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Theorem ipsubdir 18546
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 457 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  PreHil )
2 phllmod 18534 . . . . . . . 8  |-  ( W  e.  PreHil  ->  W  e.  LMod )
32adantr 465 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  LMod )
4 lmodgrp 17390 . . . . . . 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 1002 . . . . . 6  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  A  e.  V )
7 simpr2 1003 . . . . . 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 15990 . . . . . 6  |-  ( ( W  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .-  B
)  e.  V )
115, 6, 7, 10syl3anc 1228 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .-  B )  e.  V
)
12 simpr3 1004 . . . . 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 2467 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
16 eqid 2467 . . . . . 6  |-  ( +g  `  F )  =  ( +g  `  F )
1713, 14, 8, 15, 16ipdir 18543 . . . . 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 1230 . . . 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 16002 . . . . . 6  |-  ( ( W  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A  .-  B ) ( +g  `  W ) B )  =  A )
205, 6, 7, 19syl3anc 1228 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .-  B ) ( +g  `  W ) B )  =  A )
2120oveq1d 6310 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( A  .-  B
) ( +g  `  W
) B )  .,  C )  =  ( A  .,  C ) )
2218, 21eqtr3d 2510 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( A  .-  B
)  .,  C )
( +g  `  F ) ( B  .,  C
) )  =  ( A  .,  C ) )
2313lmodfgrp 17392 . . . . 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 2467 . . . . . 6  |-  ( Base `  F )  =  (
Base `  F )
2613, 14, 8, 25ipcl 18537 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  F
) )
271, 6, 12, 26syl3anc 1228 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  C )  e.  (
Base `  F )
)
2813, 14, 8, 25ipcl 18537 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .,  C )  e.  ( Base `  F
) )
291, 7, 12, 28syl3anc 1228 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( B  .,  C )  e.  (
Base `  F )
)
3013, 14, 8, 25ipcl 18537 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  .-  B )  e.  V  /\  C  e.  V )  ->  (
( A  .-  B
)  .,  C )  e.  ( Base `  F
) )
311, 11, 12, 30syl3anc 1228 . . . 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 15998 . . . 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 1230 . . 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 2475 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 973    = wceq 1379    e. wcel 1767   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572  Scalarcsca 14575   .icip 14577   Grpcgrp 15925   -gcsg 15927   LModclmod 17383   PreHilcphl 18528
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-plusg 14585  df-sca 14588  df-vsca 14589  df-ip 14590  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-ghm 16137  df-ring 17072  df-lmod 17385  df-lmhm 17539  df-lvec 17620  df-sra 17689  df-rgmod 17690  df-phl 18530
This theorem is referenced by:  ip2subdi  18548  cphsubdir  21522
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