Theorem xaddf 9844

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 8004	. . . . . . 7 |- 0 e. RR*
2	1	a1i 9	. . . . . 6 |- ( ( x e. RR* /\ y e. RR* ) -> 0 e. RR* )
3		pnfxr 8010	. . . . . . 7 |- +oo e. RR*
4	3	a1i 9	. . . . . 6 |- ( ( x e. RR* /\ y e. RR* ) -> +oo e. RR* )
5		xrmnfdc 9843	. . . . . . 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 3571	. . . . 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 8014	. . . . . . 7 |- -oo e. RR*
11	10	a1i 9	. . . . . 6 |- ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ x = -oo ) -> -oo e. RR* )
12		xrpnfdc 9842	. . . . . . 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 3571	. . . . 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 2354	. . . . . . . . . . 11 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> x =/= -oo )
21		xrnemnf 9777	. . . . . . . . . . . 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 1349	. . . . . . . . 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 2355	. . . . . . . . . . . 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 9777	. . . . . . . . . . . 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 1349	. . . . . . . . 9 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> y e. RR )
33	24, 32	readdcld 7987	. . . . . . . 8 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> ( x + y ) e. RR )
34	33	rexrd 8007	. . . . . . 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 3564	. . . . . 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 3564	. . . . 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 9843	. . . . . 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 3564	. . . 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 9842	. . . . 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 3564	. . 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 2531	. 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 9773	. . 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 6202	. 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 708  DECID wdc 834   = wceq 1353   e. wcel 2148   =/= wne 2347   A.wral 2455   ifcif 3535   X. cxp 4625   -->wf 5213  (class class class)co 5875   RRcr 7810   0cc0 7811   + caddc 7814   +oocpnf 7989   -oocmnf 7990   RR*cxr 7991   +ecxad 9770

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908  ax-rnegex 7920

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-pnf 7994  df-mnf 7995  df-xr 7996  df-xadd 9773

This theorem is referenced by:  xaddcl  9860