Theorem xaddpnf1 9912

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 8072	. . . 4 |- +oo e. RR*
2		xaddval 9911	. . . 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 8074	. . . . 5 |- +oo =/= -oo
6		ifnefalse 3568	. . . . 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 3568	. . . . 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 2193	. . . . . 6 |- +oo = +oo
10	9	iftruei 3563	. . . . 5 |- if ( +oo = +oo , +oo , if ( +oo = -oo , -oo , ( A + +oo ) ) ) = +oo
11	8, 10	eqtrdi 2242	. . . 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 3576	. . 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 9908	. . . 4 |- ( A e. RR* -> DECID A = +oo )
14		ifiddc 3591	. . . 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 2248	. 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 2226	1 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 835   = wceq 1364   e. wcel 2164   =/= wne 2364   ifcif 3557  (class class class)co 5918   0cc0 7872   + caddc 7875   +oocpnf 8051   -oocmnf 8052   RR*cxr 8053   +ecxad 9836

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969  ax-rnegex 7981

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-xadd 9839

This theorem is referenced by:  xaddnemnf  9923  xaddcom  9927  xnn0xadd0  9933  xnegdi  9934  xaddass  9935  xleadd1a  9939  xlt2add  9946  xsubge0  9947  xposdif  9948  xlesubadd  9949  xrbdtri  11419  isxmet2d  14516