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Theorem nvnncan 21221
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 21205 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  e.  X )
4 eqid 2283 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
5 eqid 2283 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
61, 4, 5, 2nvmval 21200 . . 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 9813 . . . . . . 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 21184 . . . . . . . 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 21188 . . . . . 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 21200 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) B ) ) )
1817oveq2d 5874 . . . . 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 8795 . . . . . . . . . . . 12  |-  1  e.  CC
2019, 19mul2negi 9227 . . . . . . . . . . 11  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
21 1t1e1 9870 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
2220, 21eqtri 2303 . . . . . . . . . 10  |-  ( -u
1  x.  -u 1
)  =  1
2322oveq1i 5868 . . . . . . . . 9  |-  ( (
-u 1  x.  -u 1
) ( .s OLD `  U ) B )  =  ( 1 ( .s OLD `  U
) B )
241, 5nvsid 21185 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1 ( .s OLD `  U ) B )  =  B )
2523, 24syl5eq 2327 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( -u 1  x.  -u 1
) ( .s OLD `  U ) B )  =  B )
261, 5nvsass 21186 . . . . . . . . . 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 2317 . . . . . . 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 5874 . . . . 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 2325 . . . 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 5874 . . 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 21184 . . . . . 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 21178 . . . 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 2283 . . . . . 6  |-  ( 0vec `  U )  =  (
0vec `  U )
411, 4, 5, 40nvrinv 21211 . . . . 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 5873 . . 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 2321 . 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 21194 . . 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 2319 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    x. cmul 8742   -ucneg 9038   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   0veccn0v 21144   -vcnsb 21145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156
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