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Theorem ipsubdi 18438
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 457 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  PreHil )
2 simpr1 997 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  A  e.  V )
3 phllmod 18425 . . . . . . . 8  |-  ( W  e.  PreHil  ->  W  e.  LMod )
43adantr 465 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  LMod )
5 lmodgrp 17295 . . . . . . 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 998 . . . . . 6  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  B  e.  V )
8 simpr3 999 . . . . . 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 15912 . . . . . 6  |-  ( ( W  e.  Grp  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .-  C
)  e.  V )
126, 7, 8, 11syl3anc 1223 . . . . 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 2460 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
16 eqid 2460 . . . . . 6  |-  ( +g  `  F )  =  ( +g  `  F )
1713, 14, 9, 15, 16ipdi 18435 . . . . 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 1225 . . . 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 15924 . . . . . 6  |-  ( ( W  e.  Grp  /\  B  e.  V  /\  C  e.  V )  ->  ( ( B  .-  C ) ( +g  `  W ) C )  =  B )
206, 7, 8, 19syl3anc 1223 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( B  .-  C ) ( +g  `  W ) C )  =  B )
2120oveq2d 6291 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  ( ( B  .-  C ) ( +g  `  W ) C ) )  =  ( A 
.,  B ) )
2218, 21eqtr3d 2503 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .,  ( B  .-  C ) ) ( +g  `  F ) ( A  .,  C
) )  =  ( A  .,  B ) )
2313lmodfgrp 17297 . . . . 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 2460 . . . . . 6  |-  ( Base `  F )  =  (
Base `  F )
2613, 14, 9, 25ipcl 18428 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .,  B )  e.  ( Base `  F
) )
271, 2, 7, 26syl3anc 1223 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  B )  e.  (
Base `  F )
)
2813, 14, 9, 25ipcl 18428 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  F
) )
291, 2, 8, 28syl3anc 1223 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  C )  e.  (
Base `  F )
)
3013, 14, 9, 25ipcl 18428 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  ( B  .-  C )  e.  V )  ->  ( A  .,  ( B  .-  C ) )  e.  ( Base `  F
) )
311, 2, 12, 30syl3anc 1223 . . . 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 15920 . . . 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 1225 . . 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 2468 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 968    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544  Scalarcsca 14547   .icip 14549   Grpcgrp 15716   -gcsg 15719   LModclmod 17288   PreHilcphl 18419
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-tpos 6945  df-recs 7032  df-rdg 7066  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-ip 14562  df-0g 14686  df-mnd 15721  df-mhm 15770  df-grp 15851  df-minusg 15852  df-sbg 15853  df-ghm 16053  df-mgp 16925  df-ur 16937  df-rng 16981  df-oppr 17049  df-rnghom 17141  df-staf 17270  df-srng 17271  df-lmod 17290  df-lmhm 17444  df-lvec 17525  df-sra 17594  df-rgmod 17595  df-phl 18421
This theorem is referenced by:  ip2subdi  18439  ip2eq  18448  cphsubdi  21383
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