Theorem xaddpnf1 10248

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 8378	. . . 4 |- +oo e. RR*
2		xaddval 10247	. . . 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 429	. . 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 8380	. . . . 5 |- +oo =/= -oo
6		ifnefalse 3651	. . . . 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 3651	. . . . 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 2238	. . . . . 6 |- +oo = +oo
10	9	iftruei 3646	. . . . 5 |- if ( +oo = +oo , +oo , if ( +oo = -oo , -oo , ( A + +oo ) ) ) = +oo
11	8, 10	eqtrdi 2287	. . . 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 3660	. . 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 10244	. . . 4 |- ( A e. RR* -> DECID A = +oo )
14		ifiddc 3676	. . . 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 2293	. 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 2271	1 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   ifcif 3638  (class class class)co 6085   0cc0 8179   + caddc 8182   +oocpnf 8357   -oocmnf 8358   RR*cxr 8359   +ecxad 10172

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276  ax-rnegex 8288

This proof 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-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-xadd 10175

This theorem is used by:  xaddnemnf  10259  xaddcom  10263  xnn0xadd0  10269  xnegdi  10270  xaddass  10271  xleadd1a  10275  xlt2add  10282  xsubge0  10283  xposdif  10284  xlesubadd  10285  xrbdtri  12042  isxmet2d  15449