Theorem xnegid 10093

Description: Extended real version of negid 8425. (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 10010	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 10064	. . . . 5 |- ( A e. RR -> -e A = -u A )
3	2	oveq2d 6033	. . . 4 |- ( A e. RR -> ( A +e -e A ) = ( A +e -u A ) )
4		renegcl 8439	. . . . 5 |- ( A e. RR -> -u A e. RR )
5		rexadd 10086	. . . . 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 8164	. . . . 5 |- ( A e. RR -> A e. CC )
8	7	negidd 8479	. . . 4 |- ( A e. RR -> ( A + -u A ) = 0 )
9	3, 6, 8	3eqtrd 2268	. . 3 |- ( A e. RR -> ( A +e -e A ) = 0 )
10		id 19	. . . . 5 |- ( A = +oo -> A = +oo )
11		xnegeq 10061	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
12		xnegpnf 10062	. . . . . 6 |- -e +oo = -oo
13	11, 12	eqtrdi 2280	. . . . 5 |- ( A = +oo -> -e A = -oo )
14	10, 13	oveq12d 6035	. . . 4 |- ( A = +oo -> ( A +e -e A ) = ( +oo +e -oo ) )
15		pnfaddmnf 10084	. . . 4 |- ( +oo +e -oo ) = 0
16	14, 15	eqtrdi 2280	. . 3 |- ( A = +oo -> ( A +e -e A ) = 0 )
17		id 19	. . . . 5 |- ( A = -oo -> A = -oo )
18		xnegeq 10061	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
19		xnegmnf 10063	. . . . . 6 |- -e -oo = +oo
20	18, 19	eqtrdi 2280	. . . . 5 |- ( A = -oo -> -e A = +oo )
21	17, 20	oveq12d 6035	. . . 4 |- ( A = -oo -> ( A +e -e A ) = ( -oo +e +oo ) )
22		mnfaddpnf 10085	. . . 4 |- ( -oo +e +oo ) = 0
23	21, 22	eqtrdi 2280	. . 3 |- ( A = -oo -> ( A +e -e A ) = 0 )
24	9, 16, 23	3jaoi 1339	. 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 1003   = wceq 1397   e. wcel 2202  (class class class)co 6017   RRcr 8030   0cc0 8031   + caddc 8034   +oocpnf 8210   -oocmnf 8211   RR*cxr 8212   -ucneg 8350   -ecxne 10003   +ecxad 10004

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-sub 8351  df-neg 8352  df-xneg 10006  df-xadd 10007

This theorem is referenced by: (None)