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Theorem nvnncan 21276
Description: Cancellation law for a normed complex vector space. (Contributed by NM, 17-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvsubsub23.1  |-  X  =  ( BaseSet `  U )
nvsubsub23.3  |-  M  =  ( -v `  U
)
Assertion
Ref Expression
nvnncan  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( A M B ) )  =  B )

Proof of Theorem nvnncan
StepHypRef Expression
1 nvsubsub23.1 . . . 4  |-  X  =  ( BaseSet `  U )
2 nvsubsub23.3 . . . 4  |-  M  =  ( -v `  U
)
31, 2nvmcl 21260 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  e.  X )
4 eqid 2316 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
5 eqid 2316 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
61, 4, 5, 2nvmval 21255 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  ( A M B )  e.  X )  ->  ( A M ( A M B ) )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U )
( A M B ) ) ) )
73, 6syld3an3 1227 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( A M B ) )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U )
( A M B ) ) ) )
8 simp1 955 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  U  e.  NrmCVec )
9 neg1cn 9858 . . . . . . 7  |-  -u 1  e.  CC
109a1i 10 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  -u 1  e.  CC )
11 simp2 956 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
121, 5nvscl 21239 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
139, 12mp3an2 1265 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
14133adant2 974 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
151, 4, 5nvdi 21243 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  A  e.  X  /\  ( -u 1 ( .s
OLD `  U ) B )  e.  X
) )  ->  ( -u 1 ( .s OLD `  U ) ( A ( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) )  =  ( (
-u 1 ( .s
OLD `  U ) A ) ( +v
`  U ) (
-u 1 ( .s
OLD `  U )
( -u 1 ( .s
OLD `  U ) B ) ) ) )
168, 10, 11, 14, 15syl13anc 1184 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) ( A ( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) )  =  ( (
-u 1 ( .s
OLD `  U ) A ) ( +v
`  U ) (
-u 1 ( .s
OLD `  U )
( -u 1 ( .s
OLD `  U ) B ) ) ) )
171, 4, 5, 2nvmval 21255 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) B ) ) )
1817oveq2d 5916 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) ( A M B ) )  =  ( -u 1
( .s OLD `  U
) ( A ( +v `  U ) ( -u 1 ( .s OLD `  U
) B ) ) ) )
19 ax-1cn 8840 . . . . . . . . . . . 12  |-  1  e.  CC
2019, 19mul2negi 9272 . . . . . . . . . . 11  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
21 1t1e1 9917 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
2220, 21eqtri 2336 . . . . . . . . . 10  |-  ( -u
1  x.  -u 1
)  =  1
2322oveq1i 5910 . . . . . . . . 9  |-  ( (
-u 1  x.  -u 1
) ( .s OLD `  U ) B )  =  ( 1 ( .s OLD `  U
) B )
241, 5nvsid 21240 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1 ( .s OLD `  U ) B )  =  B )
2523, 24syl5eq 2360 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( -u 1  x.  -u 1
) ( .s OLD `  U ) B )  =  B )
261, 5nvsass 21241 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  -u 1  e.  CC  /\  B  e.  X )
)  ->  ( ( -u 1  x.  -u 1
) ( .s OLD `  U ) B )  =  ( -u 1
( .s OLD `  U
) ( -u 1
( .s OLD `  U
) B ) ) )
279, 26mp3anr1 1274 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  B  e.  X )
)  ->  ( ( -u 1  x.  -u 1
) ( .s OLD `  U ) B )  =  ( -u 1
( .s OLD `  U
) ( -u 1
( .s OLD `  U
) B ) ) )
289, 27mpanr1 664 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( -u 1  x.  -u 1
) ( .s OLD `  U ) B )  =  ( -u 1
( .s OLD `  U
) ( -u 1
( .s OLD `  U
) B ) ) )
2925, 28eqtr3d 2350 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  B  =  ( -u 1
( .s OLD `  U
) ( -u 1
( .s OLD `  U
) B ) ) )
30293adant2 974 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  B  =  ( -u 1
( .s OLD `  U
) ( -u 1
( .s OLD `  U
) B ) ) )
3130oveq2d 5916 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( -u 1 ( .s
OLD `  U ) A ) ( +v
`  U ) B )  =  ( (
-u 1 ( .s
OLD `  U ) A ) ( +v
`  U ) (
-u 1 ( .s
OLD `  U )
( -u 1 ( .s
OLD `  U ) B ) ) ) )
3216, 18, 313eqtr4d 2358 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) ( A M B ) )  =  ( ( -u
1 ( .s OLD `  U ) A ) ( +v `  U
) B ) )
3332oveq2d 5916 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) ( A M B ) ) )  =  ( A ( +v `  U
) ( ( -u
1 ( .s OLD `  U ) A ) ( +v `  U
) B ) ) )
341, 5nvscl 21239 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 ( .s OLD `  U ) A )  e.  X )
359, 34mp3an2 1265 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 ( .s OLD `  U ) A )  e.  X )
36353adant3 975 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) A )  e.  X )
37 simp3 957 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  B  e.  X )
381, 4nvass 21233 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  ( -u 1 ( .s
OLD `  U ) A )  e.  X  /\  B  e.  X
) )  ->  (
( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) A ) ) ( +v `  U ) B )  =  ( A ( +v `  U ) ( (
-u 1 ( .s
OLD `  U ) A ) ( +v
`  U ) B ) ) )
398, 11, 36, 37, 38syl13anc 1184 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) A ) ) ( +v `  U ) B )  =  ( A ( +v `  U ) ( (
-u 1 ( .s
OLD `  U ) A ) ( +v
`  U ) B ) ) )
40 eqid 2316 . . . . . 6  |-  ( 0vec `  U )  =  (
0vec `  U )
411, 4, 5, 40nvrinv 21266 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) )  =  ( 0vec `  U ) )
42413adant3 975 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) )  =  ( 0vec `  U ) )
4342oveq1d 5915 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) A ) ) ( +v `  U ) B )  =  ( ( 0vec `  U
) ( +v `  U ) B ) )
4433, 39, 433eqtr2d 2354 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) ( A M B ) ) )  =  ( (
0vec `  U )
( +v `  U
) B ) )
451, 4, 40nv0lid 21249 . . 3  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( 0vec `  U )
( +v `  U
) B )  =  B )
46453adant2 974 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( 0vec `  U )
( +v `  U
) B )  =  B )
477, 44, 463eqtrd 2352 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( A M B ) )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   ` cfv 5292  (class class class)co 5900   CCcc 8780   1c1 8783    x. cmul 8787   -ucneg 9083   NrmCVeccnv 21195   +vcpv 21196   BaseSetcba 21197   .s
OLDcns 21198   0veccn0v 21199   -vcnsb 21200
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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