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Theorem nvmeq0 24044
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 24027 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  e.  X )
433expb 1188 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A M B )  e.  X
)
5 nvmeq0.5 . . . . . . 7  |-  Z  =  ( 0vec `  U
)
61, 5nvzcl 24014 . . . . . 6  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
76adantr 465 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  Z  e.  X )
8 simprr 756 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  B  e.  X )
94, 7, 83jca 1168 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( A M B )  e.  X  /\  Z  e.  X  /\  B  e.  X ) )
10 eqid 2443 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
111, 10nvrcan 24003 . . . 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 470 . . 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 1183 . 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 24040 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A M B ) ( +v `  U ) B )  =  A )
151, 10, 5nv0lid 24016 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( Z ( +v `  U ) B )  =  B )
16153adant2 1007 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( Z ( +v `  U ) B )  =  B )
1714, 16eqeq12d 2457 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( A M B ) ( +v
`  U ) B )  =  ( Z ( +v `  U
) B )  <->  A  =  B ) )
1813, 17bitr3d 255 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A M B )  =  Z  <->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   NrmCVeccnv 23962   +vcpv 23963   BaseSetcba 23964   0veccn0v 23966   -vcnsb 23967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-ltxr 9423  df-sub 9597  df-neg 9598  df-grpo 23678  df-gid 23679  df-ginv 23680  df-gdiv 23681  df-ablo 23769  df-vc 23924  df-nv 23970  df-va 23973  df-ba 23974  df-sm 23975  df-0v 23976  df-vs 23977  df-nmcv 23978
This theorem is referenced by:  nvmid  24045  ip2eqi  24257
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