Theorem xaddval 9628

Description: Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xaddval	|- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) = if ( A = +oo , if ( B = -oo , 0 , +oo ) , if ( A = -oo , if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) ) ) )

Proof of Theorem xaddval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0xr 7812	. . . . . 6 |- 0 e. RR*
2	1	a1i 9	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> 0 e. RR* )
3		pnfxr 7818	. . . . . 6 |- +oo e. RR*
4	3	a1i 9	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> +oo e. RR* )
5		xrmnfdc 9626	. . . . . 6 |- ( B e. RR* -> DECID B = -oo )
6	5	adantl 275	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> DECID B = -oo )
7	2, 4, 6	ifcldcd 3507	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> if ( B = -oo , 0 , +oo ) e. RR* )
8	7	adantr 274	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A = +oo ) -> if ( B = -oo , 0 , +oo ) e. RR* )
9		mnfxr 7822	. . . . . . 7 |- -oo e. RR*
10	9	a1i 9	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> -oo e. RR* )
11		xrpnfdc 9625	. . . . . . 7 |- ( B e. RR* -> DECID B = +oo )
12	11	adantl 275	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> DECID B = +oo )
13	2, 10, 12	ifcldcd 3507	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> if ( B = +oo , 0 , -oo ) e. RR* )
14	13	ad2antrr 479	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ A = -oo ) -> if ( B = +oo , 0 , -oo ) e. RR* )
15	3	a1i 9	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ B = +oo ) -> +oo e. RR* )
16	9	a1i 9	. . . . . 6 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ B = -oo ) -> -oo e. RR* )
17		simp-4r 531	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. A = +oo )
18		simpl 108	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> A e. RR* )
19	18	ad4antr 485	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> A e. RR* )
20		simpllr 523	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. A = -oo )
21	20	neqned 2315	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> A =/= -oo )
22		xrnemnf 9564	. . . . . . . . . . 11 |- ( ( A e. RR* /\ A =/= -oo ) <-> ( A e. RR \/ A = +oo ) )
23	22	biimpi 119	. . . . . . . . . 10 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A e. RR \/ A = +oo ) )
24	19, 21, 23	syl2anc 408	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( A e. RR \/ A = +oo ) )
25	17, 24	ecased 1327	. . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> A e. RR )
26		simplr 519	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. B = +oo )
27		simpr 109	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> B e. RR* )
28	27	ad4antr 485	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> B e. RR* )
29		simpr 109	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. B = -oo )
30	29	neqned 2315	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> B =/= -oo )
31		xrnemnf 9564	. . . . . . . . . . 11 |- ( ( B e. RR* /\ B =/= -oo ) <-> ( B e. RR \/ B = +oo ) )
32	31	biimpi 119	. . . . . . . . . 10 |- ( ( B e. RR* /\ B =/= -oo ) -> ( B e. RR \/ B = +oo ) )
33	28, 30, 32	syl2anc 408	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( B e. RR \/ B = +oo ) )
34	26, 33	ecased 1327	. . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> B e. RR )
35	25, 34	readdcld 7795	. . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( A + B ) e. RR )
36	35	rexrd 7815	. . . . . 6 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( A + B ) e. RR* )
37	6	ad3antrrr 483	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) -> DECID B = -oo )
38	16, 36, 37	ifcldadc 3501	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) -> if ( B = -oo , -oo , ( A + B ) ) e. RR* )
39	12	ad2antrr 479	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) -> DECID B = +oo )
40	15, 38, 39	ifcldadc 3501	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) -> if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) e. RR* )
41		xrmnfdc 9626	. . . . 5 |- ( A e. RR* -> DECID A = -oo )
42	41	ad2antrr 479	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) -> DECID A = -oo )
43	14, 40, 42	ifcldadc 3501	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) -> if ( A = -oo , if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) ) e. RR* )
44		xrpnfdc 9625	. . . 4 |- ( A e. RR* -> DECID A = +oo )
45	44	adantr 274	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> DECID A = +oo )
46	8, 43, 45	ifcldadc 3501	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> if ( A = +oo , if ( B = -oo , 0 , +oo ) , if ( A = -oo , if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) ) ) e. RR* )
47		simpl 108	. . . . 5 |- ( ( x = A /\ y = B ) -> x = A )
48	47	eqeq1d 2148	. . . 4 |- ( ( x = A /\ y = B ) -> ( x = +oo <-> A = +oo ) )
49		simpr 109	. . . . . 6 |- ( ( x = A /\ y = B ) -> y = B )
50	49	eqeq1d 2148	. . . . 5 |- ( ( x = A /\ y = B ) -> ( y = -oo <-> B = -oo ) )
51	50	ifbid 3493	. . . 4 |- ( ( x = A /\ y = B ) -> if ( y = -oo , 0 , +oo ) = if ( B = -oo , 0 , +oo ) )
52	47	eqeq1d 2148	. . . . 5 |- ( ( x = A /\ y = B ) -> ( x = -oo <-> A = -oo ) )
53	49	eqeq1d 2148	. . . . . 6 |- ( ( x = A /\ y = B ) -> ( y = +oo <-> B = +oo ) )
54	53	ifbid 3493	. . . . 5 |- ( ( x = A /\ y = B ) -> if ( y = +oo , 0 , -oo ) = if ( B = +oo , 0 , -oo ) )
55		oveq12 5783	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( x + y ) = ( A + B ) )
56	50, 55	ifbieq2d 3496	. . . . . 6 |- ( ( x = A /\ y = B ) -> if ( y = -oo , -oo , ( x + y ) ) = if ( B = -oo , -oo , ( A + B ) ) )
57	53, 56	ifbieq2d 3496	. . . . 5 |- ( ( x = A /\ y = B ) -> if ( y = +oo , +oo , if ( y = -oo , -oo , ( x + y ) ) ) = if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) )
58	52, 54, 57	ifbieq12d 3498	. . . 4 |- ( ( x = A /\ y = B ) -> if ( x = -oo , if ( y = +oo , 0 , -oo ) , if ( y = +oo , +oo , if ( y = -oo , -oo , ( x + y ) ) ) ) = if ( A = -oo , if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) ) )
59	48, 51, 58	ifbieq12d 3498	. . 3 |- ( ( x = A /\ y = B ) -> if ( x = +oo , if ( y = -oo , 0 , +oo ) , if ( x = -oo , if ( y = +oo , 0 , -oo ) , if ( y = +oo , +oo , if ( y = -oo , -oo , ( x + y ) ) ) ) ) = if ( A = +oo , if ( B = -oo , 0 , +oo ) , if ( A = -oo , if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) ) ) )
60		df-xadd 9560	. . 3 |- +e = ( x e. RR* , y e. RR* |-> if ( x = +oo , if ( y = -oo , 0 , +oo ) , if ( x = -oo , if ( y = +oo , 0 , -oo ) , if ( y = +oo , +oo , if ( y = -oo , -oo , ( x + y ) ) ) ) ) )
61	59, 60	ovmpoga 5900	. 2 |- ( ( A e. RR* /\ B e. RR* /\ if ( A = +oo , if ( B = -oo , 0 , +oo ) , if ( A = -oo , if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) ) ) e. RR* ) -> ( A +e B ) = if ( A = +oo , if ( B = -oo , 0 , +oo ) , if ( A = -oo , if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) ) ) )
62	46, 61	mpd3an3 1316	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) = if ( A = +oo , if ( B = -oo , 0 , +oo ) , if ( A = -oo , if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 697  DECID wdc 819   = wceq 1331   e. wcel 1480   =/= wne 2308   ifcif 3474  (class class class)co 5774   RRcr 7619   0cc0 7620   + caddc 7623   +oocpnf 7797   -oocmnf 7798   RR*cxr 7799   +ecxad 9557

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1re 7714  ax-addrcl 7717  ax-rnegex 7729

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-xadd 9560

This theorem is referenced by:  xaddpnf1  9629  xaddpnf2  9630  xaddmnf1  9631  xaddmnf2  9632  pnfaddmnf  9633  mnfaddpnf  9634  rexadd  9635