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Theorem nvmeq0 25685
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 25668 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  e.  X )
433expb 1197 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A M B )  e.  X
)
5 nvmeq0.5 . . . . . . 7  |-  Z  =  ( 0vec `  U
)
61, 5nvzcl 25655 . . . . . 6  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
76adantr 465 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  Z  e.  X )
8 simprr 757 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  B  e.  X )
94, 7, 83jca 1176 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( A M B )  e.  X  /\  Z  e.  X  /\  B  e.  X ) )
10 eqid 2457 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
111, 10nvrcan 25644 . . . 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 1192 . 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 25681 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A M B ) ( +v `  U ) B )  =  A )
151, 10, 5nv0lid 25657 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( Z ( +v `  U ) B )  =  B )
16153adant2 1015 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( Z ( +v `  U ) B )  =  B )
1714, 16eqeq12d 2479 . 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 973    = wceq 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   NrmCVeccnv 25603   +vcpv 25604   BaseSetcba 25605   0veccn0v 25607   -vcnsb 25608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-ltxr 9650  df-sub 9826  df-neg 9827  df-grpo 25319  df-gid 25320  df-ginv 25321  df-gdiv 25322  df-ablo 25410  df-vc 25565  df-nv 25611  df-va 25614  df-ba 25615  df-sm 25616  df-0v 25617  df-vs 25618  df-nmcv 25619
This theorem is referenced by:  nvmid  25686  ip2eqi  25898
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