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Theorem xadddir 11360
Description: Commuted version of xadddi 11359. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xadddir  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  (
( A +e
B ) xe C )  =  ( ( A xe C ) +e
( B xe C ) ) )

Proof of Theorem xadddir
StepHypRef Expression
1 xadddi 11359 . . 3  |-  ( ( C  e.  RR  /\  A  e.  RR*  /\  B  e.  RR* )  ->  ( C xe ( A +e B ) )  =  ( ( C xe A ) +e ( C xe B ) ) )
213coml 1195 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( C xe ( A +e B ) )  =  ( ( C xe A ) +e ( C xe B ) ) )
3 xaddcl 11308 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  e.  RR* )
433adant3 1008 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A +e B )  e.  RR* )
5 rexr 9530 . . . 4  |-  ( C  e.  RR  ->  C  e.  RR* )
653ad2ant3 1011 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  C  e.  RR* )
7 xmulcom 11330 . . 3  |-  ( ( ( A +e
B )  e.  RR*  /\  C  e.  RR* )  ->  ( ( A +e B ) xe C )  =  ( C xe ( A +e
B ) ) )
84, 6, 7syl2anc 661 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  (
( A +e
B ) xe C )  =  ( C xe ( A +e B ) ) )
9 simp1 988 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  A  e.  RR* )
10 xmulcom 11330 . . . 4  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A xe C )  =  ( C xe A ) )
119, 6, 10syl2anc 661 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A xe C )  =  ( C xe A ) )
12 simp2 989 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  B  e.  RR* )
13 xmulcom 11330 . . . 4  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B xe C )  =  ( C xe B ) )
1412, 6, 13syl2anc 661 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( B xe C )  =  ( C xe B ) )
1511, 14oveq12d 6208 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  (
( A xe C ) +e
( B xe C ) )  =  ( ( C xe A ) +e ( C xe B ) ) )
162, 8, 153eqtr4d 2502 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  (
( A +e
B ) xe C )  =  ( ( A xe C ) +e
( B xe C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758  (class class class)co 6190   RRcr 9382   RR*cxr 9518   +ecxad 11188   xecxmu 11189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-po 4739  df-so 4740  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-xneg 11190  df-xadd 11191  df-xmul 11192
This theorem is referenced by:  xrge0adddir  26289
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