ( A  .,  C ) S ( A  .,  D
) )  e.  (
Base `  F )
)
3937, 17, 20, 38syl3anc 1229 . . . 4  |-  ( ph  ->  ( ( A  .,  C ) S ( A  .,  D ) )  e.  ( Base `  F ) )
407, 14, 15, 1ipcl 18541 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  D  e.  V )  ->  ( B  .,  D )  e.  ( Base `  F
) )
414, 21, 18, 40syl3anc 1229 . . . 4  |-  ( ph  ->  ( B  .,  D
)  e.  ( Base `  F ) )
421, 2, 3, 11, 39, 23, 41ablsubsub 16702 . . 3  |-  ( ph  ->  ( ( ( A 
.,  C ) S ( A  .,  D
) ) S ( ( B  .,  C
) S ( B 
.,  D ) ) )  =  ( ( ( ( A  .,  C ) S ( A  .,  D ) ) S ( B 
.,  C ) ) 
.+  ( B  .,  D ) ) )
4330, 35, 423eqtrd 2488 . 2  |-  ( ph  ->  ( ( A  .-  B )  .,  ( C  .-  D ) )  =  ( ( ( ( A  .,  C
) S ( A 
.,  D ) ) S ( B  .,  C ) )  .+  ( B  .,  D ) ) )
441, 2ringacl 17100 . . . 4  |-  ( ( F  e.  Ring  /\  ( A  .,  D )  e.  ( Base `  F
)  /\  ( B  .,  C )  e.  (
Base `  F )
)  ->  ( ( A  .,  D )  .+  ( B  .,  C ) )  e.  ( Base `  F ) )
459, 20, 23, 44syl3anc 1229 . . 3  |-  ( ph  ->  ( ( A  .,  D )  .+  ( B  .,  C ) )  e.  ( Base `  F
) )
461, 2, 3abladdsub 16699 . . 3  |-  ( ( F  e.  Abel  /\  (
( A  .,  C
)  e.  ( Base `  F )  /\  ( B  .,  D )  e.  ( Base `  F
)  /\  ( ( A  .,  D )  .+  ( B  .,  C ) )  e.  ( Base `  F ) ) )  ->  ( ( ( A  .,  C ) 
.+  ( B  .,  D ) ) S ( ( A  .,  D )  .+  ( B  .,  C ) ) )  =  ( ( ( A  .,  C
) S ( ( A  .,  D ) 
.+  ( B  .,  C ) ) ) 
.+  ( B  .,  D ) ) )
4711, 17, 41, 45, 46syl13anc 1231 . 2  |-  ( ph  ->  ( ( ( A 
.,  C )  .+  ( B  .,  D ) ) S ( ( A  .,  D ) 
.+  ( B  .,  C ) ) )  =  ( ( ( A  .,  C ) S ( ( A 
.,  D )  .+  ( B  .,  C ) ) )  .+  ( B  .,  D ) ) )
4825, 43, 473eqtr4d 2494 1  |-  ( ph  ->  ( ( A  .-  B )  .,  ( C  .-  D ) )  =  ( ( ( A  .,  C ) 
.+  ( B  .,  D ) ) S ( ( A  .,  D )  .+  ( B  .,  C ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804   ` cfv 5578  (class class class)co 6281   Basecbs 14509   +g cplusg 14574  Scalarcsca 14577   .icip 14579   Grpcgrp 15927   -gcsg 15929   Abelcabl 16673   Ringcrg 17072   LModclmod 17386   PreHilcphl 18532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6957  df-recs 7044  df-rdg 7078  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-plusg 14587  df-mulr 14588  df-sca 14590  df-vsca 14591  df-ip 14592  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15840  df-grp 15931  df-minusg 15932  df-sbg 15933  df-ghm 16139  df-cmn 16674  df-abl 16675  df-mgp 17016  df-ur 17028  df-ring 17074  df-oppr 17146  df-rnghom 17238  df-staf 17368  df-srng 17369  df-lmod 17388  df-lmhm 17542  df-lvec 17623  df-sra 17692  df-rgmod 17693  df-phl 18534
This theorem is referenced by:  cph2subdi  21529  ipcau2  21550  tchcphlem1  21551
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