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Theorem addrcom 27555
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 27552 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A + r B )  Fn  RR )
2 addrfn 27552 . . 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 27536 . . . . . 6  |-  ( ( A `  x )  +  ( B `  x ) )  =  ( ( B `  x )  +  ( A `  x ) )
5 addrfv 27549 . . . . . 6  |-  ( ( A  e.  C  /\  B  e.  D  /\  x  e.  RR )  ->  ( ( A + r B ) `  x
)  =  ( ( A `  x )  +  ( B `  x ) ) )
6 addrfv 27549 . . . . . . 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 2470 . . . . 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 2756 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  A. x  e.  RR  ( ( A + r B ) `  x
)  =  ( ( B + r A ) `  x ) )
11 eqfnfv 5794 . . 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 1721   A.wral 2674    Fn wfn 5416   ` cfv 5421  (class class class)co 6048   RRcr 8953    + caddc 8957   + rcplusr 27537
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-addf 9033
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-ltxr 9089  df-addr 27543
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