MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvnncan Structured version   Unicode version

Theorem nvnncan 25759
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 25743 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  e.  X )
4 eqid 2454 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
5 eqid 2454 . . . 4  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
61, 4, 5, 2nvmval 25738 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  ( A M B )  e.  X )  ->  ( A M ( A M B ) )  =  ( A ( +v
`  U ) (
-u 1 ( .sOLD `  U ) ( A M B ) ) ) )
73, 6syld3an3 1271 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( A M B ) )  =  ( A ( +v
`  U ) (
-u 1 ( .sOLD `  U ) ( A M B ) ) ) )
8 simp1 994 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  U  e.  NrmCVec )
9 neg1cn 10635 . . . . . . 7  |-  -u 1  e.  CC
109a1i 11 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  -u 1  e.  CC )
11 simp2 995 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
121, 5nvscl 25722 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( -u 1 ( .sOLD `  U ) B )  e.  X )
139, 12mp3an2 1310 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 ( .sOLD `  U ) B )  e.  X )
14133adant2 1013 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .sOLD `  U ) B )  e.  X )
151, 4, 5nvdi 25726 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  A  e.  X  /\  ( -u 1 ( .sOLD `  U ) B )  e.  X
) )  ->  ( -u 1 ( .sOLD `  U ) ( A ( +v `  U
) ( -u 1
( .sOLD `  U ) B ) ) )  =  ( ( -u 1 ( .sOLD `  U
) A ) ( +v `  U ) ( -u 1 ( .sOLD `  U
) ( -u 1
( .sOLD `  U ) B ) ) ) )
168, 10, 11, 14, 15syl13anc 1228 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .sOLD `  U ) ( A ( +v `  U
) ( -u 1
( .sOLD `  U ) B ) ) )  =  ( ( -u 1 ( .sOLD `  U
) A ) ( +v `  U ) ( -u 1 ( .sOLD `  U
) ( -u 1
( .sOLD `  U ) B ) ) ) )
171, 4, 5, 2nvmval 25738 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A ( +v
`  U ) (
-u 1 ( .sOLD `  U ) B ) ) )
1817oveq2d 6286 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .sOLD `  U ) ( A M B ) )  =  ( -u 1
( .sOLD `  U ) ( A ( +v `  U
) ( -u 1
( .sOLD `  U ) B ) ) ) )
19 neg1mulneg1e1 10749 . . . . . . . . . 10  |-  ( -u
1  x.  -u 1
)  =  1
2019oveq1i 6280 . . . . . . . . 9  |-  ( (
-u 1  x.  -u 1
) ( .sOLD `  U ) B )  =  ( 1 ( .sOLD `  U
) B )
211, 5nvsid 25723 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1 ( .sOLD `  U ) B )  =  B )
2220, 21syl5eq 2507 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( -u 1  x.  -u 1
) ( .sOLD `  U ) B )  =  B )
231, 5nvsass 25724 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  -u 1  e.  CC  /\  B  e.  X )
)  ->  ( ( -u 1  x.  -u 1
) ( .sOLD `  U ) B )  =  ( -u 1
( .sOLD `  U ) ( -u
1 ( .sOLD `  U ) B ) ) )
249, 23mp3anr1 1319 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  B  e.  X )
)  ->  ( ( -u 1  x.  -u 1
) ( .sOLD `  U ) B )  =  ( -u 1
( .sOLD `  U ) ( -u
1 ( .sOLD `  U ) B ) ) )
259, 24mpanr1 681 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( -u 1  x.  -u 1
) ( .sOLD `  U ) B )  =  ( -u 1
( .sOLD `  U ) ( -u
1 ( .sOLD `  U ) B ) ) )
2622, 25eqtr3d 2497 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  B  =  ( -u 1
( .sOLD `  U ) ( -u
1 ( .sOLD `  U ) B ) ) )
27263adant2 1013 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  B  =  ( -u 1
( .sOLD `  U ) ( -u
1 ( .sOLD `  U ) B ) ) )
2827oveq2d 6286 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( -u 1 ( .sOLD `  U ) A ) ( +v
`  U ) B )  =  ( (
-u 1 ( .sOLD `  U ) A ) ( +v
`  U ) (
-u 1 ( .sOLD `  U ) ( -u 1 ( .sOLD `  U
) B ) ) ) )
2916, 18, 283eqtr4d 2505 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .sOLD `  U ) ( A M B ) )  =  ( ( -u
1 ( .sOLD `  U ) A ) ( +v `  U
) B ) )
3029oveq2d 6286 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) ( A M B ) ) )  =  ( A ( +v `  U
) ( ( -u
1 ( .sOLD `  U ) A ) ( +v `  U
) B ) ) )
311, 5nvscl 25722 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 ( .sOLD `  U ) A )  e.  X )
329, 31mp3an2 1310 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 ( .sOLD `  U ) A )  e.  X )
33323adant3 1014 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .sOLD `  U ) A )  e.  X )
34 simp3 996 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  B  e.  X )
351, 4nvass 25716 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  ( -u 1 ( .sOLD `  U ) A )  e.  X  /\  B  e.  X
) )  ->  (
( A ( +v
`  U ) (
-u 1 ( .sOLD `  U ) A ) ) ( +v `  U ) B )  =  ( A ( +v `  U ) ( (
-u 1 ( .sOLD `  U ) A ) ( +v
`  U ) B ) ) )
368, 11, 33, 34, 35syl13anc 1228 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A ( +v
`  U ) (
-u 1 ( .sOLD `  U ) A ) ) ( +v `  U ) B )  =  ( A ( +v `  U ) ( (
-u 1 ( .sOLD `  U ) A ) ( +v
`  U ) B ) ) )
37 eqid 2454 . . . . . 6  |-  ( 0vec `  U )  =  (
0vec `  U )
381, 4, 5, 37nvrinv 25749 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) A ) )  =  ( 0vec `  U ) )
39383adant3 1014 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) A ) )  =  ( 0vec `  U ) )
4039oveq1d 6285 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A ( +v
`  U ) (
-u 1 ( .sOLD `  U ) A ) ) ( +v `  U ) B )  =  ( ( 0vec `  U
) ( +v `  U ) B ) )
4130, 36, 403eqtr2d 2501 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) ( A M B ) ) )  =  ( (
0vec `  U )
( +v `  U
) B ) )
421, 4, 37nv0lid 25732 . . 3  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( 0vec `  U )
( +v `  U
) B )  =  B )
43423adant2 1013 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( 0vec `  U )
( +v `  U
) B )  =  B )
447, 41, 433eqtrd 2499 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( A M B ) )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   CCcc 9479   1c1 9482    x. cmul 9486   -ucneg 9797   NrmCVeccnv 25678   +vcpv 25679   BaseSetcba 25680   .sOLDcns 25681   0veccn0v 25682   -vcnsb 25683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9798  df-neg 9799  df-grpo 25394  df-gid 25395  df-ginv 25396  df-gdiv 25397  df-ablo 25485  df-vc 25640  df-nv 25686  df-va 25689  df-ba 25690  df-sm 25691  df-0v 25692  df-vs 25693  df-nmcv 25694
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator