Theorem xnegid 10016

Description: Extended real version of negid 8354. (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 9933	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 9987	. . . . 5 |- ( A e. RR -> -e A = -u A )
3	2	oveq2d 5983	. . . 4 |- ( A e. RR -> ( A +e -e A ) = ( A +e -u A ) )
4		renegcl 8368	. . . . 5 |- ( A e. RR -> -u A e. RR )
5		rexadd 10009	. . . . 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 8093	. . . . 5 |- ( A e. RR -> A e. CC )
8	7	negidd 8408	. . . 4 |- ( A e. RR -> ( A + -u A ) = 0 )
9	3, 6, 8	3eqtrd 2244	. . 3 |- ( A e. RR -> ( A +e -e A ) = 0 )
10		id 19	. . . . 5 |- ( A = +oo -> A = +oo )
11		xnegeq 9984	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
12		xnegpnf 9985	. . . . . 6 |- -e +oo = -oo
13	11, 12	eqtrdi 2256	. . . . 5 |- ( A = +oo -> -e A = -oo )
14	10, 13	oveq12d 5985	. . . 4 |- ( A = +oo -> ( A +e -e A ) = ( +oo +e -oo ) )
15		pnfaddmnf 10007	. . . 4 |- ( +oo +e -oo ) = 0
16	14, 15	eqtrdi 2256	. . 3 |- ( A = +oo -> ( A +e -e A ) = 0 )
17		id 19	. . . . 5 |- ( A = -oo -> A = -oo )
18		xnegeq 9984	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
19		xnegmnf 9986	. . . . . 6 |- -e -oo = +oo
20	18, 19	eqtrdi 2256	. . . . 5 |- ( A = -oo -> -e A = +oo )
21	17, 20	oveq12d 5985	. . . 4 |- ( A = -oo -> ( A +e -e A ) = ( -oo +e +oo ) )
22		mnfaddpnf 10008	. . . 4 |- ( -oo +e +oo ) = 0
23	21, 22	eqtrdi 2256	. . 3 |- ( A = -oo -> ( A +e -e A ) = 0 )
24	9, 16, 23	3jaoi 1316	. 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 980   = wceq 1373   e. wcel 2178  (class class class)co 5967   RRcr 7959   0cc0 7960   + caddc 7963   +oocpnf 8139   -oocmnf 8140   RR*cxr 8141   -ucneg 8279   -ecxne 9926   +ecxad 9927

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-sub 8280  df-neg 8281  df-xneg 9929  df-xadd 9930

This theorem is referenced by: (None)