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Theorem addrcom 30962
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 30959 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A +r
B )  Fn  RR )
2 addrfn 30959 . . 3  |-  ( ( B  e.  D  /\  A  e.  C )  ->  ( B +r
A )  Fn  RR )
32ancoms 453 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( B +r
A )  Fn  RR )
4 addcomgi 30943 . . . . . 6  |-  ( ( A `  x )  +  ( B `  x ) )  =  ( ( B `  x )  +  ( A `  x ) )
5 addrfv 30956 . . . . . 6  |-  ( ( A  e.  C  /\  B  e.  D  /\  x  e.  RR )  ->  ( ( A +r B ) `  x )  =  ( ( A `  x
)  +  ( B `
 x ) ) )
6 addrfv 30956 . . . . . . 7  |-  ( ( B  e.  D  /\  A  e.  C  /\  x  e.  RR )  ->  ( ( B +r A ) `  x )  =  ( ( B `  x
)  +  ( A `
 x ) ) )
763com12 1200 . . . . . 6  |-  ( ( A  e.  C  /\  B  e.  D  /\  x  e.  RR )  ->  ( ( B +r A ) `  x )  =  ( ( B `  x
)  +  ( A `
 x ) ) )
84, 5, 73eqtr4a 2534 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  D  /\  x  e.  RR )  ->  ( ( A +r B ) `  x )  =  ( ( B +r
A ) `  x
) )
983expia 1198 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( x  e.  RR  ->  ( ( A +r B ) `  x )  =  ( ( B +r
A ) `  x
) ) )
109ralrimiv 2876 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  A. x  e.  RR  ( ( A +r B ) `  x )  =  ( ( B +r
A ) `  x
) )
11 eqfnfv 5973 . . 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 222 . 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 679 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A +r
B )  =  ( B +r A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814    Fn wfn 5581   ` cfv 5586  (class class class)co 6282   RRcr 9487    + caddc 9491   +rcplusr 30944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-addf 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-ltxr 9629  df-addr 30950
This theorem is referenced by: (None)
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