Theorem xaddmnf1 10144

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 8295	. . . 4 |- -oo e. RR*
2		xaddval 10141	. . . 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		ifnefalse 3620	. . 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 8294	. . . . . 6 |- -oo =/= +oo
7		ifnefalse 3620	. . . . . 6 |- ( -oo =/= +oo -> if ( -oo = +oo , 0 , -oo ) = -oo )
8	6, 7	ax-mp 5	. . . . 5 |- if ( -oo = +oo , 0 , -oo ) = -oo
9		ifnefalse 3620	. . . . . . 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 2231	. . . . . . 7 |- -oo = -oo
12	11	iftruei 3615	. . . . . 6 |- if ( -oo = -oo , -oo , ( A + -oo ) ) = -oo
13	10, 12	eqtri 2252	. . . . 5 |- if ( -oo = +oo , +oo , if ( -oo = -oo , -oo , ( A + -oo ) ) ) = -oo
14		ifeq12 3626	. . . . 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 426	. . . 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 10139	. . . . 5 |- ( A e. RR* -> DECID A = -oo )
17		ifiddc 3645	. . . . 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 2276	. . 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 2286	. 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 2264	1 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 842   = wceq 1398   e. wcel 2202   =/= wne 2403   ifcif 3607  (class class class)co 6028   0cc0 8092   + caddc 8095   +oocpnf 8270   -oocmnf 8271   RR*cxr 8272   +ecxad 10066

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1re 8186  ax-addrcl 8189  ax-rnegex 8201

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-xadd 10069

This theorem is referenced by:  xaddnepnf  10154  xaddcom  10157  xnegdi  10164  xleadd1a  10169  xsubge0  10177  xposdif  10178  xlesubadd  10179  xleaddadd  10183  xblss2ps  15215  xblss2  15216