Theorem axrnegex 7963

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 8005. (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 7914	. . . . 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 7836	. . . 4 |- -1R e. R.
4		mulclsr 7838	. . . 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 7911	. . 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 5940	. . 3 |- ( A e. RR -> ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = ( <. ( 1st ` A ) , 0R >. + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) )
10		addresr 7921	. . . 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 7855	. . . . . 6 |- ( ( 1st ` A ) e. R. -> ( ( 1st ` A ) +R ( ( 1st ` A ) .R -1R ) ) = 0R )
13	12	opeq1d 3815	. . . . 5 |- ( ( 1st ` A ) e. R. -> <. ( ( 1st ` A ) +R ( ( 1st ` A ) .R -1R ) ) , 0R >. = <. 0R , 0R >. )
14		df-0 7903	. . . . 5 |- 0 = <. 0R , 0R >.
15	13, 14	eqtr4di 2247	. . . 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 2233	. 2 |- ( A e. RR -> ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = 0 )
18		oveq2 5933	. . . 4 |- ( x = <. ( ( 1st ` A ) .R -1R ) , 0R >. -> ( A + x ) = ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) )
19	18	eqeq1d 2205	. . 3 |- ( x = <. ( ( 1st ` A ) .R -1R ) , 0R >. -> ( ( A + x ) = 0 <-> ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = 0 ) )
20	19	rspcev 2868	. 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 1364   e. wcel 2167   E.wrex 2476   <.cop 3626   ` cfv 5259  (class class class)co 5925   1stc1st 6205   R.cnr 7381   0Rc0r 7382   -1Rcm1r 7384   +R cplr 7385   .R cmr 7386   RRcr 7895   0cc0 7896   + caddc 7899

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-i1p 7551  df-iplp 7552  df-imp 7553  df-enr 7810  df-nr 7811  df-plr 7812  df-mr 7813  df-0r 7815  df-1r 7816  df-m1r 7817  df-c 7902  df-0 7903  df-r 7906  df-add 7907

This theorem is referenced by: (None)