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Theorem addrcom 27341
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 27338 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A + r B )  Fn  RR )
2 addrfn 27338 . . 3  |-  ( ( B  e.  D  /\  A  e.  C )  ->  ( B + r A )  Fn  RR )
32ancoms 440 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( B + r A )  Fn  RR )
4 addcomgi 27322 . . . . . 6  |-  ( ( A `  x )  +  ( B `  x ) )  =  ( ( B `  x )  +  ( A `  x ) )
5 addrfv 27335 . . . . . 6  |-  ( ( A  e.  C  /\  B  e.  D  /\  x  e.  RR )  ->  ( ( A + r B ) `  x
)  =  ( ( A `  x )  +  ( B `  x ) ) )
6 addrfv 27335 . . . . . . 7  |-  ( ( B  e.  D  /\  A  e.  C  /\  x  e.  RR )  ->  ( ( B + r A ) `  x
)  =  ( ( B `  x )  +  ( A `  x ) ) )
763com12 1157 . . . . . 6  |-  ( ( A  e.  C  /\  B  e.  D  /\  x  e.  RR )  ->  ( ( B + r A ) `  x
)  =  ( ( B `  x )  +  ( A `  x ) ) )
84, 5, 73eqtr4a 2438 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  D  /\  x  e.  RR )  ->  ( ( A + r B ) `  x
)  =  ( ( B + r A ) `  x ) )
983expia 1155 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( x  e.  RR  ->  ( ( A + r B ) `  x
)  =  ( ( B + r A ) `  x ) ) )
109ralrimiv 2724 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  A. x  e.  RR  ( ( A + r B ) `  x
)  =  ( ( B + r A ) `  x ) )
11 eqfnfv 5759 . . 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 214 . 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 661 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A + r B )  =  ( B + r A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2642    Fn wfn 5382   ` cfv 5387  (class class class)co 6013   RRcr 8915    + caddc 8919   + rcplusr 27323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-addf 8995
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-ltxr 9051  df-addr 27329
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