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Theorem ip2eq 18455
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 6290 . . 3  |-  ( A  =  B  ->  (
x  .,  A )  =  ( x  .,  B ) )
21ralrimivw 2879 . 2  |-  ( A  =  B  ->  A. x  e.  V  ( x  .,  A )  =  ( x  .,  B ) )
3 phllmod 18432 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
4 ip2eq.v . . . . . 6  |-  V  =  ( Base `  W
)
5 eqid 2467 . . . . . 6  |-  ( -g `  W )  =  (
-g `  W )
64, 5lmodvsubcl 17338 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( A ( -g `  W
) B )  e.  V )
73, 6syl3an1 1261 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A ( -g `  W
) B )  e.  V )
8 oveq1 6289 . . . . . 6  |-  ( x  =  ( A (
-g `  W ) B )  ->  (
x  .,  A )  =  ( ( A ( -g `  W
) B )  .,  A ) )
9 oveq1 6289 . . . . . 6  |-  ( x  =  ( A (
-g `  W ) B )  ->  (
x  .,  B )  =  ( ( A ( -g `  W
) B )  .,  B ) )
108, 9eqeq12d 2489 . . . . 5  |-  ( x  =  ( A (
-g `  W ) B )  ->  (
( x  .,  A
)  =  ( x 
.,  B )  <->  ( ( A ( -g `  W
) B )  .,  A )  =  ( ( A ( -g `  W ) B ) 
.,  B ) ) )
1110rspcv 3210 . . . 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 996 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  W  e.  PreHil )
14 simp2 997 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  A  e.  V )
15 simp3 998 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  B  e.  V )
16 eqid 2467 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
17 ip2eq.h . . . . . . . 8  |-  .,  =  ( .i `  W )
18 eqid 2467 . . . . . . . 8  |-  ( -g `  (Scalar `  W )
)  =  ( -g `  (Scalar `  W )
)
1916, 17, 4, 5, 18ipsubdi 18445 . . . . . . 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 1230 . . . . . 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 2469 . . . . 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 2467 . . . . . . 7  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
23 eqid 2467 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
2416, 17, 4, 22, 23ipeq0 18440 . . . . . 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 1017 . . . . . 6  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  W  e.  LMod )
2816lmodfgrp 17304 . . . . . 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 2467 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3116, 17, 4, 30ipcl 18435 . . . . . 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 1228 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( A ( -g `  W ) B ) 
.,  A )  e.  ( Base `  (Scalar `  W ) ) )
3316, 17, 4, 30ipcl 18435 . . . . . 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 1228 . . . . 5  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (
( A ( -g `  W ) B ) 
.,  B )  e.  ( Base `  (Scalar `  W ) ) )
3530, 22, 18grpsubeq0 15925 . . . . 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 1228 . . . 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 17302 . . . . . 6  |-  ( W  e.  LMod  ->  W  e. 
Grp )
383, 37syl 16 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  Grp )
394, 23, 5grpsubeq0 15925 . . . . 5  |-  ( ( W  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A (
-g `  W ) B )  =  ( 0g `  W )  <-> 
A  =  B ) )
4038, 39syl3an1 1261 . . . 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   ` cfv 5586  (class class class)co 6282   Basecbs 14486  Scalarcsca 14554   .icip 14556   0gc0g 14691   Grpcgrp 15723   -gcsg 15726   LModclmod 17295   PreHilcphl 18426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-ip 14569  df-0g 14693  df-mnd 15728  df-mhm 15777  df-grp 15858  df-minusg 15859  df-sbg 15860  df-ghm 16060  df-mgp 16932  df-ur 16944  df-rng 16988  df-oppr 17056  df-rnghom 17148  df-staf 17277  df-srng 17278  df-lmod 17297  df-lmhm 17451  df-lvec 17532  df-sra 17601  df-rgmod 17602  df-phl 18428
This theorem is referenced by: (None)
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