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Theorem ip2subdi 17915
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 2433 . . . 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 17901 . . . . . . 7  |-  ( W  e.  PreHil  ->  W  e.  LMod )
64, 5syl 16 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
7 phlsrng.f . . . . . . 7  |-  F  =  (Scalar `  W )
87lmodrng 16880 . . . . . 6  |-  ( W  e.  LMod  ->  F  e. 
Ring )
96, 8syl 16 . . . . 5  |-  ( ph  ->  F  e.  Ring )
10 rngabl 16610 . . . . 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 17904 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  F
) )
174, 12, 13, 16syl3anc 1211 . . . 4  |-  ( ph  ->  ( A  .,  C
)  e.  ( Base `  F ) )
18 ip2subdi.5 . . . . 5  |-  ( ph  ->  D  e.  V )
197, 14, 15, 1ipcl 17904 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  D  e.  V )  ->  ( A  .,  D )  e.  ( Base `  F
) )
204, 12, 18, 19syl3anc 1211 . . . 4  |-  ( ph  ->  ( A  .,  D
)  e.  ( Base `  F ) )
21 ip2subdi.3 . . . . 5  |-  ( ph  ->  B  e.  V )
227, 14, 15, 1ipcl 17904 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .,  C )  e.  ( Base `  F
) )
234, 21, 13, 22syl3anc 1211 . . . 4  |-  ( ph  ->  ( B  .,  C
)  e.  ( Base `  F ) )
241, 2, 3, 11, 17, 20, 23ablsubsub4 16288 . . 3  |-  ( ph  ->  ( ( ( A 
.,  C ) S ( A  .,  D
) ) S ( B  .,  C ) )  =  ( ( A  .,  C ) S ( ( A 
.,  D )  .+  ( B  .,  C ) ) ) )
2524oveq1d 6095 . 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 16914 . . . . 5  |-  ( ( W  e.  LMod  /\  C  e.  V  /\  D  e.  V )  ->  ( C  .-  D )  e.  V )
286, 13, 18, 27syl3anc 1211 . . . 4  |-  ( ph  ->  ( C  .-  D
)  e.  V )
297, 14, 15, 26, 3ipsubdir 17913 . . . 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 1213 . . 3  |-  ( ph  ->  ( ( A  .-  B )  .,  ( C  .-  D ) )  =  ( ( A 
.,  ( C  .-  D ) ) S ( B  .,  ( C  .-  D ) ) ) )
317, 14, 15, 26, 3ipsubdi 17914 . . . . 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 1213 . . . 4  |-  ( ph  ->  ( A  .,  ( C  .-  D ) )  =  ( ( A 
.,  C ) S ( A  .,  D
) ) )
337, 14, 15, 26, 3ipsubdi 17914 . . . . 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 1213 . . . 4  |-  ( ph  ->  ( B  .,  ( C  .-  D ) )  =  ( ( B 
.,  C ) S ( B  .,  D
) ) )
3532, 34oveq12d 6098 . . 3  |-  ( ph  ->  ( ( A  .,  ( C  .-  D ) ) S ( B 
.,  ( C  .-  D ) ) )  =  ( ( ( A  .,  C ) S ( A  .,  D ) ) S ( ( B  .,  C ) S ( B  .,  D ) ) ) )
36 rnggrp 16586 . . . . . 6  |-  ( F  e.  Ring  ->  F  e. 
Grp )
379, 36syl 16 . . . . 5  |-  ( ph  ->  F  e.  Grp )
381, 3grpsubcl 15586 . . . . 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 1211 . . . 4  |-  ( ph  ->  ( ( A  .,  C ) S ( A  .,  D ) )  e.  ( Base `  F ) )
407, 14, 15, 1ipcl 17904 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  D  e.  V )  ->  ( B  .,  D )  e.  ( Base `  F
) )
414, 21, 18, 40syl3anc 1211 . . . 4  |-  ( ph  ->  ( B  .,  D
)  e.  ( Base `  F ) )
421, 2, 3, 11, 39, 23, 41ablsubsub 16287 . . 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 2469 . 2  |-  ( ph  ->  ( ( A  .-  B )  .,  ( C  .-  D ) )  =  ( ( ( ( A  .,  C
) S ( A 
.,  D ) ) S ( B  .,  C ) )  .+  ( B  .,  D ) ) )
441, 2rngacl 16608 . . . 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 1211 . . 3  |-  ( ph  ->  ( ( A  .,  D )  .+  ( B  .,  C ) )  e.  ( Base `  F
) )
461, 2, 3abladdsub 16284 . . 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 1213 . 2  |-  ( ph  ->  ( ( ( A 
.,  C )  .+  ( B  .,  D ) ) S ( ( A  .,  D ) 
.+  ( B  .,  C ) ) )  =  ( ( ( A  .,  C ) S ( ( A 
.,  D )  .+  ( B  .,  C ) ) )  .+  ( B  .,  D ) ) )
4825, 43, 473eqtr4d 2475 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 1362    e. wcel 1755   ` cfv 5406  (class class class)co 6080   Basecbs 14157   +g cplusg 14221  Scalarcsca 14224   .icip 14226   Grpcgrp 15393   -gcsg 15396   Abelcabel 16258   Ringcrg 16577   LModclmod 16872   PreHilcphl 17895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-tpos 6734  df-recs 6818  df-rdg 6852  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-plusg 14234  df-mulr 14235  df-sca 14237  df-vsca 14238  df-ip 14239  df-0g 14363  df-mnd 15398  df-mhm 15447  df-grp 15525  df-minusg 15526  df-sbg 15527  df-ghm 15725  df-cmn 16259  df-abl 16260  df-mgp 16566  df-rng 16580  df-ur 16582  df-oppr 16649  df-rnghom 16740  df-staf 16854  df-srng 16855  df-lmod 16874  df-lmhm 17025  df-lvec 17106  df-sra 17175  df-rgmod 17176  df-phl 17897
This theorem is referenced by:  cph2subdi  20570  ipcau2  20591  tchcphlem1  20592
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