Theorem xaddpnf1 10042

Description: Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xaddpnf1	|- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )

Proof of Theorem xaddpnf1

Step	Hyp	Ref	Expression
1		pnfxr 8199	. . . 4 |- +oo e. RR*
2		xaddval 10041	. . . 4 |- ( ( A e. RR* /\ +oo e. RR* ) -> ( A +e +oo ) = if ( A = +oo , if ( +oo = -oo , 0 , +oo ) , if ( A = -oo , if ( +oo = +oo , 0 , -oo ) , if ( +oo = +oo , +oo , if ( +oo = -oo , -oo , ( A + +oo ) ) ) ) ) )
3	1, 2	mpan2 425	. . 3 |- ( A e. RR* -> ( A +e +oo ) = if ( A = +oo , if ( +oo = -oo , 0 , +oo ) , if ( A = -oo , if ( +oo = +oo , 0 , -oo ) , if ( +oo = +oo , +oo , if ( +oo = -oo , -oo , ( A + +oo ) ) ) ) ) )
4	3	adantr 276	. 2 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = if ( A = +oo , if ( +oo = -oo , 0 , +oo ) , if ( A = -oo , if ( +oo = +oo , 0 , -oo ) , if ( +oo = +oo , +oo , if ( +oo = -oo , -oo , ( A + +oo ) ) ) ) ) )
5		pnfnemnf 8201	. . . . 5 |- +oo =/= -oo
6		ifnefalse 3613	. . . . 5 |- ( +oo =/= -oo -> if ( +oo = -oo , 0 , +oo ) = +oo )
7	5, 6	mp1i 10	. . . 4 |- ( A =/= -oo -> if ( +oo = -oo , 0 , +oo ) = +oo )
8		ifnefalse 3613	. . . . 5 |- ( A =/= -oo -> if ( A = -oo , if ( +oo = +oo , 0 , -oo ) , if ( +oo = +oo , +oo , if ( +oo = -oo , -oo , ( A + +oo ) ) ) ) = if ( +oo = +oo , +oo , if ( +oo = -oo , -oo , ( A + +oo ) ) ) )
9		eqid 2229	. . . . . 6 |- +oo = +oo
10	9	iftruei 3608	. . . . 5 |- if ( +oo = +oo , +oo , if ( +oo = -oo , -oo , ( A + +oo ) ) ) = +oo
11	8, 10	eqtrdi 2278	. . . 4 |- ( A =/= -oo -> if ( A = -oo , if ( +oo = +oo , 0 , -oo ) , if ( +oo = +oo , +oo , if ( +oo = -oo , -oo , ( A + +oo ) ) ) ) = +oo )
12	7, 11	ifeq12d 3622	. . 3 |- ( A =/= -oo -> if ( A = +oo , if ( +oo = -oo , 0 , +oo ) , if ( A = -oo , if ( +oo = +oo , 0 , -oo ) , if ( +oo = +oo , +oo , if ( +oo = -oo , -oo , ( A + +oo ) ) ) ) ) = if ( A = +oo , +oo , +oo ) )
13		xrpnfdc 10038	. . . 4 |- ( A e. RR* -> DECID A = +oo )
14		ifiddc 3638	. . . 4 |- (DECID A = +oo -> if ( A = +oo , +oo , +oo ) = +oo )
15	13, 14	syl 14	. . 3 |- ( A e. RR* -> if ( A = +oo , +oo , +oo ) = +oo )
16	12, 15	sylan9eqr 2284	. 2 |- ( ( A e. RR* /\ A =/= -oo ) -> if ( A = +oo , if ( +oo = -oo , 0 , +oo ) , if ( A = -oo , if ( +oo = +oo , 0 , -oo ) , if ( +oo = +oo , +oo , if ( +oo = -oo , -oo , ( A + +oo ) ) ) ) ) = +oo )
17	4, 16	eqtrd 2262	1 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 839   = wceq 1395   e. wcel 2200   =/= wne 2400   ifcif 3602  (class class class)co 6001   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-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-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-xadd 9969

This theorem is referenced by:  xaddnemnf  10053  xaddcom  10057  xnn0xadd0  10063  xnegdi  10064  xaddass  10065  xleadd1a  10069  xlt2add  10076  xsubge0  10077  xposdif  10078  xlesubadd  10079  xrbdtri  11787  isxmet2d  15022