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Theorem ipsubdi 18031
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
ipsubdi  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  ( B  .-  C
) )  =  ( ( A  .,  B
) S ( A 
.,  C ) ) )

Proof of Theorem ipsubdi
StepHypRef Expression
1 simpl 454 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  PreHil )
2 simpr1 989 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  A  e.  V )
3 phllmod 18018 . . . . . . . 8  |-  ( W  e.  PreHil  ->  W  e.  LMod )
43adantr 462 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  LMod )
5 lmodgrp 16935 . . . . . . 7  |-  ( W  e.  LMod  ->  W  e. 
Grp )
64, 5syl 16 . . . . . 6  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  Grp )
7 simpr2 990 . . . . . 6  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  B  e.  V )
8 simpr3 991 . . . . . 6  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  C  e.  V )
9 phllmhm.v . . . . . . 7  |-  V  =  ( Base `  W
)
10 ipsubdir.m . . . . . . 7  |-  .-  =  ( -g `  W )
119, 10grpsubcl 15599 . . . . . 6  |-  ( ( W  e.  Grp  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .-  C
)  e.  V )
126, 7, 8, 11syl3anc 1213 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( B  .-  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, 9, 15, 16ipdi 18028 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  ( B  .-  C )  e.  V  /\  C  e.  V ) )  -> 
( A  .,  (
( B  .-  C
) ( +g  `  W
) C ) )  =  ( ( A 
.,  ( B  .-  C ) ) ( +g  `  F ) ( A  .,  C
) ) )
181, 2, 12, 8, 17syl13anc 1215 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  ( ( B  .-  C ) ( +g  `  W ) C ) )  =  ( ( A  .,  ( B 
.-  C ) ) ( +g  `  F
) ( A  .,  C ) ) )
199, 15, 10grpnpcan 15610 . . . . . 6  |-  ( ( W  e.  Grp  /\  B  e.  V  /\  C  e.  V )  ->  ( ( B  .-  C ) ( +g  `  W ) C )  =  B )
206, 7, 8, 19syl3anc 1213 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( B  .-  C ) ( +g  `  W ) C )  =  B )
2120oveq2d 6106 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  ( ( B  .-  C ) ( +g  `  W ) C ) )  =  ( A 
.,  B ) )
2218, 21eqtr3d 2475 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .,  ( B  .-  C ) ) ( +g  `  F ) ( A  .,  C
) )  =  ( A  .,  B ) )
2313lmodfgrp 16937 . . . . 5  |-  ( W  e.  LMod  ->  F  e. 
Grp )
244, 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, 9, 25ipcl 18021 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .,  B )  e.  ( Base `  F
) )
271, 2, 7, 26syl3anc 1213 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  B )  e.  (
Base `  F )
)
2813, 14, 9, 25ipcl 18021 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  F
) )
291, 2, 8, 28syl3anc 1213 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  C )  e.  (
Base `  F )
)
3013, 14, 9, 25ipcl 18021 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  ( B  .-  C )  e.  V )  ->  ( A  .,  ( B  .-  C ) )  e.  ( Base `  F
) )
311, 2, 12, 30syl3anc 1213 . . . 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 15606 . . . 4  |-  ( ( F  e.  Grp  /\  ( ( A  .,  B )  e.  (
Base `  F )  /\  ( A  .,  C
)  e.  ( Base `  F )  /\  ( A  .,  ( B  .-  C ) )  e.  ( Base `  F
) ) )  -> 
( ( ( A 
.,  B ) S ( A  .,  C
) )  =  ( A  .,  ( B 
.-  C ) )  <-> 
( ( A  .,  ( B  .-  C ) ) ( +g  `  F
) ( A  .,  C ) )  =  ( A  .,  B
) ) )
3424, 27, 29, 31, 33syl13anc 1215 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( A  .,  B
) S ( A 
.,  C ) )  =  ( A  .,  ( B  .-  C ) )  <->  ( ( A 
.,  ( B  .-  C ) ) ( +g  `  F ) ( A  .,  C
) )  =  ( A  .,  B ) ) )
3522, 34mpbird 232 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .,  B ) S ( A  .,  C
) )  =  ( A  .,  ( B 
.-  C ) ) )
3635eqcomd 2446 1  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  ( B  .-  C
) )  =  ( ( A  .,  B
) S ( A 
.,  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234  Scalarcsca 14237   .icip 14239   Grpcgrp 15406   -gcsg 15409   LModclmod 16928   PreHilcphl 18012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-ip 14252  df-0g 14376  df-mnd 15411  df-mhm 15460  df-grp 15538  df-minusg 15539  df-sbg 15540  df-ghm 15738  df-mgp 16582  df-ur 16594  df-rng 16637  df-oppr 16705  df-rnghom 16796  df-staf 16910  df-srng 16911  df-lmod 16930  df-lmhm 17081  df-lvec 17162  df-sra 17231  df-rgmod 17232  df-phl 18014
This theorem is referenced by:  ip2subdi  18032  ip2eq  18041  cphsubdi  20686
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