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Theorem addasssrg 7773
Description: Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
Assertion
Ref Expression
addasssrg  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  (
( A  +R  B
)  +R  C )  =  ( A  +R  ( B  +R  C
) ) )

Proof of Theorem addasssrg
Dummy variables  u  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 7744 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 addsrpr 7762 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  +R  [ <. z ,  w >. ]  ~R  )  =  [ <. (
x  +P.  z ) ,  ( y  +P.  w ) >. ]  ~R  )
3 addsrpr 7762 . 2  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  +R  [ <. v ,  u >. ]  ~R  )  =  [ <. (
z  +P.  v ) ,  ( w  +P.  u ) >. ]  ~R  )
4 addsrpr 7762 . 2  |-  ( ( ( ( x  +P.  z )  e.  P.  /\  ( y  +P.  w
)  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. ( x  +P.  z
) ,  ( y  +P.  w ) >. ]  ~R  +R  [ <. v ,  u >. ]  ~R  )  =  [ <. (
( x  +P.  z
)  +P.  v ) ,  ( ( y  +P.  w )  +P.  u ) >. ]  ~R  )
5 addsrpr 7762 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( ( z  +P.  v )  e.  P.  /\  ( w  +P.  u
)  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  +R  [ <. ( z  +P.  v ) ,  ( w  +P.  u ) >. ]  ~R  )  =  [ <. (
x  +P.  ( z  +P.  v ) ) ,  ( y  +P.  (
w  +P.  u )
) >. ]  ~R  )
6 addclpr 7554 . . . 4  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  +P.  z
)  e.  P. )
7 addclpr 7554 . . . 4  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  +P.  w
)  e.  P. )
86, 7anim12i 338 . . 3  |-  ( ( ( x  e.  P.  /\  z  e.  P. )  /\  ( y  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  +P.  z )  e.  P.  /\  ( y  +P.  w )  e. 
P. ) )
98an4s 588 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  +P.  z )  e.  P.  /\  ( y  +P.  w )  e. 
P. ) )
10 addclpr 7554 . . . 4  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  +P.  v
)  e.  P. )
11 addclpr 7554 . . . 4  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  +P.  u
)  e.  P. )
1210, 11anim12i 338 . . 3  |-  ( ( ( z  e.  P.  /\  v  e.  P. )  /\  ( w  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  +P.  v )  e.  P.  /\  ( w  +P.  u )  e. 
P. ) )
1312an4s 588 . 2  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  +P.  v )  e.  P.  /\  ( w  +P.  u )  e. 
P. ) )
14 addassprg 7596 . . . . 5  |-  ( ( x  e.  P.  /\  z  e.  P.  /\  v  e.  P. )  ->  (
( x  +P.  z
)  +P.  v )  =  ( x  +P.  ( z  +P.  v
) ) )
15143adant1r 1233 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  z  e.  P.  /\  v  e.  P. )  ->  ( ( x  +P.  z )  +P.  v
)  =  ( x  +P.  ( z  +P.  v ) ) )
16153adant2r 1235 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  v  e.  P. )  ->  ( ( x  +P.  z )  +P.  v )  =  ( x  +P.  ( z  +P.  v ) ) )
17163adant3r 1237 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  +P.  z )  +P.  v )  =  ( x  +P.  ( z  +P.  v ) ) )
18 addassprg 7596 . . . . 5  |-  ( ( y  e.  P.  /\  w  e.  P.  /\  u  e.  P. )  ->  (
( y  +P.  w
)  +P.  u )  =  ( y  +P.  ( w  +P.  u
) ) )
19183adant1l 1232 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  w  e.  P.  /\  u  e.  P. )  ->  ( ( y  +P.  w )  +P.  u
)  =  ( y  +P.  ( w  +P.  u ) ) )
20193adant2l 1234 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  u  e.  P. )  ->  ( ( y  +P.  w )  +P.  u )  =  ( y  +P.  ( w  +P.  u ) ) )
21203adant3l 1236 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
y  +P.  w )  +P.  u )  =  ( y  +P.  ( w  +P.  u ) ) )
221, 2, 3, 4, 5, 9, 13, 17, 21ecoviass 6663 1  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  (
( A  +R  B
)  +R  C )  =  ( A  +R  ( B  +R  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    = wceq 1364    e. wcel 2160  (class class class)co 5891   P.cnp 7308    +P. cpp 7310    ~R cer 7313   R.cnr 7314    +R cplr 7318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-1o 6435  df-2o 6436  df-oadd 6439  df-omul 6440  df-er 6553  df-ec 6555  df-qs 6559  df-ni 7321  df-pli 7322  df-mi 7323  df-lti 7324  df-plpq 7361  df-mpq 7362  df-enq 7364  df-nqqs 7365  df-plqqs 7366  df-mqqs 7367  df-1nqqs 7368  df-rq 7369  df-ltnqqs 7370  df-enq0 7441  df-nq0 7442  df-0nq0 7443  df-plq0 7444  df-mq0 7445  df-inp 7483  df-iplp 7485  df-enr 7743  df-nr 7744  df-plr 7745
This theorem is referenced by:  ltm1sr  7794  caucvgsrlemoffval  7813  caucvgsrlemoffcau  7815  caucvgsrlemoffres  7817  caucvgsr  7819  map2psrprg  7822  axaddass  7889  axmulass  7890  axdistr  7891
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