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Theorem ipsubdi 18769
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 455 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  PreHil )
2 simpr1 1000 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  A  e.  V )
3 phllmod 18756 . . . . . . . 8  |-  ( W  e.  PreHil  ->  W  e.  LMod )
43adantr 463 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  LMod )
5 lmodgrp 17632 . . . . . . 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 1001 . . . . . 6  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  B  e.  V )
8 simpr3 1002 . . . . . 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 16235 . . . . . 6  |-  ( ( W  e.  Grp  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .-  C
)  e.  V )
126, 7, 8, 11syl3anc 1226 . . . . 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 2382 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
16 eqid 2382 . . . . . 6  |-  ( +g  `  F )  =  ( +g  `  F )
1713, 14, 9, 15, 16ipdi 18766 . . . . 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 1228 . . . 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 16247 . . . . . 6  |-  ( ( W  e.  Grp  /\  B  e.  V  /\  C  e.  V )  ->  ( ( B  .-  C ) ( +g  `  W ) C )  =  B )
206, 7, 8, 19syl3anc 1226 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( B  .-  C ) ( +g  `  W ) C )  =  B )
2120oveq2d 6212 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  ( ( B  .-  C ) ( +g  `  W ) C ) )  =  ( A 
.,  B ) )
2218, 21eqtr3d 2425 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .,  ( B  .-  C ) ) ( +g  `  F ) ( A  .,  C
) )  =  ( A  .,  B ) )
2313lmodfgrp 17634 . . . . 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 2382 . . . . . 6  |-  ( Base `  F )  =  (
Base `  F )
2613, 14, 9, 25ipcl 18759 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .,  B )  e.  ( Base `  F
) )
271, 2, 7, 26syl3anc 1226 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  B )  e.  (
Base `  F )
)
2813, 14, 9, 25ipcl 18759 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  F
) )
291, 2, 8, 28syl3anc 1226 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  C )  e.  (
Base `  F )
)
3013, 14, 9, 25ipcl 18759 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  ( B  .-  C )  e.  V )  ->  ( A  .,  ( B  .-  C ) )  e.  ( Base `  F
) )
311, 2, 12, 30syl3anc 1226 . . . 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 16243 . . . 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 1228 . . 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 2390 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   ` cfv 5496  (class class class)co 6196   Basecbs 14634   +g cplusg 14702  Scalarcsca 14705   .icip 14707   Grpcgrp 16170   -gcsg 16172   LModclmod 17625   PreHilcphl 18750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-tpos 6873  df-recs 6960  df-rdg 6994  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-plusg 14715  df-mulr 14716  df-sca 14718  df-vsca 14719  df-ip 14720  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-mhm 16083  df-grp 16174  df-minusg 16175  df-sbg 16176  df-ghm 16382  df-mgp 17255  df-ur 17267  df-ring 17313  df-oppr 17385  df-rnghom 17477  df-staf 17607  df-srng 17608  df-lmod 17627  df-lmhm 17781  df-lvec 17862  df-sra 17931  df-rgmod 17932  df-phl 18752
This theorem is referenced by:  ip2subdi  18770  ip2eq  18779  cphsubdi  21740
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