Theorem axrnegex 8087

Description: Existence of negative of real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 8129. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
axrnegex	|- ( A e. RR -> E. x e. RR ( A + x ) = 0 )

Distinct variable group:   x, A

Proof of Theorem axrnegex

Step	Hyp	Ref	Expression
1		elreal2 8038	. . . . 5 |- ( A e. RR <-> ( ( 1st ` A ) e. R. /\ A = <. ( 1st ` A ) , 0R >. ) )
2	1	simplbi 274	. . . 4 |- ( A e. RR -> ( 1st ` A ) e. R. )
3		m1r 7960	. . . 4 |- -1R e. R.
4		mulclsr 7962	. . . 4 |- ( ( ( 1st ` A ) e. R. /\ -1R e. R. ) -> ( ( 1st ` A ) .R -1R ) e. R. )
5	2, 3, 4	sylancl 413	. . 3 |- ( A e. RR -> ( ( 1st ` A ) .R -1R ) e. R. )
6		opelreal 8035	. . 3 |- ( <. ( ( 1st ` A ) .R -1R ) , 0R >. e. RR <-> ( ( 1st ` A ) .R -1R ) e. R. )
7	5, 6	sylibr 134	. 2 |- ( A e. RR -> <. ( ( 1st ` A ) .R -1R ) , 0R >. e. RR )
8	1	simprbi 275	. . . 4 |- ( A e. RR -> A = <. ( 1st ` A ) , 0R >. )
9	8	oveq1d 6026	. . 3 |- ( A e. RR -> ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = ( <. ( 1st ` A ) , 0R >. + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) )
10		addresr 8045	. . . 4 |- ( ( ( 1st ` A ) e. R. /\ ( ( 1st ` A ) .R -1R ) e. R. ) -> ( <. ( 1st ` A ) , 0R >. + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = <. ( ( 1st ` A ) +R ( ( 1st ` A ) .R -1R ) ) , 0R >. )
11	2, 5, 10	syl2anc 411	. . 3 |- ( A e. RR -> ( <. ( 1st ` A ) , 0R >. + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = <. ( ( 1st ` A ) +R ( ( 1st ` A ) .R -1R ) ) , 0R >. )
12		pn0sr 7979	. . . . . 6 |- ( ( 1st ` A ) e. R. -> ( ( 1st ` A ) +R ( ( 1st ` A ) .R -1R ) ) = 0R )
13	12	opeq1d 3864	. . . . 5 |- ( ( 1st ` A ) e. R. -> <. ( ( 1st ` A ) +R ( ( 1st ` A ) .R -1R ) ) , 0R >. = <. 0R , 0R >. )
14		df-0 8027	. . . . 5 |- 0 = <. 0R , 0R >.
15	13, 14	eqtr4di 2280	. . . 4 |- ( ( 1st ` A ) e. R. -> <. ( ( 1st ` A ) +R ( ( 1st ` A ) .R -1R ) ) , 0R >. = 0 )
16	2, 15	syl 14	. . 3 |- ( A e. RR -> <. ( ( 1st ` A ) +R ( ( 1st ` A ) .R -1R ) ) , 0R >. = 0 )
17	9, 11, 16	3eqtrd 2266	. 2 |- ( A e. RR -> ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = 0 )
18		oveq2 6019	. . . 4 |- ( x = <. ( ( 1st ` A ) .R -1R ) , 0R >. -> ( A + x ) = ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) )
19	18	eqeq1d 2238	. . 3 |- ( x = <. ( ( 1st ` A ) .R -1R ) , 0R >. -> ( ( A + x ) = 0 <-> ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = 0 ) )
20	19	rspcev 2908	. 2 |- ( ( <. ( ( 1st ` A ) .R -1R ) , 0R >. e. RR /\ ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = 0 ) -> E. x e. RR ( A + x ) = 0 )
21	7, 17, 20	syl2anc 411	1 |- ( A e. RR -> E. x e. RR ( A + x ) = 0 )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   E.wrex 2509   <.cop 3670   ` cfv 5322  (class class class)co 6011   1stc1st 6294   R.cnr 7505   0Rc0r 7506   -1Rcm1r 7508   +R cplr 7509   .R cmr 7510   RRcr 8019   0cc0 8020   + caddc 8023

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-eprel 4382  df-id 4386  df-po 4389  df-iso 4390  df-iord 4459  df-on 4461  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-irdg 6529  df-1o 6575  df-2o 6576  df-oadd 6579  df-omul 6580  df-er 6695  df-ec 6697  df-qs 6701  df-ni 7512  df-pli 7513  df-mi 7514  df-lti 7515  df-plpq 7552  df-mpq 7553  df-enq 7555  df-nqqs 7556  df-plqqs 7557  df-mqqs 7558  df-1nqqs 7559  df-rq 7560  df-ltnqqs 7561  df-enq0 7632  df-nq0 7633  df-0nq0 7634  df-plq0 7635  df-mq0 7636  df-inp 7674  df-i1p 7675  df-iplp 7676  df-imp 7677  df-enr 7934  df-nr 7935  df-plr 7936  df-mr 7937  df-0r 7939  df-1r 7940  df-m1r 7941  df-c 8026  df-0 8027  df-r 8030  df-add 8031

This theorem is referenced by: (None)