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Theorem ip2eq 16885
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 6090 . . 3  |-  ( A  =  B  ->  (
x  .,  A )  =  ( x  .,  B ) )
21ralrimivw 2791 . 2  |-  ( A  =  B  ->  A. x  e.  V  ( x  .,  A )  =  ( x  .,  B ) )
3 phllmod 16862 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
4 ip2eq.v . . . . . 6  |-  V  =  ( Base `  W
)
5 eqid 2437 . . . . . 6  |-  ( -g `  W )  =  (
-g `  W )
64, 5lmodvsubcl 15990 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( A ( -g `  W
) B )  e.  V )
73, 6syl3an1 1218 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A ( -g `  W
) B )  e.  V )
8 oveq1 6089 . . . . . 6  |-  ( x  =  ( A (
-g `  W ) B )  ->  (
x  .,  A )  =  ( ( A ( -g `  W
) B )  .,  A ) )
9 oveq1 6089 . . . . . 6  |-  ( x  =  ( A (
-g `  W ) B )  ->  (
x  .,  B )  =  ( ( A ( -g `  W
) B )  .,  B ) )
108, 9eqeq12d 2451 . . . . 5  |-  ( x  =  ( A (
-g `  W ) B )  ->  (
( x  .,  A
)  =  ( x 
.,  B )  <->  ( ( A ( -g `  W
) B )  .,  A )  =  ( ( A ( -g `  W ) B ) 
.,  B ) ) )
1110rspcv 3049 . . . 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 958 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  W  e.  PreHil )
14 simp2 959 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  A  e.  V )
15 simp3 960 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  B  e.  V )
16 eqid 2437 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
17 ip2eq.h . . . . . . . 8  |-  .,  =  ( .i `  W )
18 eqid 2437 . . . . . . . 8  |-  ( -g `  (Scalar `  W )
)  =  ( -g `  (Scalar `  W )
)
1916, 17, 4, 5, 18ipsubdi 16875 . . . . . . 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 1187 . . . . . 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 2437 . . . . . . 7  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
23 eqid 2437 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
2416, 17, 4, 22, 23ipeq0 16870 . . . . . 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 644 . . . . 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 248 . . . 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 979 . . . . . 6  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  W  e.  LMod )
2816lmodfgrp 15960 . . . . . 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 2437 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3116, 17, 4, 30ipcl 16865 . . . . . 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 1185 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( A ( -g `  W ) B ) 
.,  A )  e.  ( Base `  (Scalar `  W ) ) )
3316, 17, 4, 30ipcl 16865 . . . . . 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 1185 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( A ( -g `  W ) B ) 
.,  B )  e.  ( Base `  (Scalar `  W ) ) )
3530, 22, 18grpsubeq0 14876 . . . . 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 1185 . . . 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 15958 . . . . . 6  |-  ( W  e.  LMod  ->  W  e. 
Grp )
383, 37syl 16 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  Grp )
394, 23, 5grpsubeq0 14876 . . . . 5  |-  ( ( W  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A (
-g `  W ) B )  =  ( 0g `  W )  <-> 
A  =  B ) )
4038, 39syl3an1 1218 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( A ( -g `  W ) B )  =  ( 0g `  W )  <->  A  =  B ) )
4126, 36, 403bitr3d 276 . . 3  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( ( A (
-g `  W ) B )  .,  A
)  =  ( ( A ( -g `  W
) B )  .,  B )  <->  A  =  B ) )
4212, 41sylibd 207 . 2  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A. x  e.  V  ( x  .,  A )  =  ( x  .,  B )  ->  A  =  B ) )
432, 42impbid2 197 1  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  =  B  <->  A. x  e.  V  ( x  .,  A )  =  ( x  .,  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2706   ` cfv 5455  (class class class)co 6082   Basecbs 13470  Scalarcsca 13533   .icip 13535   0gc0g 13724   Grpcgrp 14686   -gcsg 14689   LModclmod 15951   PreHilcphl 16856
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-tpos 6480  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-plusg 13543  df-mulr 13544  df-sca 13546  df-vsca 13547  df-0g 13728  df-mnd 14691  df-mhm 14739  df-grp 14813  df-minusg 14814  df-sbg 14815  df-ghm 15005  df-mgp 15650  df-rng 15664  df-ur 15666  df-oppr 15729  df-rnghom 15820  df-staf 15934  df-srng 15935  df-lmod 15953  df-lmhm 16099  df-lvec 16176  df-sra 16245  df-rgmod 16246  df-phl 16858
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