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Theorem ipsubdir 18544
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 18532 . . . . . . . 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 17387 . . . . . . 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 1001 . . . . . 6  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  A  e.  V )
7 simpr2 1002 . . . . . 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 15987 . . . . . 6  |-  ( ( W  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .-  B
)  e.  V )
115, 6, 7, 10syl3anc 1227 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .-  B )  e.  V
)
12 simpr3 1003 . . . . 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 2441 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
16 eqid 2441 . . . . . 6  |-  ( +g  `  F )  =  ( +g  `  F )
1713, 14, 8, 15, 16ipdir 18541 . . . . 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 1229 . . . 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 15999 . . . . . 6  |-  ( ( W  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A  .-  B ) ( +g  `  W ) B )  =  A )
205, 6, 7, 19syl3anc 1227 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .-  B ) ( +g  `  W ) B )  =  A )
2120oveq1d 6292 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( A  .-  B
) ( +g  `  W
) B )  .,  C )  =  ( A  .,  C ) )
2218, 21eqtr3d 2484 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( A  .-  B
)  .,  C )
( +g  `  F ) ( B  .,  C
) )  =  ( A  .,  C ) )
2313lmodfgrp 17389 . . . . 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 2441 . . . . . 6  |-  ( Base `  F )  =  (
Base `  F )
2613, 14, 8, 25ipcl 18535 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  F
) )
271, 6, 12, 26syl3anc 1227 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  C )  e.  (
Base `  F )
)
2813, 14, 8, 25ipcl 18535 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .,  C )  e.  ( Base `  F
) )
291, 7, 12, 28syl3anc 1227 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( B  .,  C )  e.  (
Base `  F )
)
3013, 14, 8, 25ipcl 18535 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  .-  B )  e.  V  /\  C  e.  V )  ->  (
( A  .-  B
)  .,  C )  e.  ( Base `  F
) )
311, 11, 12, 30syl3anc 1227 . . . 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 15995 . . . 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 1229 . . 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 2449 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 972    = wceq 1381    e. wcel 1802   ` cfv 5574  (class class class)co 6277   Basecbs 14504   +g cplusg 14569  Scalarcsca 14572   .icip 14574   Grpcgrp 15922   -gcsg 15924   LModclmod 17380   PreHilcphl 18526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-plusg 14582  df-sca 14585  df-vsca 14586  df-ip 14587  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-minusg 15927  df-sbg 15928  df-ghm 16134  df-ring 17068  df-lmod 17382  df-lmhm 17536  df-lvec 17617  df-sra 17686  df-rgmod 17687  df-phl 18528
This theorem is referenced by:  ip2subdi  18546  cphsubdir  21520
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