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Theorem ip2eq 17924
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 6088 . . 3  |-  ( A  =  B  ->  (
x  .,  A )  =  ( x  .,  B ) )
21ralrimivw 2790 . 2  |-  ( A  =  B  ->  A. x  e.  V  ( x  .,  A )  =  ( x  .,  B ) )
3 phllmod 17901 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
4 ip2eq.v . . . . . 6  |-  V  =  ( Base `  W
)
5 eqid 2433 . . . . . 6  |-  ( -g `  W )  =  (
-g `  W )
64, 5lmodvsubcl 16914 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( A ( -g `  W
) B )  e.  V )
73, 6syl3an1 1244 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A ( -g `  W
) B )  e.  V )
8 oveq1 6087 . . . . . 6  |-  ( x  =  ( A (
-g `  W ) B )  ->  (
x  .,  A )  =  ( ( A ( -g `  W
) B )  .,  A ) )
9 oveq1 6087 . . . . . 6  |-  ( x  =  ( A (
-g `  W ) B )  ->  (
x  .,  B )  =  ( ( A ( -g `  W
) B )  .,  B ) )
108, 9eqeq12d 2447 . . . . 5  |-  ( x  =  ( A (
-g `  W ) B )  ->  (
( x  .,  A
)  =  ( x 
.,  B )  <->  ( ( A ( -g `  W
) B )  .,  A )  =  ( ( A ( -g `  W ) B ) 
.,  B ) ) )
1110rspcv 3058 . . . 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 981 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  W  e.  PreHil )
14 simp2 982 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  A  e.  V )
15 simp3 983 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  B  e.  V )
16 eqid 2433 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
17 ip2eq.h . . . . . . . 8  |-  .,  =  ( .i `  W )
18 eqid 2433 . . . . . . . 8  |-  ( -g `  (Scalar `  W )
)  =  ( -g `  (Scalar `  W )
)
1916, 17, 4, 5, 18ipsubdi 17914 . . . . . . 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 1213 . . . . . 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 2441 . . . . 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 2433 . . . . . . 7  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
23 eqid 2433 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
2416, 17, 4, 22, 23ipeq0 17909 . . . . . 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 654 . . . . 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 1002 . . . . . 6  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  W  e.  LMod )
2816lmodfgrp 16881 . . . . . 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 2433 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3116, 17, 4, 30ipcl 17904 . . . . . 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 1211 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( A ( -g `  W ) B ) 
.,  A )  e.  ( Base `  (Scalar `  W ) ) )
3316, 17, 4, 30ipcl 17904 . . . . . 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 1211 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( A ( -g `  W ) B ) 
.,  B )  e.  ( Base `  (Scalar `  W ) ) )
3530, 22, 18grpsubeq0 15592 . . . . 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 1211 . . . 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 16879 . . . . . 6  |-  ( W  e.  LMod  ->  W  e. 
Grp )
383, 37syl 16 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  Grp )
394, 23, 5grpsubeq0 15592 . . . . 5  |-  ( ( W  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A (
-g `  W ) B )  =  ( 0g `  W )  <-> 
A  =  B ) )
4038, 39syl3an1 1244 . . . 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 958    = wceq 1362    e. wcel 1755   A.wral 2705   ` cfv 5406  (class class class)co 6080   Basecbs 14157  Scalarcsca 14224   .icip 14226   0gc0g 14361   Grpcgrp 15393   -gcsg 15396   LModclmod 16872   PreHilcphl 17895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-tpos 6734  df-recs 6818  df-rdg 6852  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-plusg 14234  df-mulr 14235  df-sca 14237  df-vsca 14238  df-ip 14239  df-0g 14363  df-mnd 15398  df-mhm 15447  df-grp 15525  df-minusg 15526  df-sbg 15527  df-ghm 15725  df-mgp 16566  df-rng 16580  df-ur 16582  df-oppr 16649  df-rnghom 16740  df-staf 16854  df-srng 16855  df-lmod 16874  df-lmhm 17025  df-lvec 17106  df-sra 17175  df-rgmod 17176  df-phl 17897
This theorem is referenced by: (None)
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