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Theorem addrcom 27783
Description: Vector addition is commutative. (Contributed by Andrew Salmon, 28-Jan-2012.)
Assertion
Ref Expression
addrcom  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A + r B )  =  ( B + r A ) )

Proof of Theorem addrcom
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 addrfn 27780 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A + r B )  Fn  RR )
2 addrfn 27780 . . 3  |-  ( ( B  e.  D  /\  A  e.  C )  ->  ( B + r A )  Fn  RR )
32ancoms 439 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( B + r A )  Fn  RR )
4 addcomgi 27764 . . . . . 6  |-  ( ( A `  x )  +  ( B `  x ) )  =  ( ( B `  x )  +  ( A `  x ) )
5 addrfv 27777 . . . . . 6  |-  ( ( A  e.  C  /\  B  e.  D  /\  x  e.  RR )  ->  ( ( A + r B ) `  x
)  =  ( ( A `  x )  +  ( B `  x ) ) )
6 addrfv 27777 . . . . . . 7  |-  ( ( B  e.  D  /\  A  e.  C  /\  x  e.  RR )  ->  ( ( B + r A ) `  x
)  =  ( ( B `  x )  +  ( A `  x ) ) )
763com12 1155 . . . . . 6  |-  ( ( A  e.  C  /\  B  e.  D  /\  x  e.  RR )  ->  ( ( B + r A ) `  x
)  =  ( ( B `  x )  +  ( A `  x ) ) )
84, 5, 73eqtr4a 2354 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  D  /\  x  e.  RR )  ->  ( ( A + r B ) `  x
)  =  ( ( B + r A ) `  x ) )
983expia 1153 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( x  e.  RR  ->  ( ( A + r B ) `  x
)  =  ( ( B + r A ) `  x ) ) )
109ralrimiv 2638 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  A. x  e.  RR  ( ( A + r B ) `  x
)  =  ( ( B + r A ) `  x ) )
11 eqfnfv 5638 . . 3  |-  ( ( ( A + r B )  Fn  RR  /\  ( B + r A )  Fn  RR )  ->  ( ( A + r B )  =  ( B + r A )  <->  A. x  e.  RR  ( ( A + r B ) `
 x )  =  ( ( B + r A ) `  x
) ) )
1210, 11syl5ibrcom 213 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( ( A + r B )  Fn  RR  /\  ( B + r A )  Fn  RR )  -> 
( A + r B )  =  ( B + r A ) ) )
131, 3, 12mp2and 660 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A + r B )  =  ( B + r A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   RRcr 8752    + caddc 8756   + rcplusr 27765
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  ax-addf 8832
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-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-addr 27771
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