Theorem axrnegex 7687

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 7729. (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 7638	. . . . 5 |- ( A e. RR <-> ( ( 1st ` A ) e. R. /\ A = <. ( 1st ` A ) , 0R >. ) )
2	1	simplbi 272	. . . 4 |- ( A e. RR -> ( 1st ` A ) e. R. )
3		m1r 7560	. . . 4 |- -1R e. R.
4		mulclsr 7562	. . . 4 |- ( ( ( 1st ` A ) e. R. /\ -1R e. R. ) -> ( ( 1st ` A ) .R -1R ) e. R. )
5	2, 3, 4	sylancl 409	. . 3 |- ( A e. RR -> ( ( 1st ` A ) .R -1R ) e. R. )
6		opelreal 7635	. . 3 |- ( <. ( ( 1st ` A ) .R -1R ) , 0R >. e. RR <-> ( ( 1st ` A ) .R -1R ) e. R. )
7	5, 6	sylibr 133	. 2 |- ( A e. RR -> <. ( ( 1st ` A ) .R -1R ) , 0R >. e. RR )
8	1	simprbi 273	. . . 4 |- ( A e. RR -> A = <. ( 1st ` A ) , 0R >. )
9	8	oveq1d 5789	. . 3 |- ( A e. RR -> ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = ( <. ( 1st ` A ) , 0R >. + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) )
10		addresr 7645	. . . 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 408	. . 3 |- ( A e. RR -> ( <. ( 1st ` A ) , 0R >. + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = <. ( ( 1st ` A ) +R ( ( 1st ` A ) .R -1R ) ) , 0R >. )
12		pn0sr 7579	. . . . . 6 |- ( ( 1st ` A ) e. R. -> ( ( 1st ` A ) +R ( ( 1st ` A ) .R -1R ) ) = 0R )
13	12	opeq1d 3711	. . . . 5 |- ( ( 1st ` A ) e. R. -> <. ( ( 1st ` A ) +R ( ( 1st ` A ) .R -1R ) ) , 0R >. = <. 0R , 0R >. )
14		df-0 7627	. . . . 5 |- 0 = <. 0R , 0R >.
15	13, 14	syl6eqr 2190	. . . 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 2176	. 2 |- ( A e. RR -> ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = 0 )
18		oveq2 5782	. . . 4 |- ( x = <. ( ( 1st ` A ) .R -1R ) , 0R >. -> ( A + x ) = ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) )
19	18	eqeq1d 2148	. . 3 |- ( x = <. ( ( 1st ` A ) .R -1R ) , 0R >. -> ( ( A + x ) = 0 <-> ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = 0 ) )
20	19	rspcev 2789	. 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 408	1 |- ( A e. RR -> E. x e. RR ( A + x ) = 0 )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   E.wrex 2417   <.cop 3530   ` cfv 5123  (class class class)co 5774   1stc1st 6036   R.cnr 7105   0Rc0r 7106   -1Rcm1r 7108   +R cplr 7109   .R cmr 7110   RRcr 7619   0cc0 7620   + caddc 7623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-i1p 7275  df-iplp 7276  df-imp 7277  df-enr 7534  df-nr 7535  df-plr 7536  df-mr 7537  df-0r 7539  df-1r 7540  df-m1r 7541  df-c 7626  df-0 7627  df-r 7630  df-add 7631

This theorem is referenced by: (None)