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Theorem addasssrg 6841
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 6812 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 addsrpr 6830 . 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 6830 . 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 6830 . 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 6830 . 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 6635 . . . 4  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  +P.  z
)  e.  P. )
7 addclpr 6635 . . . 4  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  +P.  w
)  e.  P. )
86, 7anim12i 321 . . 3  |-  ( ( ( x  e.  P.  /\  z  e.  P. )  /\  ( y  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  +P.  z )  e.  P.  /\  ( y  +P.  w )  e. 
P. ) )
98an4s 522 . 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 6635 . . . 4  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  +P.  v
)  e.  P. )
11 addclpr 6635 . . . 4  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  +P.  u
)  e.  P. )
1210, 11anim12i 321 . . 3  |-  ( ( ( z  e.  P.  /\  v  e.  P. )  /\  ( w  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  +P.  v )  e.  P.  /\  ( w  +P.  u )  e. 
P. ) )
1312an4s 522 . 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 6677 . . . . 5  |-  ( ( x  e.  P.  /\  z  e.  P.  /\  v  e.  P. )  ->  (
( x  +P.  z
)  +P.  v )  =  ( x  +P.  ( z  +P.  v
) ) )
15143adant1r 1128 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  z  e.  P.  /\  v  e.  P. )  ->  ( ( x  +P.  z )  +P.  v
)  =  ( x  +P.  ( z  +P.  v ) ) )
16153adant2r 1130 . . 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 1132 . 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 6677 . . . . 5  |-  ( ( y  e.  P.  /\  w  e.  P.  /\  u  e.  P. )  ->  (
( y  +P.  w
)  +P.  u )  =  ( y  +P.  ( w  +P.  u
) ) )
19183adant1l 1127 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  w  e.  P.  /\  u  e.  P. )  ->  ( ( y  +P.  w )  +P.  u
)  =  ( y  +P.  ( w  +P.  u ) ) )
20193adant2l 1129 . . 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 1131 . 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 6216 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 97    /\ w3a 885    = wceq 1243    e. wcel 1393  (class class class)co 5512   P.cnp 6389    +P. cpp 6391    ~R cer 6394   R.cnr 6395    +R cplr 6399
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566  df-enr 6811  df-nr 6812  df-plr 6813
This theorem is referenced by:  caucvgsrlemoffval  6880  caucvgsrlemoffcau  6882  caucvgsrlemoffres  6884  caucvgsr  6886  axaddass  6946  axmulass  6947  axdistr  6948
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