Theorem xaddf 10040

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 8193	. . . . . . 7 |- 0 e. RR*
2	1	a1i 9	. . . . . 6 |- ( ( x e. RR* /\ y e. RR* ) -> 0 e. RR* )
3		pnfxr 8199	. . . . . . 7 |- +oo e. RR*
4	3	a1i 9	. . . . . 6 |- ( ( x e. RR* /\ y e. RR* ) -> +oo e. RR* )
5		xrmnfdc 10039	. . . . . . 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 3640	. . . . 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 8203	. . . . . . 7 |- -oo e. RR*
11	10	a1i 9	. . . . . 6 |- ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ x = -oo ) -> -oo e. RR* )
12		xrpnfdc 10038	. . . . . . 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 3640	. . . . 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 2407	. . . . . . . . . . 11 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> x =/= -oo )
21		xrnemnf 9973	. . . . . . . . . . . 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 1383	. . . . . . . . 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 2408	. . . . . . . . . . . 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 9973	. . . . . . . . . . . 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 1383	. . . . . . . . 9 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> y e. RR )
33	24, 32	readdcld 8176	. . . . . . . 8 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> ( x + y ) e. RR )
34	33	rexrd 8196	. . . . . . 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 3632	. . . . . 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 3632	. . . . 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 10039	. . . . . 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 3632	. . . 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 10038	. . . . 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 3632	. . 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 2584	. 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 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 ) ) ) ) ) )
47	46	fmpo 6347	. 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 713  DECID wdc 839   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   ifcif 3602   X. cxp 4717   -->wf 5314  (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-csb 3125  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-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-pnf 8183  df-mnf 8184  df-xr 8185  df-xadd 9969

This theorem is referenced by:  xaddcl  10056