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Theorem ip2subdi 16548
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 2283 . . . 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 16534 . . . . . . 7  |-  ( W  e.  PreHil  ->  W  e.  LMod )
64, 5syl 15 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
7 phlsrng.f . . . . . . 7  |-  F  =  (Scalar `  W )
87lmodrng 15635 . . . . . 6  |-  ( W  e.  LMod  ->  F  e. 
Ring )
96, 8syl 15 . . . . 5  |-  ( ph  ->  F  e.  Ring )
10 rngabl 15370 . . . . 5  |-  ( F  e.  Ring  ->  F  e. 
Abel )
119, 10syl 15 . . . 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 16537 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  F
) )
174, 12, 13, 16syl3anc 1182 . . . 4  |-  ( ph  ->  ( A  .,  C
)  e.  ( Base `  F ) )
18 ip2subdi.5 . . . . 5  |-  ( ph  ->  D  e.  V )
197, 14, 15, 1ipcl 16537 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  D  e.  V )  ->  ( A  .,  D )  e.  ( Base `  F
) )
204, 12, 18, 19syl3anc 1182 . . . 4  |-  ( ph  ->  ( A  .,  D
)  e.  ( Base `  F ) )
21 ip2subdi.3 . . . . 5  |-  ( ph  ->  B  e.  V )
227, 14, 15, 1ipcl 16537 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .,  C )  e.  ( Base `  F
) )
234, 21, 13, 22syl3anc 1182 . . . 4  |-  ( ph  ->  ( B  .,  C
)  e.  ( Base `  F ) )
241, 2, 3, 11, 17, 20, 23ablsubsub4 15120 . . 3  |-  ( ph  ->  ( ( ( A 
.,  C ) S ( A  .,  D
) ) S ( B  .,  C ) )  =  ( ( A  .,  C ) S ( ( A 
.,  D )  .+  ( B  .,  C ) ) ) )
2524oveq1d 5873 . 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 15670 . . . . 5  |-  ( ( W  e.  LMod  /\  C  e.  V  /\  D  e.  V )  ->  ( C  .-  D )  e.  V )
286, 13, 18, 27syl3anc 1182 . . . 4  |-  ( ph  ->  ( C  .-  D
)  e.  V )
297, 14, 15, 26, 3ipsubdir 16546 . . . 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 1184 . . 3  |-  ( ph  ->  ( ( A  .-  B )  .,  ( C  .-  D ) )  =  ( ( A 
.,  ( C  .-  D ) ) S ( B  .,  ( C  .-  D ) ) ) )
317, 14, 15, 26, 3ipsubdi 16547 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( A  .,  ( C  .-  D ) )  =  ( ( A 
.,  C ) S ( A  .,  D
) ) )
337, 14, 15, 26, 3ipsubdi 16547 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( B  .,  ( C  .-  D ) )  =  ( ( B 
.,  C ) S ( B  .,  D
) ) )
3532, 34oveq12d 5876 . . 3  |-  ( ph  ->  ( ( A  .,  ( C  .-  D ) ) S ( B 
.,  ( C  .-  D ) ) )  =  ( ( ( A  .,  C ) S ( A  .,  D ) ) S ( ( B  .,  C ) S ( B  .,  D ) ) ) )
36 rnggrp 15346 . . . . . 6  |-  ( F  e.  Ring  ->  F  e. 
Grp )
379, 36syl 15 . . . . 5  |-  ( ph  ->  F  e.  Grp )
381, 3grpsubcl 14546 . . . . 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 1182 . . . 4  |-  ( ph  ->  ( ( A  .,  C ) S ( A  .,  D ) )  e.  ( Base `  F ) )
407, 14, 15, 1ipcl 16537 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  D  e.  V )  ->  ( B  .,  D )  e.  ( Base `  F
) )
414, 21, 18, 40syl3anc 1182 . . . 4  |-  ( ph  ->  ( B  .,  D
)  e.  ( Base `  F ) )
421, 2, 3, 11, 39, 23, 41ablsubsub 15119 . . 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 2319 . 2  |-  ( ph  ->  ( ( A  .-  B )  .,  ( C  .-  D ) )  =  ( ( ( ( A  .,  C
) S ( A 
.,  D ) ) S ( B  .,  C ) )  .+  ( B  .,  D ) ) )
441, 2rngacl 15368 . . . 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 1182 . . 3  |-  ( ph  ->  ( ( A  .,  D )  .+  ( B  .,  C ) )  e.  ( Base `  F
) )
461, 2, 3abladdsub 15116 . . 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 1184 . 2  |-  ( ph  ->  ( ( ( A 
.,  C )  .+  ( B  .,  D ) ) S ( ( A  .,  D ) 
.+  ( B  .,  C ) ) )  =  ( ( ( A  .,  C ) S ( ( A 
.,  D )  .+  ( B  .,  C ) ) )  .+  ( B  .,  D ) ) )
4825, 43, 473eqtr4d 2325 1  |-  ( ph  ->  ( ( A  .-  B )  .,  ( C  .-  D ) )  =  ( ( ( A  .,  C ) 
.+  ( B  .,  D ) ) S ( ( A  .,  D )  .+  ( B  .,  C ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .icip 13213   Grpcgrp 14362   -gcsg 14365   Abelcabel 15090   Ringcrg 15337   LModclmod 15627   PreHilcphl 16528
This theorem is referenced by:  cph2subdi  18645  ipcau2  18664  tchcphlem1  18665
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-minusg 14490  df-sbg 14491  df-ghm 14681  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-rnghom 15496  df-staf 15610  df-srng 15611  df-lmod 15629  df-lmhm 15779  df-lvec 15856  df-sra 15925  df-rgmod 15926  df-phl 16530
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