Theorem addasssrg 8071

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 8042	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		addsrpr 8060	. 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 8060	. 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 8060	. 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 8060	. 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 7852	. . . 4 |- ( ( x e. P. /\ z e. P. ) -> ( x +P. z ) e. P. )
7		addclpr 7852	. . . 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 592	. 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 7852	. . . 4 |- ( ( z e. P. /\ v e. P. ) -> ( z +P. v ) e. P. )
11		addclpr 7852	. . . 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 592	. 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 7894	. . . . 5 |- ( ( x e. P. /\ z e. P. /\ v e. P. ) -> ( ( x +P. z ) +P. v ) = ( x +P. ( z +P. v ) ) )
15	14	3adant1r 1258	. . . 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 1260	. . 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 1262	. 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 7894	. . . . 5 |- ( ( y e. P. /\ w e. P. /\ u e. P. ) -> ( ( y +P. w ) +P. u ) = ( y +P. ( w +P. u ) ) )
19	18	3adant1l 1257	. . . 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 1259	. . 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 1261	. 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 6879	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 1005   = wceq 1398   e. wcel 2203  (class class class)co 6050   P.cnp 7606   +P. cpp 7608   ~R cer 7611   R.cnr 7612   +R cplr 7616

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-iplp 7783  df-enr 8041  df-nr 8042  df-plr 8043

This theorem is referenced by:  ltm1sr  8092  caucvgsrlemoffval  8111  caucvgsrlemoffcau  8113  caucvgsrlemoffres  8115  caucvgsr  8117  map2psrprg  8120  axaddass  8187  axmulass  8188  axdistr  8189