Theorem addasssrg 6995

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.

Step	Hyp	Ref	Expression
1		df-nr 6966	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		addsrpr 6984	. 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 6984	. 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 6984	. 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 6984	. 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 6789	. . . 4 |- ( ( x e. P. /\ z e. P. ) -> ( x +P. z ) e. P. )
7		addclpr 6789	. . . 4 |- ( ( y e. P. /\ w e. P. ) -> ( y +P. w ) e. P. )
8	6, 7	anim12i 331	. . 3 |- ( ( ( x e. P. /\ z e. P. ) /\ ( y e. P. /\ w e. P. ) ) -> ( ( x +P. z ) e. P. /\ ( y +P. w ) e. P. ) )
9	8	an4s 553	. 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 6789	. . . 4 |- ( ( z e. P. /\ v e. P. ) -> ( z +P. v ) e. P. )
11		addclpr 6789	. . . 4 |- ( ( w e. P. /\ u e. P. ) -> ( w +P. u ) e. P. )
12	10, 11	anim12i 331	. . 3 |- ( ( ( z e. P. /\ v e. P. ) /\ ( w e. P. /\ u e. P. ) ) -> ( ( z +P. v ) e. P. /\ ( w +P. u ) e. P. ) )
13	12	an4s 553	. 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 6831	. . . . 5 |- ( ( x e. P. /\ z e. P. /\ v e. P. ) -> ( ( x +P. z ) +P. v ) = ( x +P. ( z +P. v ) ) )
15	14	3adant1r 1163	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ z e. P. /\ v e. P. ) -> ( ( x +P. z ) +P. v ) = ( x +P. ( z +P. v ) ) )
16	15	3adant2r 1165	. . 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 ) ) )
17	16	3adant3r 1167	. 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 6831	. . . . 5 |- ( ( y e. P. /\ w e. P. /\ u e. P. ) -> ( ( y +P. w ) +P. u ) = ( y +P. ( w +P. u ) ) )
19	18	3adant1l 1162	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ w e. P. /\ u e. P. ) -> ( ( y +P. w ) +P. u ) = ( y +P. ( w +P. u ) ) )
20	19	3adant2l 1164	. . 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 ) ) )
21	20	3adant3l 1166	. 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 ) ) )
22	1, 2, 3, 4, 5, 9, 13, 17, 21	ecoviass 6282	1 |- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( ( A +R B ) +R C ) = ( A +R ( B +R C ) ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 920   = wceq 1285   e. wcel 1434  (class class class)co 5543   P.cnp 6543   +P. cpp 6545   ~R cer 6548   R.cnr 6549   +R cplr 6553

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-enr 6965  df-nr 6966  df-plr 6967

This theorem is referenced by:  caucvgsrlemoffval  7034  caucvgsrlemoffcau  7036  caucvgsrlemoffres  7038  caucvgsr  7040  axaddass  7100  axmulass  7101  axdistr  7102