Theorem axrnegex 7941

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 7983. (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 7892	. . . . 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 7814	. . . 4 |- -1R e. R.
4		mulclsr 7816	. . . 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 7889	. . 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 5934	. . 3 |- ( A e. RR -> ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = ( <. ( 1st ` A ) , 0R >. + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) )
10		addresr 7899	. . . 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 7833	. . . . . 6 |- ( ( 1st ` A ) e. R. -> ( ( 1st ` A ) +R ( ( 1st ` A ) .R -1R ) ) = 0R )
13	12	opeq1d 3811	. . . . 5 |- ( ( 1st ` A ) e. R. -> <. ( ( 1st ` A ) +R ( ( 1st ` A ) .R -1R ) ) , 0R >. = <. 0R , 0R >. )
14		df-0 7881	. . . . 5 |- 0 = <. 0R , 0R >.
15	13, 14	eqtr4di 2244	. . . 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 2230	. 2 |- ( A e. RR -> ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = 0 )
18		oveq2 5927	. . . 4 |- ( x = <. ( ( 1st ` A ) .R -1R ) , 0R >. -> ( A + x ) = ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) )
19	18	eqeq1d 2202	. . 3 |- ( x = <. ( ( 1st ` A ) .R -1R ) , 0R >. -> ( ( A + x ) = 0 <-> ( A + <. ( ( 1st ` A ) .R -1R ) , 0R >. ) = 0 ) )
20	19	rspcev 2865	. 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 2164   E.wrex 2473   <.cop 3622   ` cfv 5255  (class class class)co 5919   1stc1st 6193   R.cnr 7359   0Rc0r 7360   -1Rcm1r 7362   +R cplr 7363   .R cmr 7364   RRcr 7873   0cc0 7874   + caddc 7877

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-2o 6472  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415  df-enq0 7486  df-nq0 7487  df-0nq0 7488  df-plq0 7489  df-mq0 7490  df-inp 7528  df-i1p 7529  df-iplp 7530  df-imp 7531  df-enr 7788  df-nr 7789  df-plr 7790  df-mr 7791  df-0r 7793  df-1r 7794  df-m1r 7795  df-c 7880  df-0 7881  df-r 7884  df-add 7885

This theorem is referenced by: (None)