Theorem xaddf 9910

Description: The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
xaddf	|- +e : ( RR* X. RR* ) --> RR*

Proof of Theorem xaddf

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

Step	Hyp	Ref	Expression
1		0xr 8066	. . . . . . 7 |- 0 e. RR*
2	1	a1i 9	. . . . . 6 |- ( ( x e. RR* /\ y e. RR* ) -> 0 e. RR* )
3		pnfxr 8072	. . . . . . 7 |- +oo e. RR*
4	3	a1i 9	. . . . . 6 |- ( ( x e. RR* /\ y e. RR* ) -> +oo e. RR* )
5		xrmnfdc 9909	. . . . . . 7 |- ( y e. RR* -> DECID y = -oo )
6	5	adantl 277	. . . . . 6 |- ( ( x e. RR* /\ y e. RR* ) -> DECID y = -oo )
7	2, 4, 6	ifcldcd 3593	. . . . 5 |- ( ( x e. RR* /\ y e. RR* ) -> if ( y = -oo , 0 , +oo ) e. RR* )
8	7	adantr 276	. . . 4 |- ( ( ( x e. RR* /\ y e. RR* ) /\ x = +oo ) -> if ( y = -oo , 0 , +oo ) e. RR* )
9	1	a1i 9	. . . . . 6 |- ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ x = -oo ) -> 0 e. RR* )
10		mnfxr 8076	. . . . . . 7 |- -oo e. RR*
11	10	a1i 9	. . . . . 6 |- ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ x = -oo ) -> -oo e. RR* )
12		xrpnfdc 9908	. . . . . . 7 |- ( y e. RR* -> DECID y = +oo )
13	12	ad3antlr 493	. . . . . 6 |- ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ x = -oo ) -> DECID y = +oo )
14	9, 11, 13	ifcldcd 3593	. . . . 5 |- ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ x = -oo ) -> if ( y = +oo , 0 , -oo ) e. RR* )
15	3	a1i 9	. . . . . 6 |- ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ y = +oo ) -> +oo e. RR* )
16	10	a1i 9	. . . . . . 7 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ y = -oo ) -> -oo e. RR* )
17		simp-4r 542	. . . . . . . . . 10 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> -. x = +oo )
18		simp-5l 543	. . . . . . . . . . 11 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> x e. RR* )
19		simpllr 534	. . . . . . . . . . . 12 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> -. x = -oo )
20	19	neqned 2371	. . . . . . . . . . 11 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> x =/= -oo )
21		xrnemnf 9843	. . . . . . . . . . . 12 |- ( ( x e. RR* /\ x =/= -oo ) <-> ( x e. RR \/ x = +oo ) )
22	21	biimpi 120	. . . . . . . . . . 11 |- ( ( x e. RR* /\ x =/= -oo ) -> ( x e. RR \/ x = +oo ) )
23	18, 20, 22	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> ( x e. RR \/ x = +oo ) )
24	17, 23	ecased 1360	. . . . . . . . 9 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> x e. RR )
25		simplr 528	. . . . . . . . . 10 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> -. y = +oo )
26		simp-5r 544	. . . . . . . . . . 11 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> y e. RR* )
27		neqne 2372	. . . . . . . . . . . 12 |- ( -. y = -oo -> y =/= -oo )
28	27	adantl 277	. . . . . . . . . . 11 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> y =/= -oo )
29		xrnemnf 9843	. . . . . . . . . . . 12 |- ( ( y e. RR* /\ y =/= -oo ) <-> ( y e. RR \/ y = +oo ) )
30	29	biimpi 120	. . . . . . . . . . 11 |- ( ( y e. RR* /\ y =/= -oo ) -> ( y e. RR \/ y = +oo ) )
31	26, 28, 30	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> ( y e. RR \/ y = +oo ) )
32	25, 31	ecased 1360	. . . . . . . . 9 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> y e. RR )
33	24, 32	readdcld 8049	. . . . . . . 8 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> ( x + y ) e. RR )
34	33	rexrd 8069	. . . . . . 7 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> ( x + y ) e. RR* )
35	6	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) -> DECID y = -oo )
36	16, 34, 35	ifcldadc 3586	. . . . . 6 |- ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) -> if ( y = -oo , -oo , ( x + y ) ) e. RR* )
37	12	ad3antlr 493	. . . . . 6 |- ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) -> DECID y = +oo )
38	15, 36, 37	ifcldadc 3586	. . . . 5 |- ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) -> if ( y = +oo , +oo , if ( y = -oo , -oo , ( x + y ) ) ) e. RR* )
39		xrmnfdc 9909	. . . . . 6 |- ( x e. RR* -> DECID x = -oo )
40	39	ad2antrr 488	. . . . 5 |- ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) -> DECID x = -oo )
41	14, 38, 40	ifcldadc 3586	. . . 4 |- ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) -> if ( x = -oo , if ( y = +oo , 0 , -oo ) , if ( y = +oo , +oo , if ( y = -oo , -oo , ( x + y ) ) ) ) e. RR* )
42		xrpnfdc 9908	. . . . 5 |- ( x e. RR* -> DECID x = +oo )
43	42	adantr 276	. . . 4 |- ( ( x e. RR* /\ y e. RR* ) -> DECID x = +oo )
44	8, 41, 43	ifcldadc 3586	. . 3 |- ( ( 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 ) ) ) ) ) e. RR* )
45	44	rgen2a 2548	. 2 |- A. x e. RR* A. 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 ) ) ) ) ) e. RR*
46		df-xadd 9839	. . 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 ) ) ) ) ) )
47	46	fmpo 6254	. 2 |- ( A. x e. RR* A. 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 ) ) ) ) ) e. RR* <-> +e : ( RR* X. RR* ) --> RR* )
48	45, 47	mpbi 145	1 |- +e : ( RR* X. RR* ) --> RR*

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2164   =/= wne 2364   A.wral 2472   ifcif 3557   X. cxp 4657   -->wf 5250  (class class class)co 5918   RRcr 7871   0cc0 7872   + caddc 7875   +oocpnf 8051   -oocmnf 8052   RR*cxr 8053   +ecxad 9836

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969  ax-rnegex 7981

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-pnf 8056  df-mnf 8057  df-xr 8058  df-xadd 9839

This theorem is referenced by:  xaddcl  9926