Theorem xaddf 9966

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 8119	. . . . . . 7 |- 0 e. RR*
2	1	a1i 9	. . . . . 6 |- ( ( x e. RR* /\ y e. RR* ) -> 0 e. RR* )
3		pnfxr 8125	. . . . . . 7 |- +oo e. RR*
4	3	a1i 9	. . . . . 6 |- ( ( x e. RR* /\ y e. RR* ) -> +oo e. RR* )
5		xrmnfdc 9965	. . . . . . 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 3608	. . . . 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 8129	. . . . . . 7 |- -oo e. RR*
11	10	a1i 9	. . . . . 6 |- ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ x = -oo ) -> -oo e. RR* )
12		xrpnfdc 9964	. . . . . . 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 3608	. . . . 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 2383	. . . . . . . . . . 11 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> x =/= -oo )
21		xrnemnf 9899	. . . . . . . . . . . 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 1362	. . . . . . . . 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 2384	. . . . . . . . . . . 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 9899	. . . . . . . . . . . 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 1362	. . . . . . . . 9 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> y e. RR )
33	24, 32	readdcld 8102	. . . . . . . 8 |- ( ( ( ( ( ( x e. RR* /\ y e. RR* ) /\ -. x = +oo ) /\ -. x = -oo ) /\ -. y = +oo ) /\ -. y = -oo ) -> ( x + y ) e. RR )
34	33	rexrd 8122	. . . . . . 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 3600	. . . . . 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 3600	. . . . 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 9965	. . . . . 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 3600	. . . 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 9964	. . . . 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 3600	. . 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 2560	. 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 9895	. . 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 6287	. 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 710  DECID wdc 836   = wceq 1373   e. wcel 2176   =/= wne 2376   A.wral 2484   ifcif 3571   X. cxp 4673   -->wf 5267  (class class class)co 5944   RRcr 7924   0cc0 7925   + caddc 7928   +oocpnf 8104   -oocmnf 8105   RR*cxr 8106   +ecxad 9892

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022  ax-rnegex 8034

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-pnf 8109  df-mnf 8110  df-xr 8111  df-xadd 9895

This theorem is referenced by:  xaddcl  9982