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Theorem ipsubdir 18206
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 18194 . . . . . . . 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 17088 . . . . . . 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 994 . . . . . 6  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  A  e.  V )
7 simpr2 995 . . . . . 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 15729 . . . . . 6  |-  ( ( W  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .-  B
)  e.  V )
115, 6, 7, 10syl3anc 1219 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .-  B )  e.  V
)
12 simpr3 996 . . . . 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 2454 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
16 eqid 2454 . . . . . 6  |-  ( +g  `  F )  =  ( +g  `  F )
1713, 14, 8, 15, 16ipdir 18203 . . . . 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 1221 . . . 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 15740 . . . . . 6  |-  ( ( W  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A  .-  B ) ( +g  `  W ) B )  =  A )
205, 6, 7, 19syl3anc 1219 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .-  B ) ( +g  `  W ) B )  =  A )
2120oveq1d 6218 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( A  .-  B
) ( +g  `  W
) B )  .,  C )  =  ( A  .,  C ) )
2218, 21eqtr3d 2497 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( A  .-  B
)  .,  C )
( +g  `  F ) ( B  .,  C
) )  =  ( A  .,  C ) )
2313lmodfgrp 17090 . . . . 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 2454 . . . . . 6  |-  ( Base `  F )  =  (
Base `  F )
2613, 14, 8, 25ipcl 18197 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  F
) )
271, 6, 12, 26syl3anc 1219 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  C )  e.  (
Base `  F )
)
2813, 14, 8, 25ipcl 18197 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .,  C )  e.  ( Base `  F
) )
291, 7, 12, 28syl3anc 1219 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( B  .,  C )  e.  (
Base `  F )
)
3013, 14, 8, 25ipcl 18197 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  .-  B )  e.  V  /\  C  e.  V )  ->  (
( A  .-  B
)  .,  C )  e.  ( Base `  F
) )
311, 11, 12, 30syl3anc 1219 . . . 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 15736 . . . 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 1221 . . 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 2462 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 965    = wceq 1370    e. wcel 1758   ` cfv 5529  (class class class)co 6203   Basecbs 14296   +g cplusg 14361  Scalarcsca 14364   .icip 14366   Grpcgrp 15533   -gcsg 15536   LModclmod 17081   PreHilcphl 18188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-plusg 14374  df-sca 14377  df-vsca 14378  df-ip 14379  df-0g 14503  df-mnd 15538  df-grp 15668  df-minusg 15669  df-sbg 15670  df-ghm 15868  df-rng 16780  df-lmod 17083  df-lmhm 17236  df-lvec 17317  df-sra 17386  df-rgmod 17387  df-phl 18190
This theorem is referenced by:  ip2subdi  18208  cphsubdir  20868
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