Theorem elreal2 8187

Description: Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.)

Assertion

Ref	Expression
elreal2	|- ( A e. RR <-> ( ( 1st ` A ) e. R. /\ A = <. ( 1st ` A ) , 0R >. ) )

Proof of Theorem elreal2

Step	Hyp	Ref	Expression
1		df-r 8179	. . 3 |- RR = ( R. X. { 0R } )
2	1	eleq2i 2305	. 2 |- ( A e. RR <-> A e. ( R. X. { 0R } ) )
3		xp1st 6389	. . . 4 |- ( A e. ( R. X. { 0R } ) -> ( 1st ` A ) e. R. )
4		1st2nd2 6399	. . . . 5 |- ( A e. ( R. X. { 0R } ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
5		xp2nd 6390	. . . . . . 7 |- ( A e. ( R. X. { 0R } ) -> ( 2nd ` A ) e. { 0R } )
6		elsni 3723	. . . . . . 7 |- ( ( 2nd ` A ) e. { 0R } -> ( 2nd ` A ) = 0R )
7	5, 6	syl 14	. . . . . 6 |- ( A e. ( R. X. { 0R } ) -> ( 2nd ` A ) = 0R )
8	7	opeq2d 3906	. . . . 5 |- ( A e. ( R. X. { 0R } ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. ( 1st ` A ) , 0R >. )
9	4, 8	eqtrd 2271	. . . 4 |- ( A e. ( R. X. { 0R } ) -> A = <. ( 1st ` A ) , 0R >. )
10	3, 9	jca 306	. . 3 |- ( A e. ( R. X. { 0R } ) -> ( ( 1st ` A ) e. R. /\ A = <. ( 1st ` A ) , 0R >. ) )
11		eleq1 2301	. . . . 5 |- ( A = <. ( 1st ` A ) , 0R >. -> ( A e. ( R. X. { 0R } ) <-> <. ( 1st ` A ) , 0R >. e. ( R. X. { 0R } ) ) )
12		0r 8107	. . . . . . . 8 |- 0R e. R.
13	12	elexi 2834	. . . . . . 7 |- 0R e. _V
14	13	snid 3736	. . . . . 6 |- 0R e. { 0R }
15		opelxp 4799	. . . . . 6 |- ( <. ( 1st ` A ) , 0R >. e. ( R. X. { 0R } ) <-> ( ( 1st ` A ) e. R. /\ 0R e. { 0R } ) )
16	14, 15	mpbiran2 954	. . . . 5 |- ( <. ( 1st ` A ) , 0R >. e. ( R. X. { 0R } ) <-> ( 1st ` A ) e. R. )
17	11, 16	bitrdi 196	. . . 4 |- ( A = <. ( 1st ` A ) , 0R >. -> ( A e. ( R. X. { 0R } ) <-> ( 1st ` A ) e. R. ) )
18	17	biimparc 299	. . 3 |- ( ( ( 1st ` A ) e. R. /\ A = <. ( 1st ` A ) , 0R >. ) -> A e. ( R. X. { 0R } ) )
19	10, 18	impbii 126	. 2 |- ( A e. ( R. X. { 0R } ) <-> ( ( 1st ` A ) e. R. /\ A = <. ( 1st ` A ) , 0R >. ) )
20	2, 19	bitri 184	1 |- ( A e. RR <-> ( ( 1st ` A ) e. R. /\ A = <. ( 1st ` A ) , 0R >. ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {csn 3705   <.cop 3708   X. cxp 4767   ` cfv 5372   1stc1st 6362   2ndc2nd 6363   R.cnr 7654   0Rc0r 7655   RRcr 8168

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-inp 7823  df-i1p 7824  df-enr 8083  df-nr 8084  df-0r 8088  df-r 8179

This theorem is referenced by:  ltresr2  8197  axrnegex  8236  axpre-suploclemres  8258