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Theorem ipsubdi 16874
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 444 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  PreHil )
2 simpr1 963 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  A  e.  V )
3 phllmod 16861 . . . . . . . 8  |-  ( W  e.  PreHil  ->  W  e.  LMod )
43adantr 452 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  LMod )
5 lmodgrp 15957 . . . . . . 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 964 . . . . . 6  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  B  e.  V )
8 simpr3 965 . . . . . 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 14869 . . . . . 6  |-  ( ( W  e.  Grp  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .-  C
)  e.  V )
126, 7, 8, 11syl3anc 1184 . . . . 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 2436 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
16 eqid 2436 . . . . . 6  |-  ( +g  `  F )  =  ( +g  `  F )
1713, 14, 9, 15, 16ipdi 16871 . . . . 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 1186 . . . 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 14880 . . . . . 6  |-  ( ( W  e.  Grp  /\  B  e.  V  /\  C  e.  V )  ->  ( ( B  .-  C ) ( +g  `  W ) C )  =  B )
206, 7, 8, 19syl3anc 1184 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( B  .-  C ) ( +g  `  W ) C )  =  B )
2120oveq2d 6097 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  ( ( B  .-  C ) ( +g  `  W ) C ) )  =  ( A 
.,  B ) )
2218, 21eqtr3d 2470 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .,  ( B  .-  C ) ) ( +g  `  F ) ( A  .,  C
) )  =  ( A  .,  B ) )
2313lmodfgrp 15959 . . . . 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 2436 . . . . . 6  |-  ( Base `  F )  =  (
Base `  F )
2613, 14, 9, 25ipcl 16864 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .,  B )  e.  ( Base `  F
) )
271, 2, 7, 26syl3anc 1184 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  B )  e.  (
Base `  F )
)
2813, 14, 9, 25ipcl 16864 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  F
) )
291, 2, 8, 28syl3anc 1184 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  C )  e.  (
Base `  F )
)
3013, 14, 9, 25ipcl 16864 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  ( B  .-  C )  e.  V )  ->  ( A  .,  ( B  .-  C ) )  e.  ( Base `  F
) )
311, 2, 12, 30syl3anc 1184 . . . 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 14876 . . . 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 1186 . . 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 224 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .,  B ) S ( A  .,  C
) )  =  ( A  .,  ( B 
.-  C ) ) )
3635eqcomd 2441 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529  Scalarcsca 13532   .icip 13534   Grpcgrp 14685   -gcsg 14688   LModclmod 15950   PreHilcphl 16855
This theorem is referenced by:  ip2subdi  16875  ip2eq  16884  cphsubdi  19171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-0g 13727  df-mnd 14690  df-mhm 14738  df-grp 14812  df-minusg 14813  df-sbg 14814  df-ghm 15004  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-rnghom 15819  df-staf 15933  df-srng 15934  df-lmod 15952  df-lmhm 16098  df-lvec 16175  df-sra 16244  df-rgmod 16245  df-phl 16857
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