Theorem xaddmnf1 9848

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 8014	. . . 4 |- -oo e. RR*
2		xaddval 9845	. . . 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 3546	. . 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 8013	. . . . . 6 |- -oo =/= +oo
7		ifnefalse 3546	. . . . . 6 |- ( -oo =/= +oo -> if ( -oo = +oo , 0 , -oo ) = -oo )
8	6, 7	ax-mp 5	. . . . 5 |- if ( -oo = +oo , 0 , -oo ) = -oo
9		ifnefalse 3546	. . . . . . 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 2177	. . . . . . 7 |- -oo = -oo
12	11	iftruei 3541	. . . . . 6 |- if ( -oo = -oo , -oo , ( A + -oo ) ) = -oo
13	10, 12	eqtri 2198	. . . . 5 |- if ( -oo = +oo , +oo , if ( -oo = -oo , -oo , ( A + -oo ) ) ) = -oo
14		ifeq12 3551	. . . . 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 9843	. . . . 5 |- ( A e. RR* -> DECID A = -oo )
17		ifiddc 3569	. . . . 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 2222	. . 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 2232	. 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 2210	1 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 834   = wceq 1353   e. wcel 2148   =/= wne 2347   ifcif 3535  (class class class)co 5875   0cc0 7811   + caddc 7814   +oocpnf 7989   -oocmnf 7990   RR*cxr 7991   +ecxad 9770

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908  ax-rnegex 7920

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-xr 7996  df-xadd 9773

This theorem is referenced by:  xaddnepnf  9858  xaddcom  9861  xnegdi  9868  xleadd1a  9873  xsubge0  9881  xposdif  9882  xlesubadd  9883  xleaddadd  9887  xblss2ps  13907  xblss2  13908