Theorem xnegid 9635

Description: Extended real version of negid 8002. (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 9556	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 9606	. . . . 5 |- ( A e. RR -> -e A = -u A )
3	2	oveq2d 5783	. . . 4 |- ( A e. RR -> ( A +e -e A ) = ( A +e -u A ) )
4		renegcl 8016	. . . . 5 |- ( A e. RR -> -u A e. RR )
5		rexadd 9628	. . . . 5 |- ( ( A e. RR /\ -u A e. RR ) -> ( A +e -u A ) = ( A + -u A ) )
6	4, 5	mpdan 417	. . . 4 |- ( A e. RR -> ( A +e -u A ) = ( A + -u A ) )
7		recn 7746	. . . . 5 |- ( A e. RR -> A e. CC )
8	7	negidd 8056	. . . 4 |- ( A e. RR -> ( A + -u A ) = 0 )
9	3, 6, 8	3eqtrd 2174	. . 3 |- ( A e. RR -> ( A +e -e A ) = 0 )
10		id 19	. . . . 5 |- ( A = +oo -> A = +oo )
11		xnegeq 9603	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
12		xnegpnf 9604	. . . . . 6 |- -e +oo = -oo
13	11, 12	syl6eq 2186	. . . . 5 |- ( A = +oo -> -e A = -oo )
14	10, 13	oveq12d 5785	. . . 4 |- ( A = +oo -> ( A +e -e A ) = ( +oo +e -oo ) )
15		pnfaddmnf 9626	. . . 4 |- ( +oo +e -oo ) = 0
16	14, 15	syl6eq 2186	. . 3 |- ( A = +oo -> ( A +e -e A ) = 0 )
17		id 19	. . . . 5 |- ( A = -oo -> A = -oo )
18		xnegeq 9603	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
19		xnegmnf 9605	. . . . . 6 |- -e -oo = +oo
20	18, 19	syl6eq 2186	. . . . 5 |- ( A = -oo -> -e A = +oo )
21	17, 20	oveq12d 5785	. . . 4 |- ( A = -oo -> ( A +e -e A ) = ( -oo +e +oo ) )
22		mnfaddpnf 9627	. . . 4 |- ( -oo +e +oo ) = 0
23	21, 22	syl6eq 2186	. . 3 |- ( A = -oo -> ( A +e -e A ) = 0 )
24	9, 16, 23	3jaoi 1281	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( A +e -e A ) = 0 )
25	1, 24	sylbi 120	1 |- ( A e. RR* -> ( A +e -e A ) = 0 )

Syntax hints:   -> wi 4   \/ w3o 961   = wceq 1331   e. wcel 1480  (class class class)co 5767   RRcr 7612   0cc0 7613   + caddc 7616   +oocpnf 7790   -oocmnf 7791   RR*cxr 7792   -ucneg 7927   -ecxne 9549   +ecxad 9550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-sub 7928  df-neg 7929  df-xneg 9552  df-xadd 9553

This theorem is referenced by: (None)