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Theorem nvmeq0 22150
Description: The difference between two vectors is zero iff they are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvmeq0.1  |-  X  =  ( BaseSet `  U )
nvmeq0.3  |-  M  =  ( -v `  U
)
nvmeq0.5  |-  Z  =  ( 0vec `  U
)
Assertion
Ref Expression
nvmeq0  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A M B )  =  Z  <->  A  =  B ) )

Proof of Theorem nvmeq0
StepHypRef Expression
1 nvmeq0.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
2 nvmeq0.3 . . . . . . 7  |-  M  =  ( -v `  U
)
31, 2nvmcl 22133 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  e.  X )
433expb 1155 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A M B )  e.  X
)
5 nvmeq0.5 . . . . . . 7  |-  Z  =  ( 0vec `  U
)
61, 5nvzcl 22120 . . . . . 6  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
76adantr 453 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  Z  e.  X )
8 simprr 735 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  B  e.  X )
94, 7, 83jca 1135 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( A M B )  e.  X  /\  Z  e.  X  /\  B  e.  X ) )
10 eqid 2438 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
111, 10nvrcan 22109 . . . 4  |-  ( ( U  e.  NrmCVec  /\  (
( A M B )  e.  X  /\  Z  e.  X  /\  B  e.  X )
)  ->  ( (
( A M B ) ( +v `  U ) B )  =  ( Z ( +v `  U ) B )  <->  ( A M B )  =  Z ) )
129, 11syldan 458 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( (
( A M B ) ( +v `  U ) B )  =  ( Z ( +v `  U ) B )  <->  ( A M B )  =  Z ) )
13123impb 1150 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( A M B ) ( +v
`  U ) B )  =  ( Z ( +v `  U
) B )  <->  ( A M B )  =  Z ) )
141, 10, 2nvnpcan 22146 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A M B ) ( +v `  U ) B )  =  A )
151, 10, 5nv0lid 22122 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( Z ( +v `  U ) B )  =  B )
16153adant2 977 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( Z ( +v `  U ) B )  =  B )
1714, 16eqeq12d 2452 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( A M B ) ( +v
`  U ) B )  =  ( Z ( +v `  U
) B )  <->  A  =  B ) )
1813, 17bitr3d 248 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A M B )  =  Z  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5457  (class class class)co 6084   NrmCVeccnv 22068   +vcpv 22069   BaseSetcba 22070   0veccn0v 22072   -vcnsb 22073
This theorem is referenced by:  nvmid  22151  ip2eqi  22363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-ltxr 9130  df-sub 9298  df-neg 9299  df-grpo 21784  df-gid 21785  df-ginv 21786  df-gdiv 21787  df-ablo 21875  df-vc 22030  df-nv 22076  df-va 22079  df-ba 22080  df-sm 22081  df-0v 22082  df-vs 22083  df-nmcv 22084
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