Theorem addasssrg 7966

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 7937	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		addsrpr 7955	. 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 7955	. 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 7955	. 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 7955	. 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 7747	. . . 4 |- ( ( x e. P. /\ z e. P. ) -> ( x +P. z ) e. P. )
7		addclpr 7747	. . . 4 |- ( ( y e. P. /\ w e. P. ) -> ( y +P. w ) e. P. )
8	6, 7	anim12i 338	. . 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 590	. 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 7747	. . . 4 |- ( ( z e. P. /\ v e. P. ) -> ( z +P. v ) e. P. )
11		addclpr 7747	. . . 4 |- ( ( w e. P. /\ u e. P. ) -> ( w +P. u ) e. P. )
12	10, 11	anim12i 338	. . 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 590	. 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 7789	. . . . 5 |- ( ( x e. P. /\ z e. P. /\ v e. P. ) -> ( ( x +P. z ) +P. v ) = ( x +P. ( z +P. v ) ) )
15	14	3adant1r 1255	. . . 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 1257	. . 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 1259	. 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 7789	. . . . 5 |- ( ( y e. P. /\ w e. P. /\ u e. P. ) -> ( ( y +P. w ) +P. u ) = ( y +P. ( w +P. u ) ) )
19	18	3adant1l 1254	. . . 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 1256	. . 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 1258	. 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 6809	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 104   /\ w3a 1002   = wceq 1395   e. wcel 2200  (class class class)co 6013   P.cnp 7501   +P. cpp 7503   ~R cer 7506   R.cnr 7507   +R cplr 7511

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-enq0 7634  df-nq0 7635  df-0nq0 7636  df-plq0 7637  df-mq0 7638  df-inp 7676  df-iplp 7678  df-enr 7936  df-nr 7937  df-plr 7938

This theorem is referenced by:  ltm1sr  7987  caucvgsrlemoffval  8006  caucvgsrlemoffcau  8008  caucvgsrlemoffres  8010  caucvgsr  8012  map2psrprg  8015  axaddass  8082  axmulass  8083  axdistr  8084