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Theorem ipsubdir 16828
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 444 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  PreHil )
2 phllmod 16816 . . . . . . . 8  |-  ( W  e.  PreHil  ->  W  e.  LMod )
32adantr 452 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  LMod )
4 lmodgrp 15912 . . . . . . 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 963 . . . . . 6  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  A  e.  V )
7 simpr2 964 . . . . . 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 14824 . . . . . 6  |-  ( ( W  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .-  B
)  e.  V )
115, 6, 7, 10syl3anc 1184 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .-  B )  e.  V
)
12 simpr3 965 . . . . 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 2404 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
16 eqid 2404 . . . . . 6  |-  ( +g  `  F )  =  ( +g  `  F )
1713, 14, 8, 15, 16ipdir 16825 . . . . 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 1186 . . . 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 14835 . . . . . 6  |-  ( ( W  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A  .-  B ) ( +g  `  W ) B )  =  A )
205, 6, 7, 19syl3anc 1184 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .-  B ) ( +g  `  W ) B )  =  A )
2120oveq1d 6055 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( A  .-  B
) ( +g  `  W
) B )  .,  C )  =  ( A  .,  C ) )
2218, 21eqtr3d 2438 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( A  .-  B
)  .,  C )
( +g  `  F ) ( B  .,  C
) )  =  ( A  .,  C ) )
2313lmodfgrp 15914 . . . . 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 2404 . . . . . 6  |-  ( Base `  F )  =  (
Base `  F )
2613, 14, 8, 25ipcl 16819 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  F
) )
271, 6, 12, 26syl3anc 1184 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  C )  e.  (
Base `  F )
)
2813, 14, 8, 25ipcl 16819 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .,  C )  e.  ( Base `  F
) )
291, 7, 12, 28syl3anc 1184 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( B  .,  C )  e.  (
Base `  F )
)
3013, 14, 8, 25ipcl 16819 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  .-  B )  e.  V  /\  C  e.  V )  ->  (
( A  .-  B
)  .,  C )  e.  ( Base `  F
) )
311, 11, 12, 30syl3anc 1184 . . . 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 14831 . . . 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 1186 . . 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 224 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .,  C ) S ( B  .,  C
) )  =  ( ( A  .-  B
)  .,  C )
)
3635eqcomd 2409 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484  Scalarcsca 13487   .icip 13489   Grpcgrp 14640   -gcsg 14643   LModclmod 15905   PreHilcphl 16810
This theorem is referenced by:  ip2subdi  16830  cphsubdir  19123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-plusg 13497  df-sca 13500  df-vsca 13501  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-ghm 14959  df-rng 15618  df-lmod 15907  df-lmhm 16053  df-lvec 16130  df-sra 16199  df-rgmod 16200  df-phl 16812
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