Theorem addasssrg 7588

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 7559	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		addsrpr 7577	. 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 7577	. 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 7577	. 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 7577	. 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 7369	. . . 4 |- ( ( x e. P. /\ z e. P. ) -> ( x +P. z ) e. P. )
7		addclpr 7369	. . . 4 |- ( ( y e. P. /\ w e. P. ) -> ( y +P. w ) e. P. )
8	6, 7	anim12i 336	. . 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 578	. 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 7369	. . . 4 |- ( ( z e. P. /\ v e. P. ) -> ( z +P. v ) e. P. )
11		addclpr 7369	. . . 4 |- ( ( w e. P. /\ u e. P. ) -> ( w +P. u ) e. P. )
12	10, 11	anim12i 336	. . 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 578	. 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 7411	. . . . 5 |- ( ( x e. P. /\ z e. P. /\ v e. P. ) -> ( ( x +P. z ) +P. v ) = ( x +P. ( z +P. v ) ) )
15	14	3adant1r 1210	. . . 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 1212	. . 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 1214	. 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 7411	. . . . 5 |- ( ( y e. P. /\ w e. P. /\ u e. P. ) -> ( ( y +P. w ) +P. u ) = ( y +P. ( w +P. u ) ) )
19	18	3adant1l 1209	. . . 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 1211	. . 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 1213	. 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 6547	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 103   /\ w3a 963   = wceq 1332   e. wcel 1481  (class class class)co 5782   P.cnp 7123   +P. cpp 7125   ~R cer 7128   R.cnr 7129   +R cplr 7133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-iplp 7300  df-enr 7558  df-nr 7559  df-plr 7560

This theorem is referenced by:  ltm1sr  7609  caucvgsrlemoffval  7628  caucvgsrlemoffcau  7630  caucvgsrlemoffres  7632  caucvgsr  7634  map2psrprg  7637  axaddass  7704  axmulass  7705  axdistr  7706