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Theorem ip2eq 18062
Description: Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
ip2eq.h  |-  .,  =  ( .i `  W )
ip2eq.v  |-  V  =  ( Base `  W
)
Assertion
Ref Expression
ip2eq  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  =  B  <->  A. x  e.  V  ( x  .,  A )  =  ( x  .,  B ) ) )
Distinct variable groups:    x, A    x, B    x,  .,    x, V   
x, W

Proof of Theorem ip2eq
StepHypRef Expression
1 oveq2 6094 . . 3  |-  ( A  =  B  ->  (
x  .,  A )  =  ( x  .,  B ) )
21ralrimivw 2795 . 2  |-  ( A  =  B  ->  A. x  e.  V  ( x  .,  A )  =  ( x  .,  B ) )
3 phllmod 18039 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
4 ip2eq.v . . . . . 6  |-  V  =  ( Base `  W
)
5 eqid 2438 . . . . . 6  |-  ( -g `  W )  =  (
-g `  W )
64, 5lmodvsubcl 16970 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( A ( -g `  W
) B )  e.  V )
73, 6syl3an1 1251 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A ( -g `  W
) B )  e.  V )
8 oveq1 6093 . . . . . 6  |-  ( x  =  ( A (
-g `  W ) B )  ->  (
x  .,  A )  =  ( ( A ( -g `  W
) B )  .,  A ) )
9 oveq1 6093 . . . . . 6  |-  ( x  =  ( A (
-g `  W ) B )  ->  (
x  .,  B )  =  ( ( A ( -g `  W
) B )  .,  B ) )
108, 9eqeq12d 2452 . . . . 5  |-  ( x  =  ( A (
-g `  W ) B )  ->  (
( x  .,  A
)  =  ( x 
.,  B )  <->  ( ( A ( -g `  W
) B )  .,  A )  =  ( ( A ( -g `  W ) B ) 
.,  B ) ) )
1110rspcv 3064 . . . 4  |-  ( ( A ( -g `  W
) B )  e.  V  ->  ( A. x  e.  V  (
x  .,  A )  =  ( x  .,  B )  ->  (
( A ( -g `  W ) B ) 
.,  A )  =  ( ( A (
-g `  W ) B )  .,  B
) ) )
127, 11syl 16 . . 3  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A. x  e.  V  ( x  .,  A )  =  ( x  .,  B )  ->  (
( A ( -g `  W ) B ) 
.,  A )  =  ( ( A (
-g `  W ) B )  .,  B
) ) )
13 simp1 988 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  W  e.  PreHil )
14 simp2 989 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  A  e.  V )
15 simp3 990 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  B  e.  V )
16 eqid 2438 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
17 ip2eq.h . . . . . . . 8  |-  .,  =  ( .i `  W )
18 eqid 2438 . . . . . . . 8  |-  ( -g `  (Scalar `  W )
)  =  ( -g `  (Scalar `  W )
)
1916, 17, 4, 5, 18ipsubdi 18052 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  (
( A ( -g `  W ) B )  e.  V  /\  A  e.  V  /\  B  e.  V ) )  -> 
( ( A (
-g `  W ) B )  .,  ( A ( -g `  W
) B ) )  =  ( ( ( A ( -g `  W
) B )  .,  A ) ( -g `  (Scalar `  W )
) ( ( A ( -g `  W
) B )  .,  B ) ) )
2013, 7, 14, 15, 19syl13anc 1220 . . . . . 6  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( A ( -g `  W ) B ) 
.,  ( A (
-g `  W ) B ) )  =  ( ( ( A ( -g `  W
) B )  .,  A ) ( -g `  (Scalar `  W )
) ( ( A ( -g `  W
) B )  .,  B ) ) )
2120eqeq1d 2446 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( ( A (
-g `  W ) B )  .,  ( A ( -g `  W
) B ) )  =  ( 0g `  (Scalar `  W ) )  <-> 
( ( ( A ( -g `  W
) B )  .,  A ) ( -g `  (Scalar `  W )
) ( ( A ( -g `  W
) B )  .,  B ) )  =  ( 0g `  (Scalar `  W ) ) ) )
22 eqid 2438 . . . . . . 7  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
23 eqid 2438 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
2416, 17, 4, 22, 23ipeq0 18047 . . . . . 6  |-  ( ( W  e.  PreHil  /\  ( A ( -g `  W
) B )  e.  V )  ->  (
( ( A (
-g `  W ) B )  .,  ( A ( -g `  W
) B ) )  =  ( 0g `  (Scalar `  W ) )  <-> 
( A ( -g `  W ) B )  =  ( 0g `  W ) ) )
2513, 7, 24syl2anc 661 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( ( A (
-g `  W ) B )  .,  ( A ( -g `  W
) B ) )  =  ( 0g `  (Scalar `  W ) )  <-> 
( A ( -g `  W ) B )  =  ( 0g `  W ) ) )
2621, 25bitr3d 255 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( ( ( A ( -g `  W
) B )  .,  A ) ( -g `  (Scalar `  W )
) ( ( A ( -g `  W
) B )  .,  B ) )  =  ( 0g `  (Scalar `  W ) )  <->  ( A
( -g `  W ) B )  =  ( 0g `  W ) ) )
2733ad2ant1 1009 . . . . . 6  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  W  e.  LMod )
2816lmodfgrp 16937 . . . . . 6  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Grp )
2927, 28syl 16 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (Scalar `  W )  e.  Grp )
30 eqid 2438 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3116, 17, 4, 30ipcl 18042 . . . . . 6  |-  ( ( W  e.  PreHil  /\  ( A ( -g `  W
) B )  e.  V  /\  A  e.  V )  ->  (
( A ( -g `  W ) B ) 
.,  A )  e.  ( Base `  (Scalar `  W ) ) )
3213, 7, 14, 31syl3anc 1218 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( A ( -g `  W ) B ) 
.,  A )  e.  ( Base `  (Scalar `  W ) ) )
3316, 17, 4, 30ipcl 18042 . . . . . 6  |-  ( ( W  e.  PreHil  /\  ( A ( -g `  W
) B )  e.  V  /\  B  e.  V )  ->  (
( A ( -g `  W ) B ) 
.,  B )  e.  ( Base `  (Scalar `  W ) ) )
3413, 7, 15, 33syl3anc 1218 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( A ( -g `  W ) B ) 
.,  B )  e.  ( Base `  (Scalar `  W ) ) )
3530, 22, 18grpsubeq0 15603 . . . . 5  |-  ( ( (Scalar `  W )  e.  Grp  /\  ( ( A ( -g `  W
) B )  .,  A )  e.  (
Base `  (Scalar `  W
) )  /\  (
( A ( -g `  W ) B ) 
.,  B )  e.  ( Base `  (Scalar `  W ) ) )  ->  ( ( ( ( A ( -g `  W ) B ) 
.,  A ) (
-g `  (Scalar `  W
) ) ( ( A ( -g `  W
) B )  .,  B ) )  =  ( 0g `  (Scalar `  W ) )  <->  ( ( A ( -g `  W
) B )  .,  A )  =  ( ( A ( -g `  W ) B ) 
.,  B ) ) )
3629, 32, 34, 35syl3anc 1218 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( ( ( A ( -g `  W
) B )  .,  A ) ( -g `  (Scalar `  W )
) ( ( A ( -g `  W
) B )  .,  B ) )  =  ( 0g `  (Scalar `  W ) )  <->  ( ( A ( -g `  W
) B )  .,  A )  =  ( ( A ( -g `  W ) B ) 
.,  B ) ) )
37 lmodgrp 16935 . . . . . 6  |-  ( W  e.  LMod  ->  W  e. 
Grp )
383, 37syl 16 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  Grp )
394, 23, 5grpsubeq0 15603 . . . . 5  |-  ( ( W  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A (
-g `  W ) B )  =  ( 0g `  W )  <-> 
A  =  B ) )
4038, 39syl3an1 1251 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( A ( -g `  W ) B )  =  ( 0g `  W )  <->  A  =  B ) )
4126, 36, 403bitr3d 283 . . 3  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( ( A (
-g `  W ) B )  .,  A
)  =  ( ( A ( -g `  W
) B )  .,  B )  <->  A  =  B ) )
4212, 41sylibd 214 . 2  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A. x  e.  V  ( x  .,  A )  =  ( x  .,  B )  ->  A  =  B ) )
432, 42impbid2 204 1  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  =  B  <->  A. x  e.  V  ( x  .,  A )  =  ( x  .,  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   ` cfv 5413  (class class class)co 6086   Basecbs 14166  Scalarcsca 14233   .icip 14235   0gc0g 14370   Grpcgrp 15402   -gcsg 15405   LModclmod 16928   PreHilcphl 18033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-tpos 6740  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-plusg 14243  df-mulr 14244  df-sca 14246  df-vsca 14247  df-ip 14248  df-0g 14372  df-mnd 15407  df-mhm 15456  df-grp 15536  df-minusg 15537  df-sbg 15538  df-ghm 15736  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 17083  df-lvec 17164  df-sra 17233  df-rgmod 17234  df-phl 18035
This theorem is referenced by: (None)
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