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Theorem addcomv 25758
Description: Vector addition is commutative. (Contributed by FL, 15-Sep-2013.)
Hypothesis
Ref Expression
addcomv.1  |-  + w  =  (  + cv `  N )
Assertion
Ref Expression
addcomv  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) )  /\  B  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( A + w B )  =  ( B + w A ) )

Proof of Theorem addcomv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cnex 8834 . . . . . . . . 9  |-  CC  e.  _V
2 ovex 5899 . . . . . . . . 9  |-  ( 1 ... N )  e. 
_V
31, 2elmap 6812 . . . . . . . 8  |-  ( A  e.  ( CC  ^m  ( 1 ... N
) )  <->  A :
( 1 ... N
) --> CC )
41, 2elmap 6812 . . . . . . . . . 10  |-  ( B  e.  ( CC  ^m  ( 1 ... N
) )  <->  B :
( 1 ... N
) --> CC )
5 ffvelrn 5679 . . . . . . . . . . . . 13  |-  ( ( A : ( 1 ... N ) --> CC 
/\  x  e.  ( 1 ... N ) )  ->  ( A `  x )  e.  CC )
653adant1 973 . . . . . . . . . . . 12  |-  ( ( B : ( 1 ... N ) --> CC 
/\  A : ( 1 ... N ) --> CC  /\  x  e.  ( 1 ... N
) )  ->  ( A `  x )  e.  CC )
7 ffvelrn 5679 . . . . . . . . . . . . 13  |-  ( ( B : ( 1 ... N ) --> CC 
/\  x  e.  ( 1 ... N ) )  ->  ( B `  x )  e.  CC )
873adant2 974 . . . . . . . . . . . 12  |-  ( ( B : ( 1 ... N ) --> CC 
/\  A : ( 1 ... N ) --> CC  /\  x  e.  ( 1 ... N
) )  ->  ( B `  x )  e.  CC )
96, 8jca 518 . . . . . . . . . . 11  |-  ( ( B : ( 1 ... N ) --> CC 
/\  A : ( 1 ... N ) --> CC  /\  x  e.  ( 1 ... N
) )  ->  (
( A `  x
)  e.  CC  /\  ( B `  x )  e.  CC ) )
1093exp 1150 . . . . . . . . . 10  |-  ( B : ( 1 ... N ) --> CC  ->  ( A : ( 1 ... N ) --> CC 
->  ( x  e.  ( 1 ... N )  ->  ( ( A `
 x )  e.  CC  /\  ( B `
 x )  e.  CC ) ) ) )
114, 10sylbi 187 . . . . . . . . 9  |-  ( B  e.  ( CC  ^m  ( 1 ... N
) )  ->  ( A : ( 1 ... N ) --> CC  ->  ( x  e.  ( 1 ... N )  -> 
( ( A `  x )  e.  CC  /\  ( B `  x
)  e.  CC ) ) ) )
1211com12 27 . . . . . . . 8  |-  ( A : ( 1 ... N ) --> CC  ->  ( B  e.  ( CC 
^m  ( 1 ... N ) )  -> 
( x  e.  ( 1 ... N )  ->  ( ( A `
 x )  e.  CC  /\  ( B `
 x )  e.  CC ) ) ) )
133, 12sylbi 187 . . . . . . 7  |-  ( A  e.  ( CC  ^m  ( 1 ... N
) )  ->  ( B  e.  ( CC  ^m  ( 1 ... N
) )  ->  (
x  e.  ( 1 ... N )  -> 
( ( A `  x )  e.  CC  /\  ( B `  x
)  e.  CC ) ) ) )
1413imp 418 . . . . . 6  |-  ( ( A  e.  ( CC 
^m  ( 1 ... N ) )  /\  B  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( x  e.  ( 1 ... N )  ->  ( ( A `
 x )  e.  CC  /\  ( B `
 x )  e.  CC ) ) )
15143adant1 973 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) )  /\  B  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( x  e.  ( 1 ... N
)  ->  ( ( A `  x )  e.  CC  /\  ( B `
 x )  e.  CC ) ) )
1615imp 418 . . . 4  |-  ( ( ( N  e.  NN  /\  A  e.  ( CC 
^m  ( 1 ... N ) )  /\  B  e.  ( CC  ^m  ( 1 ... N
) ) )  /\  x  e.  ( 1 ... N ) )  ->  ( ( A `
 x )  e.  CC  /\  ( B `
 x )  e.  CC ) )
17 addcom 9014 . . . 4  |-  ( ( ( A `  x
)  e.  CC  /\  ( B `  x )  e.  CC )  -> 
( ( A `  x )  +  ( B `  x ) )  =  ( ( B `  x )  +  ( A `  x ) ) )
1816, 17syl 15 . . 3  |-  ( ( ( N  e.  NN  /\  A  e.  ( CC 
^m  ( 1 ... N ) )  /\  B  e.  ( CC  ^m  ( 1 ... N
) ) )  /\  x  e.  ( 1 ... N ) )  ->  ( ( A `
 x )  +  ( B `  x
) )  =  ( ( B `  x
)  +  ( A `
 x ) ) )
1918mpteq2dva 4122 . 2  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) )  /\  B  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( x  e.  ( 1 ... N
)  |->  ( ( A `
 x )  +  ( B `  x
) ) )  =  ( x  e.  ( 1 ... N ) 
|->  ( ( B `  x )  +  ( A `  x ) ) ) )
20 addcomv.1 . . 3  |-  + w  =  (  + cv `  N )
2120isaddrv 25749 . 2  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) )  /\  B  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( A + w B )  =  ( x  e.  ( 1 ... N ) 
|->  ( ( A `  x )  +  ( B `  x ) ) ) )
2220isaddrv 25749 . . 3  |-  ( ( N  e.  NN  /\  B  e.  ( CC  ^m  ( 1 ... N
) )  /\  A  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( B + w A )  =  ( x  e.  ( 1 ... N ) 
|->  ( ( B `  x )  +  ( A `  x ) ) ) )
23223com23 1157 . 2  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) )  /\  B  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( B + w A )  =  ( x  e.  ( 1 ... N ) 
|->  ( ( B `  x )  +  ( A `  x ) ) ) )
2419, 21, 233eqtr4d 2338 1  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) )  /\  B  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( A + w B )  =  ( B + w A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   CCcc 8751   1c1 8754    + caddc 8756   NNcn 9762   ...cfz 10798    + cvcplcv 25747
This theorem is referenced by:  addidrv2  25761  negveud  25771  negveudr  25772  tcnvec  25793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-addcv 25748
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