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Theorem ip2eq 18561
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 6289 . . 3  |-  ( A  =  B  ->  (
x  .,  A )  =  ( x  .,  B ) )
21ralrimivw 2858 . 2  |-  ( A  =  B  ->  A. x  e.  V  ( x  .,  A )  =  ( x  .,  B ) )
3 phllmod 18538 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
4 ip2eq.v . . . . . 6  |-  V  =  ( Base `  W
)
5 eqid 2443 . . . . . 6  |-  ( -g `  W )  =  (
-g `  W )
64, 5lmodvsubcl 17429 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( A ( -g `  W
) B )  e.  V )
73, 6syl3an1 1262 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A ( -g `  W
) B )  e.  V )
8 oveq1 6288 . . . . . 6  |-  ( x  =  ( A (
-g `  W ) B )  ->  (
x  .,  A )  =  ( ( A ( -g `  W
) B )  .,  A ) )
9 oveq1 6288 . . . . . 6  |-  ( x  =  ( A (
-g `  W ) B )  ->  (
x  .,  B )  =  ( ( A ( -g `  W
) B )  .,  B ) )
108, 9eqeq12d 2465 . . . . 5  |-  ( x  =  ( A (
-g `  W ) B )  ->  (
( x  .,  A
)  =  ( x 
.,  B )  <->  ( ( A ( -g `  W
) B )  .,  A )  =  ( ( A ( -g `  W ) B ) 
.,  B ) ) )
1110rspcv 3192 . . . 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 997 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  W  e.  PreHil )
14 simp2 998 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  A  e.  V )
15 simp3 999 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  B  e.  V )
16 eqid 2443 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
17 ip2eq.h . . . . . . . 8  |-  .,  =  ( .i `  W )
18 eqid 2443 . . . . . . . 8  |-  ( -g `  (Scalar `  W )
)  =  ( -g `  (Scalar `  W )
)
1916, 17, 4, 5, 18ipsubdi 18551 . . . . . . 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 1231 . . . . . 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 2445 . . . . 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 2443 . . . . . . 7  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
23 eqid 2443 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
2416, 17, 4, 22, 23ipeq0 18546 . . . . . 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 1018 . . . . . 6  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  W  e.  LMod )
2816lmodfgrp 17395 . . . . . 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 2443 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3116, 17, 4, 30ipcl 18541 . . . . . 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 1229 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( A ( -g `  W ) B ) 
.,  A )  e.  ( Base `  (Scalar `  W ) ) )
3316, 17, 4, 30ipcl 18541 . . . . . 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 1229 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( A ( -g `  W ) B ) 
.,  B )  e.  ( Base `  (Scalar `  W ) ) )
3530, 22, 18grpsubeq0 15998 . . . . 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 1229 . . . 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 17393 . . . . . 6  |-  ( W  e.  LMod  ->  W  e. 
Grp )
383, 37syl 16 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  Grp )
394, 23, 5grpsubeq0 15998 . . . . 5  |-  ( ( W  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A (
-g `  W ) B )  =  ( 0g `  W )  <-> 
A  =  B ) )
4038, 39syl3an1 1262 . . . 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 974    = wceq 1383    e. wcel 1804   A.wral 2793   ` cfv 5578  (class class class)co 6281   Basecbs 14509  Scalarcsca 14577   .icip 14579   0gc0g 14714   Grpcgrp 15927   -gcsg 15929   LModclmod 17386   PreHilcphl 18532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6957  df-recs 7044  df-rdg 7078  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-plusg 14587  df-mulr 14588  df-sca 14590  df-vsca 14591  df-ip 14592  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15840  df-grp 15931  df-minusg 15932  df-sbg 15933  df-ghm 16139  df-mgp 17016  df-ur 17028  df-ring 17074  df-oppr 17146  df-rnghom 17238  df-staf 17368  df-srng 17369  df-lmod 17388  df-lmhm 17542  df-lvec 17623  df-sra 17692  df-rgmod 17693  df-phl 18534
This theorem is referenced by: (None)
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