Theorem xaddmnf1 10250

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

Assertion

Ref	Expression
xaddmnf1	|- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )

Proof of Theorem xaddmnf1

Step	Hyp	Ref	Expression
1		mnfxr 8382	. . . 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		ifnefalse 3651	. . 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 , if ( -oo = +oo , 0 , -oo ) , if ( -oo = +oo , +oo , if ( -oo = -oo , -oo , ( A + -oo ) ) ) ) )
6		mnfnepnf 8381	. . . . . 6 |- -oo =/= +oo
7		ifnefalse 3651	. . . . . 6 |- ( -oo =/= +oo -> if ( -oo = +oo , 0 , -oo ) = -oo )
8	6, 7	ax-mp 5	. . . . 5 |- if ( -oo = +oo , 0 , -oo ) = -oo
9		ifnefalse 3651	. . . . . . 7 |- ( -oo =/= +oo -> if ( -oo = +oo , +oo , if ( -oo = -oo , -oo , ( A + -oo ) ) ) = if ( -oo = -oo , -oo , ( A + -oo ) ) )
10	6, 9	ax-mp 5	. . . . . 6 |- if ( -oo = +oo , +oo , if ( -oo = -oo , -oo , ( A + -oo ) ) ) = if ( -oo = -oo , -oo , ( A + -oo ) )
11		eqid 2238	. . . . . . 7 |- -oo = -oo
12	11	iftruei 3646	. . . . . 6 |- if ( -oo = -oo , -oo , ( A + -oo ) ) = -oo
13	10, 12	eqtri 2259	. . . . 5 |- if ( -oo = +oo , +oo , if ( -oo = -oo , -oo , ( A + -oo ) ) ) = -oo
14		ifeq12 3657	. . . . 5 |- ( ( if ( -oo = +oo , 0 , -oo ) = -oo /\ if ( -oo = +oo , +oo , if ( -oo = -oo , -oo , ( A + -oo ) ) ) = -oo ) -> if ( A = -oo , if ( -oo = +oo , 0 , -oo ) , if ( -oo = +oo , +oo , if ( -oo = -oo , -oo , ( A + -oo ) ) ) ) = if ( A = -oo , -oo , -oo ) )
15	8, 13, 14	mp2an 430	. . . 4 |- if ( A = -oo , if ( -oo = +oo , 0 , -oo ) , if ( -oo = +oo , +oo , if ( -oo = -oo , -oo , ( A + -oo ) ) ) ) = if ( A = -oo , -oo , -oo )
16		xrmnfdc 10245	. . . . 5 |- ( A e. RR* -> DECID A = -oo )
17		ifiddc 3676	. . . . 5 |- (DECID A = -oo -> if ( A = -oo , -oo , -oo ) = -oo )
18	16, 17	syl 14	. . . 4 |- ( A e. RR* -> if ( A = -oo , -oo , -oo ) = -oo )
19	15, 18	eqtrid 2283	. . 3 |- ( A e. RR* -> if ( A = -oo , if ( -oo = +oo , 0 , -oo ) , if ( -oo = +oo , +oo , if ( -oo = -oo , -oo , ( A + -oo ) ) ) ) = -oo )
20	5, 19	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 )
21	4, 20	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:  xaddnepnf  10260  xaddcom  10263  xnegdi  10270  xleadd1a  10275  xsubge0  10283  xposdif  10284  xlesubadd  10285  xleaddadd  10289  xblss2ps  15505  xblss2  15506