Theorem xnegid 9981

Description: Extended real version of negid 8319. (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 9898	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 9952	. . . . 5 |- ( A e. RR -> -e A = -u A )
3	2	oveq2d 5960	. . . 4 |- ( A e. RR -> ( A +e -e A ) = ( A +e -u A ) )
4		renegcl 8333	. . . . 5 |- ( A e. RR -> -u A e. RR )
5		rexadd 9974	. . . . 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 8058	. . . . 5 |- ( A e. RR -> A e. CC )
8	7	negidd 8373	. . . 4 |- ( A e. RR -> ( A + -u A ) = 0 )
9	3, 6, 8	3eqtrd 2242	. . 3 |- ( A e. RR -> ( A +e -e A ) = 0 )
10		id 19	. . . . 5 |- ( A = +oo -> A = +oo )
11		xnegeq 9949	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
12		xnegpnf 9950	. . . . . 6 |- -e +oo = -oo
13	11, 12	eqtrdi 2254	. . . . 5 |- ( A = +oo -> -e A = -oo )
14	10, 13	oveq12d 5962	. . . 4 |- ( A = +oo -> ( A +e -e A ) = ( +oo +e -oo ) )
15		pnfaddmnf 9972	. . . 4 |- ( +oo +e -oo ) = 0
16	14, 15	eqtrdi 2254	. . 3 |- ( A = +oo -> ( A +e -e A ) = 0 )
17		id 19	. . . . 5 |- ( A = -oo -> A = -oo )
18		xnegeq 9949	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
19		xnegmnf 9951	. . . . . 6 |- -e -oo = +oo
20	18, 19	eqtrdi 2254	. . . . 5 |- ( A = -oo -> -e A = +oo )
21	17, 20	oveq12d 5962	. . . 4 |- ( A = -oo -> ( A +e -e A ) = ( -oo +e +oo ) )
22		mnfaddpnf 9973	. . . 4 |- ( -oo +e +oo ) = 0
23	21, 22	eqtrdi 2254	. . 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 2176  (class class class)co 5944   RRcr 7924   0cc0 7925   + caddc 7928   +oocpnf 8104   -oocmnf 8105   RR*cxr 8106   -ucneg 8244   -ecxne 9891   +ecxad 9892

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-sub 8245  df-neg 8246  df-xneg 9894  df-xadd 9895

This theorem is referenced by: (None)