Theorem xnegid 9951

Description: Extended real version of negid 8290. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xnegid	|- ( A e. RR* -> ( A +e -e A ) = 0 )

Proof of Theorem xnegid

Step	Hyp	Ref	Expression
1		elxr 9868	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 9922	. . . . 5 |- ( A e. RR -> -e A = -u A )
3	2	oveq2d 5941	. . . 4 |- ( A e. RR -> ( A +e -e A ) = ( A +e -u A ) )
4		renegcl 8304	. . . . 5 |- ( A e. RR -> -u A e. RR )
5		rexadd 9944	. . . . 5 |- ( ( A e. RR /\ -u A e. RR ) -> ( A +e -u A ) = ( A + -u A ) )
6	4, 5	mpdan 421	. . . 4 |- ( A e. RR -> ( A +e -u A ) = ( A + -u A ) )
7		recn 8029	. . . . 5 |- ( A e. RR -> A e. CC )
8	7	negidd 8344	. . . 4 |- ( A e. RR -> ( A + -u A ) = 0 )
9	3, 6, 8	3eqtrd 2233	. . 3 |- ( A e. RR -> ( A +e -e A ) = 0 )
10		id 19	. . . . 5 |- ( A = +oo -> A = +oo )
11		xnegeq 9919	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
12		xnegpnf 9920	. . . . . 6 |- -e +oo = -oo
13	11, 12	eqtrdi 2245	. . . . 5 |- ( A = +oo -> -e A = -oo )
14	10, 13	oveq12d 5943	. . . 4 |- ( A = +oo -> ( A +e -e A ) = ( +oo +e -oo ) )
15		pnfaddmnf 9942	. . . 4 |- ( +oo +e -oo ) = 0
16	14, 15	eqtrdi 2245	. . 3 |- ( A = +oo -> ( A +e -e A ) = 0 )
17		id 19	. . . . 5 |- ( A = -oo -> A = -oo )
18		xnegeq 9919	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
19		xnegmnf 9921	. . . . . 6 |- -e -oo = +oo
20	18, 19	eqtrdi 2245	. . . . 5 |- ( A = -oo -> -e A = +oo )
21	17, 20	oveq12d 5943	. . . 4 |- ( A = -oo -> ( A +e -e A ) = ( -oo +e +oo ) )
22		mnfaddpnf 9943	. . . 4 |- ( -oo +e +oo ) = 0
23	21, 22	eqtrdi 2245	. . 3 |- ( A = -oo -> ( A +e -e A ) = 0 )
24	9, 16, 23	3jaoi 1314	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( A +e -e A ) = 0 )
25	1, 24	sylbi 121	1 |- ( A e. RR* -> ( A +e -e A ) = 0 )

Syntax hints:   -> wi 4   \/ w3o 979   = wceq 1364   e. wcel 2167  (class class class)co 5925   RRcr 7895   0cc0 7896   + caddc 7899   +oocpnf 8075   -oocmnf 8076   RR*cxr 8077   -ucneg 8215   -ecxne 9861   +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-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007

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-reu 2482  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-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-sub 8216  df-neg 8217  df-xneg 9864  df-xadd 9865

This theorem is referenced by: (None)