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Theorem nvmeq0 21330
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 21313 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  e.  X )
433expb 1152 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A M B )  e.  X
)
5 nvmeq0.5 . . . . . . 7  |-  Z  =  ( 0vec `  U
)
61, 5nvzcl 21300 . . . . . 6  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
76adantr 451 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  Z  e.  X )
8 simprr 733 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  B  e.  X )
94, 7, 83jca 1132 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( A M B )  e.  X  /\  Z  e.  X  /\  B  e.  X ) )
10 eqid 2358 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
111, 10nvrcan 21289 . . . 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 456 . . 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 1147 . 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 21326 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A M B ) ( +v `  U ) B )  =  A )
151, 10, 5nv0lid 21302 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( Z ( +v `  U ) B )  =  B )
16153adant2 974 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( Z ( +v `  U ) B )  =  B )
1714, 16eqeq12d 2372 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( A M B ) ( +v
`  U ) B )  =  ( Z ( +v `  U
) B )  <->  A  =  B ) )
1813, 17bitr3d 246 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 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   ` cfv 5334  (class class class)co 5942   NrmCVeccnv 21248   +vcpv 21249   BaseSetcba 21250   0veccn0v 21252   -vcnsb 21253
This theorem is referenced by:  nvmid  21331  ip2eqi  21543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-po 4393  df-so 4394  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-ltxr 8959  df-sub 9126  df-neg 9127  df-grpo 20964  df-gid 20965  df-ginv 20966  df-gdiv 20967  df-ablo 21055  df-vc 21210  df-nv 21256  df-va 21259  df-ba 21260  df-sm 21261  df-0v 21262  df-vs 21263  df-nmcv 21264
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