Theorem xaddf 10225

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 8362	. . . . . . 7 |- 0 e. RR*
2	1	a1i 9	. . . . . 6 |- ( ( x e. RR* /\ y e. RR* ) -> 0 e. RR* )
3		pnfxr 8368	. . . . . . 7 |- +oo e. RR*
4	3	a1i 9	. . . . . 6 |- ( ( x e. RR* /\ y e. RR* ) -> +oo e. RR* )
5		xrmnfdc 10224	. . . . . . 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 3675	. . . . 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 8372	. . . . . . 7 |- -oo e. RR*
11	10	a1i 9	. . . . . 6 |- ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ x = -oo ) -> -oo e. RR* )
12		xrpnfdc 10223	. . . . . . 7 |- ( y e. RR* -> DECID y = +oo )
13	12	ad3antlr 497	. . . . . 6 |- ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ x = -oo ) -> DECID y = +oo )
14	9, 11, 13	ifcldcd 3675	. . . . 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 548	. . . . . . . . . 10 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> -. x = +oo )
18		simp-5l 549	. . . . . . . . . . 11 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> x e. RR* )
19		simpllr 540	. . . . . . . . . . . 12 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> -. x = -oo )
20	19	neqned 2427	. . . . . . . . . . 11 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> x =/= -oo )
21		xrnemnf 10158	. . . . . . . . . . . 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 415	. . . . . . . . . 10 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> ( x e. RR \/ x = +oo ) )
24	17, 23	ecased 1390	. . . . . . . . 9 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> x e. RR )
25		simplr 533	. . . . . . . . . 10 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> -. y = +oo )
26		simp-5r 550	. . . . . . . . . . 11 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> y e. RR* )
27		neqne 2428	. . . . . . . . . . . 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 10158	. . . . . . . . . . . 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 415	. . . . . . . . . 10 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> ( y e. RR \/ y = +oo ) )
32	25, 31	ecased 1390	. . . . . . . . 9 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> y e. RR )
33	24, 32	readdcld 8345	. . . . . . . 8 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> ( x + y ) e. RR )
34	33	rexrd 8365	. . . . . . 7 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> ( x + y ) e. RR* )
35	6	ad3antrrr 496	. . . . . . 7 |- ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) -> DECID y = -oo )
36	16, 34, 35	ifcldadc 3667	. . . . . 6 |- ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) -> if ( y = -oo , -oo , ( x + y ) ) e. RR* )
37	12	ad3antlr 497	. . . . . 6 |- ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) -> DECID y = +oo )
38	15, 36, 37	ifcldadc 3667	. . . . 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 10224	. . . . . 6 |- ( x e. RR* -> DECID x = -oo )
40	39	ad2antrr 492	. . . . 5 |- ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) -> DECID x = -oo )
41	14, 38, 40	ifcldadc 3667	. . . 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 10223	. . . . 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 3667	. . 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 2604	. 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 10154	. . 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 6427	. 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 720  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   ifcif 3635   X. cxp 4767   -->wf 5368  (class class class)co 6075   RRcr 8168   0cc0 8169   + caddc 8172   +oocpnf 8347   -oocmnf 8348   RR*cxr 8349   +ecxad 10151

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266  ax-rnegex 8278

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pnf 8352  df-mnf 8353  df-xr 8354  df-xadd 10154

This theorem is referenced by:  xaddcl  10241