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Theorem nvpncan2 21269
Description: Cancellation law for vector subtraction. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvsubadd.1  |-  X  =  ( BaseSet `  U )
nvsubadd.2  |-  G  =  ( +v `  U
)
nvsubadd.3  |-  M  =  ( -v `  U
)
Assertion
Ref Expression
nvpncan2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) M A )  =  B )

Proof of Theorem nvpncan2
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  U  e.  NrmCVec )
2 nvsubadd.1 . . . 4  |-  X  =  ( BaseSet `  U )
3 nvsubadd.2 . . . 4  |-  G  =  ( +v `  U
)
42, 3nvgcl 21231 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
5 simp2 956 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
6 eqid 2316 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
7 nvsubadd.3 . . . 4  |-  M  =  ( -v `  U
)
82, 3, 6, 7nvmval 21255 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A G B )  e.  X  /\  A  e.  X )  ->  (
( A G B ) M A )  =  ( ( A G B ) G ( -u 1 ( .s OLD `  U
) A ) ) )
91, 4, 5, 8syl3anc 1182 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) M A )  =  ( ( A G B ) G ( -u 1 ( .s OLD `  U
) A ) ) )
10 simp3 957 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  B  e.  X )
11 neg1cn 9858 . . . . . 6  |-  -u 1  e.  CC
122, 6nvscl 21239 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 ( .s OLD `  U ) A )  e.  X )
1311, 12mp3an2 1265 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 ( .s OLD `  U ) A )  e.  X )
14133adant3 975 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) A )  e.  X )
152, 3nvadd32 21235 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  ( -u 1 ( .s
OLD `  U ) A )  e.  X
) )  ->  (
( A G B ) G ( -u
1 ( .s OLD `  U ) A ) )  =  ( ( A G ( -u
1 ( .s OLD `  U ) A ) ) G B ) )
161, 5, 10, 14, 15syl13anc 1184 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) G ( -u
1 ( .s OLD `  U ) A ) )  =  ( ( A G ( -u
1 ( .s OLD `  U ) A ) ) G B ) )
17 eqid 2316 . . . . . . 7  |-  ( 0vec `  U )  =  (
0vec `  U )
182, 3, 6, 17nvrinv 21266 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G ( -u 1
( .s OLD `  U
) A ) )  =  ( 0vec `  U
) )
19183adant3 975 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( -u 1
( .s OLD `  U
) A ) )  =  ( 0vec `  U
) )
2019oveq1d 5915 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G (
-u 1 ( .s
OLD `  U ) A ) ) G B )  =  ( ( 0vec `  U
) G B ) )
212, 3, 17nv0lid 21249 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( 0vec `  U ) G B )  =  B )
22213adant2 974 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( 0vec `  U ) G B )  =  B )
2320, 22eqtrd 2348 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G (
-u 1 ( .s
OLD `  U ) A ) ) G B )  =  B )
2416, 23eqtrd 2348 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) G ( -u
1 ( .s OLD `  U ) A ) )  =  B )
259, 24eqtrd 2348 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) M A )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1633    e. wcel 1701   ` cfv 5292  (class class class)co 5900   CCcc 8780   1c1 8783   -ucneg 9083   NrmCVeccnv 21195   +vcpv 21196   BaseSetcba 21197   .s
OLDcns 21198   0veccn0v 21199   -vcnsb 21200
This theorem is referenced by:  nvpncan  21270  nvelbl2  21318  blocnilem  21437  ubthlem2  21505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-ltxr 8917  df-sub 9084  df-neg 9085  df-grpo 20911  df-gid 20912  df-ginv 20913  df-gdiv 20914  df-ablo 21002  df-vc 21157  df-nv 21203  df-va 21206  df-ba 21207  df-sm 21208  df-0v 21209  df-vs 21210  df-nmcv 21211
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