Theorem xnegid 9645

Description: Extended real version of negid 8012. (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 9566	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 9616	. . . . 5 |- ( A e. RR -> -e A = -u A )
3	2	oveq2d 5790	. . . 4 |- ( A e. RR -> ( A +e -e A ) = ( A +e -u A ) )
4		renegcl 8026	. . . . 5 |- ( A e. RR -> -u A e. RR )
5		rexadd 9638	. . . . 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 7756	. . . . 5 |- ( A e. RR -> A e. CC )
8	7	negidd 8066	. . . 4 |- ( A e. RR -> ( A + -u A ) = 0 )
9	3, 6, 8	3eqtrd 2176	. . 3 |- ( A e. RR -> ( A +e -e A ) = 0 )
10		id 19	. . . . 5 |- ( A = +oo -> A = +oo )
11		xnegeq 9613	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
12		xnegpnf 9614	. . . . . 6 |- -e +oo = -oo
13	11, 12	syl6eq 2188	. . . . 5 |- ( A = +oo -> -e A = -oo )
14	10, 13	oveq12d 5792	. . . 4 |- ( A = +oo -> ( A +e -e A ) = ( +oo +e -oo ) )
15		pnfaddmnf 9636	. . . 4 |- ( +oo +e -oo ) = 0
16	14, 15	syl6eq 2188	. . 3 |- ( A = +oo -> ( A +e -e A ) = 0 )
17		id 19	. . . . 5 |- ( A = -oo -> A = -oo )
18		xnegeq 9613	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
19		xnegmnf 9615	. . . . . 6 |- -e -oo = +oo
20	18, 19	syl6eq 2188	. . . . 5 |- ( A = -oo -> -e A = +oo )
21	17, 20	oveq12d 5792	. . . 4 |- ( A = -oo -> ( A +e -e A ) = ( -oo +e +oo ) )
22		mnfaddpnf 9637	. . . 4 |- ( -oo +e +oo ) = 0
23	21, 22	syl6eq 2188	. . 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 5774   RRcr 7622   0cc0 7623   + caddc 7626   +oocpnf 7800   -oocmnf 7801   RR*cxr 7802   -ucneg 7937   -ecxne 9559   +ecxad 9560

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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7714  ax-resscn 7715  ax-1cn 7716  ax-1re 7717  ax-icn 7718  ax-addcl 7719  ax-addrcl 7720  ax-mulcl 7721  ax-addcom 7723  ax-addass 7725  ax-distr 7727  ax-i2m1 7728  ax-0id 7731  ax-rnegex 7732  ax-cnre 7734

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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7805  df-mnf 7806  df-xr 7807  df-sub 7938  df-neg 7939  df-xneg 9562  df-xadd 9563

This theorem is referenced by: (None)