Theorem xaddmnf1 9940

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 8100	. . . 4 |- -oo e. RR*
2		xaddval 9937	. . . 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 3573	. . 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 8099	. . . . . 6 |- -oo =/= +oo
7		ifnefalse 3573	. . . . . 6 |- ( -oo =/= +oo -> if ( -oo = +oo , 0 , -oo ) = -oo )
8	6, 7	ax-mp 5	. . . . 5 |- if ( -oo = +oo , 0 , -oo ) = -oo
9		ifnefalse 3573	. . . . . . 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 2196	. . . . . . 7 |- -oo = -oo
12	11	iftruei 3568	. . . . . 6 |- if ( -oo = -oo , -oo , ( A + -oo ) ) = -oo
13	10, 12	eqtri 2217	. . . . 5 |- if ( -oo = +oo , +oo , if ( -oo = -oo , -oo , ( A + -oo ) ) ) = -oo
14		ifeq12 3578	. . . . 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 9935	. . . . 5 |- ( A e. RR* -> DECID A = -oo )
17		ifiddc 3596	. . . . 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 2241	. . 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 2251	. 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 2229	1 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 835   = wceq 1364   e. wcel 2167   =/= wne 2367   ifcif 3562  (class class class)co 5925   0cc0 7896   + caddc 7899   +oocpnf 8075   -oocmnf 8076   RR*cxr 8077   +ecxad 9862

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993  ax-rnegex 8005

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-xadd 9865

This theorem is referenced by:  xaddnepnf  9950  xaddcom  9953  xnegdi  9960  xleadd1a  9965  xsubge0  9973  xposdif  9974  xlesubadd  9975  xleaddadd  9979  xblss2ps  14724  xblss2  14725