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Theorem ipsubdir 17913
Description: Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-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
)
Assertion
Ref Expression
ipsubdir  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .-  B )  .,  C )  =  ( ( A  .,  C
) S ( B 
.,  C ) ) )

Proof of Theorem ipsubdir
StepHypRef Expression
1 simpl 454 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  PreHil )
2 phllmod 17901 . . . . . . . 8  |-  ( W  e.  PreHil  ->  W  e.  LMod )
32adantr 462 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  LMod )
4 lmodgrp 16879 . . . . . . 7  |-  ( W  e.  LMod  ->  W  e. 
Grp )
53, 4syl 16 . . . . . 6  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  Grp )
6 simpr1 987 . . . . . 6  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  A  e.  V )
7 simpr2 988 . . . . . 6  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  B  e.  V )
8 phllmhm.v . . . . . . 7  |-  V  =  ( Base `  W
)
9 ipsubdir.m . . . . . . 7  |-  .-  =  ( -g `  W )
108, 9grpsubcl 15586 . . . . . 6  |-  ( ( W  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .-  B
)  e.  V )
115, 6, 7, 10syl3anc 1211 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .-  B )  e.  V
)
12 simpr3 989 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  C  e.  V )
13 phlsrng.f . . . . . 6  |-  F  =  (Scalar `  W )
14 phllmhm.h . . . . . 6  |-  .,  =  ( .i `  W )
15 eqid 2433 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
16 eqid 2433 . . . . . 6  |-  ( +g  `  F )  =  ( +g  `  F )
1713, 14, 8, 15, 16ipdir 17910 . . . . 5  |-  ( ( W  e.  PreHil  /\  (
( A  .-  B
)  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( A  .-  B
) ( +g  `  W
) B )  .,  C )  =  ( ( ( A  .-  B )  .,  C
) ( +g  `  F
) ( B  .,  C ) ) )
181, 11, 7, 12, 17syl13anc 1213 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( A  .-  B
) ( +g  `  W
) B )  .,  C )  =  ( ( ( A  .-  B )  .,  C
) ( +g  `  F
) ( B  .,  C ) ) )
198, 15, 9grpnpcan 15597 . . . . . 6  |-  ( ( W  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A  .-  B ) ( +g  `  W ) B )  =  A )
205, 6, 7, 19syl3anc 1211 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .-  B ) ( +g  `  W ) B )  =  A )
2120oveq1d 6095 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( A  .-  B
) ( +g  `  W
) B )  .,  C )  =  ( A  .,  C ) )
2218, 21eqtr3d 2467 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( A  .-  B
)  .,  C )
( +g  `  F ) ( B  .,  C
) )  =  ( A  .,  C ) )
2313lmodfgrp 16881 . . . . 5  |-  ( W  e.  LMod  ->  F  e. 
Grp )
243, 23syl 16 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  F  e.  Grp )
25 eqid 2433 . . . . . 6  |-  ( Base `  F )  =  (
Base `  F )
2613, 14, 8, 25ipcl 17904 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  F
) )
271, 6, 12, 26syl3anc 1211 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  C )  e.  (
Base `  F )
)
2813, 14, 8, 25ipcl 17904 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .,  C )  e.  ( Base `  F
) )
291, 7, 12, 28syl3anc 1211 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( B  .,  C )  e.  (
Base `  F )
)
3013, 14, 8, 25ipcl 17904 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  .-  B )  e.  V  /\  C  e.  V )  ->  (
( A  .-  B
)  .,  C )  e.  ( Base `  F
) )
311, 11, 12, 30syl3anc 1211 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .-  B )  .,  C )  e.  (
Base `  F )
)
32 ipsubdir.s . . . . 5  |-  S  =  ( -g `  F
)
3325, 16, 32grpsubadd 15593 . . . 4  |-  ( ( F  e.  Grp  /\  ( ( A  .,  C )  e.  (
Base `  F )  /\  ( B  .,  C
)  e.  ( Base `  F )  /\  (
( A  .-  B
)  .,  C )  e.  ( Base `  F
) ) )  -> 
( ( ( A 
.,  C ) S ( B  .,  C
) )  =  ( ( A  .-  B
)  .,  C )  <->  ( ( ( A  .-  B )  .,  C
) ( +g  `  F
) ( B  .,  C ) )  =  ( A  .,  C
) ) )
3424, 27, 29, 31, 33syl13anc 1213 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( A  .,  C
) S ( B 
.,  C ) )  =  ( ( A 
.-  B )  .,  C )  <->  ( (
( A  .-  B
)  .,  C )
( +g  `  F ) ( B  .,  C
) )  =  ( A  .,  C ) ) )
3522, 34mpbird 232 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .,  C ) S ( B  .,  C
) )  =  ( ( A  .-  B
)  .,  C )
)
3635eqcomd 2438 1  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .-  B )  .,  C )  =  ( ( A  .,  C
) S ( B 
.,  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 958    = 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   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-recs 6818  df-rdg 6852  df-er 7089  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-sca 14237  df-vsca 14238  df-ip 14239  df-0g 14363  df-mnd 15398  df-grp 15525  df-minusg 15526  df-sbg 15527  df-ghm 15725  df-rng 16580  df-lmod 16874  df-lmhm 17025  df-lvec 17106  df-sra 17175  df-rgmod 17176  df-phl 17897
This theorem is referenced by:  ip2subdi  17915  cphsubdir  20568
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