Theorem xaddval 10041

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 8193	. . . . . 6 |- 0 e. RR*
2	1	a1i 9	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> 0 e. RR* )
3		pnfxr 8199	. . . . . 6 |- +oo e. RR*
4	3	a1i 9	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> +oo e. RR* )
5		xrmnfdc 10039	. . . . . 6 |- ( B e. RR* -> DECID B = -oo )
6	5	adantl 277	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> DECID B = -oo )
7	2, 4, 6	ifcldcd 3640	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> if ( B = -oo , 0 , +oo ) e. RR* )
8	7	adantr 276	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A = +oo ) -> if ( B = -oo , 0 , +oo ) e. RR* )
9		mnfxr 8203	. . . . . . 7 |- -oo e. RR*
10	9	a1i 9	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> -oo e. RR* )
11		xrpnfdc 10038	. . . . . . 7 |- ( B e. RR* -> DECID B = +oo )
12	11	adantl 277	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> DECID B = +oo )
13	2, 10, 12	ifcldcd 3640	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> if ( B = +oo , 0 , -oo ) e. RR* )
14	13	ad2antrr 488	. . . 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 542	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. A = +oo )
18		simpl 109	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> A e. RR* )
19	18	ad4antr 494	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> A e. RR* )
20		simpllr 534	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. A = -oo )
21	20	neqned 2407	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> A =/= -oo )
22		xrnemnf 9973	. . . . . . . . . . 11 |- ( ( A e. RR* /\ A =/= -oo ) <-> ( A e. RR \/ A = +oo ) )
23	22	biimpi 120	. . . . . . . . . 10 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A e. RR \/ A = +oo ) )
24	19, 21, 23	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( A e. RR \/ A = +oo ) )
25	17, 24	ecased 1383	. . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> A e. RR )
26		simplr 528	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. B = +oo )
27		simpr 110	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> B e. RR* )
28	27	ad4antr 494	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> B e. RR* )
29		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. B = -oo )
30	29	neqned 2407	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> B =/= -oo )
31		xrnemnf 9973	. . . . . . . . . . 11 |- ( ( B e. RR* /\ B =/= -oo ) <-> ( B e. RR \/ B = +oo ) )
32	31	biimpi 120	. . . . . . . . . 10 |- ( ( B e. RR* /\ B =/= -oo ) -> ( B e. RR \/ B = +oo ) )
33	28, 30, 32	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( B e. RR \/ B = +oo ) )
34	26, 33	ecased 1383	. . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> B e. RR )
35	25, 34	readdcld 8176	. . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( A + B ) e. RR )
36	35	rexrd 8196	. . . . . 6 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( A + B ) e. RR* )
37	6	ad3antrrr 492	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) -> DECID B = -oo )
38	16, 36, 37	ifcldadc 3632	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) -> if ( B = -oo , -oo , ( A + B ) ) e. RR* )
39	12	ad2antrr 488	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) -> DECID B = +oo )
40	15, 38, 39	ifcldadc 3632	. . . 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 10039	. . . . 5 |- ( A e. RR* -> DECID A = -oo )
42	41	ad2antrr 488	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) -> DECID A = -oo )
43	14, 40, 42	ifcldadc 3632	. . 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 10038	. . . 4 |- ( A e. RR* -> DECID A = +oo )
45	44	adantr 276	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> DECID A = +oo )
46	8, 43, 45	ifcldadc 3632	. 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 109	. . . . 5 |- ( ( x = A /\ y = B ) -> x = A )
48	47	eqeq1d 2238	. . . 4 |- ( ( x = A /\ y = B ) -> ( x = +oo <-> A = +oo ) )
49		simpr 110	. . . . . 6 |- ( ( x = A /\ y = B ) -> y = B )
50	49	eqeq1d 2238	. . . . 5 |- ( ( x = A /\ y = B ) -> ( y = -oo <-> B = -oo ) )
51	50	ifbid 3624	. . . 4 |- ( ( x = A /\ y = B ) -> if ( y = -oo , 0 , +oo ) = if ( B = -oo , 0 , +oo ) )
52	47	eqeq1d 2238	. . . . 5 |- ( ( x = A /\ y = B ) -> ( x = -oo <-> A = -oo ) )
53	49	eqeq1d 2238	. . . . . 6 |- ( ( x = A /\ y = B ) -> ( y = +oo <-> B = +oo ) )
54	53	ifbid 3624	. . . . 5 |- ( ( x = A /\ y = B ) -> if ( y = +oo , 0 , -oo ) = if ( B = +oo , 0 , -oo ) )
55		oveq12 6010	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( x + y ) = ( A + B ) )
56	50, 55	ifbieq2d 3627	. . . . . 6 |- ( ( x = A /\ y = B ) -> if ( y = -oo , -oo , ( x + y ) ) = if ( B = -oo , -oo , ( A + B ) ) )
57	53, 56	ifbieq2d 3627	. . . . 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 3629	. . . 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 3629	. . 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 9969	. . 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 6134	. 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 1372	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 104   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   =/= wne 2400   ifcif 3602  (class class class)co 6001   RRcr 7998   0cc0 7999   + caddc 8002   +oocpnf 8178   -oocmnf 8179   RR*cxr 8180   +ecxad 9966

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-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096  ax-rnegex 8108

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-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-xadd 9969

This theorem is referenced by:  xaddpnf1  10042  xaddpnf2  10043  xaddmnf1  10044  xaddmnf2  10045  pnfaddmnf  10046  mnfaddpnf  10047  rexadd  10048