Theorem axrnegex 8194

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 8236. (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 8145	. . . . 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 8067	. . . 4 |- -1R e. R.
4		mulclsr 8069	. . . 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 8142	. . 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 6065	. . 3 |- ( A e. RR -> ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = ( <. ( 1st ` A ) , 0R >. + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) )
10		addresr 8152	. . . 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 8086	. . . . . 6 |- ( ( 1st ` A ) e. R. -> ( ( 1st ` A ) +R ( ( 1st ` A ) .R -1R ) ) = 0R )
13	12	opeq1d 3889	. . . . 5 |- ( ( 1st ` A ) e. R. -> <. ( ( 1st ` A ) +R ( ( 1st ` A ) .R -1R ) ) , 0R >. = <. 0R , 0R >. )
14		df-0 8134	. . . . 5 |- 0 = <. 0R , 0R >.
15	13, 14	eqtr4di 2283	. . . 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 2269	. 2 |- ( A e. RR -> ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = 0 )
18		oveq2 6058	. . . 4 |- ( x = <. ( ( 1st ` A ) .R -1R ) , 0R >. -> ( A + x ) = ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) )
19	18	eqeq1d 2241	. . 3 |- ( x = <. ( ( 1st ` A ) .R -1R ) , 0R >. -> ( ( A + x ) = 0 <-> ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = 0 ) )
20	19	rspcev 2921	. 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 1398   e. wcel 2203   E.wrex 2521   <.cop 3692   ` cfv 5352  (class class class)co 6050   1stc1st 6332   R.cnr 7612   0Rc0r 7613   -1Rcm1r 7615   +R cplr 7616   .R cmr 7617   RRcr 8126   0cc0 8127   + caddc 8130

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-i1p 7782  df-iplp 7783  df-imp 7784  df-enr 8041  df-nr 8042  df-plr 8043  df-mr 8044  df-0r 8046  df-1r 8047  df-m1r 8048  df-c 8133  df-0 8134  df-r 8137  df-add 8138

This theorem is referenced by: (None)