Theorem renegcl 7272

Description: Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)

Assertion

Ref	Expression
renegcl	|- ( A e. RR -> -u A e. RR )

Proof of Theorem renegcl

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-rnegex 6993	. 2 |- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
2		recn 7014	. . . . 5 |- ( x e. RR -> x e. CC )
3		df-neg 7185	. . . . . . 7 |- -u A = ( 0 - A )
4	3	eqeq1i 2047	. . . . . 6 |- ( -u A = x <-> ( 0 - A ) = x )
5		recn 7014	. . . . . . 7 |- ( A e. RR -> A e. CC )
6		0cn 7019	. . . . . . . 8 |- 0 e. CC
7		subadd 7214	. . . . . . . 8 |- ( ( 0 e. CC /\ A e. CC /\ x e. CC ) -> ( ( 0 - A ) = x <-> ( A + x ) = 0 ) )
8	6, 7	mp3an1 1219	. . . . . . 7 |- ( ( A e. CC /\ x e. CC ) -> ( ( 0 - A ) = x <-> ( A + x ) = 0 ) )
9	5, 8	sylan 267	. . . . . 6 |- ( ( A e. RR /\ x e. CC ) -> ( ( 0 - A ) = x <-> ( A + x ) = 0 ) )
10	4, 9	syl5bb 181	. . . . 5 |- ( ( A e. RR /\ x e. CC ) -> ( -u A = x <-> ( A + x ) = 0 ) )
11	2, 10	sylan2 270	. . . 4 |- ( ( A e. RR /\ x e. RR ) -> ( -u A = x <-> ( A + x ) = 0 ) )
12		eleq1a 2109	. . . . 5 |- ( x e. RR -> ( -u A = x -> -u A e. RR ) )
13	12	adantl 262	. . . 4 |- ( ( A e. RR /\ x e. RR ) -> ( -u A = x -> -u A e. RR ) )
14	11, 13	sylbird 159	. . 3 |- ( ( A e. RR /\ x e. RR ) -> ( ( A + x ) = 0 -> -u A e. RR ) )
15	14	rexlimdva 2433	. 2 |- ( A e. RR -> ( E. x e. RR ( A + x ) = 0 -> -u A e. RR ) )
16	1, 15	mpd 13	1 |- ( A e. RR -> -u A e. RR )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   E.wrex 2307  (class class class)co 5512   CCcc 6887   RRcr 6888   0cc0 6889   + caddc 6892   - cmin 7182   -ucneg 7183

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-sub 7184  df-neg 7185

This theorem is referenced by:  renegcli  7273  resubcl  7275  negreb  7276  renegcld  7378  ltnegcon1  7458  ltnegcon2  7459  lenegcon1  7461  lenegcon2  7462  mullt0  7475  recexre  7569  elnnz  8255  btwnz  8357  ublbneg  8548  negm  8550  rpnegap  8615  xnegcl  8745  xnegneg  8746  xltnegi  8748  iooneg  8856  iccneg  8857  icoshftf1o  8859  crim  9458  absnid  9671  absdiflt  9688  absdifle  9689