Theorem elreal2 8013

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

Assertion

Ref	Expression
elreal2	⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1st ‘𝐴), 0R⟩))

Proof of Theorem elreal2

Step	Hyp	Ref	Expression
1		df-r 8005	. . 3 ⊢ ℝ = (R × {0R})
2	1	eleq2i 2296	. 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R}))
3		xp1st 6309	. . . 4 ⊢ (𝐴 ∈ (R × {0R}) → (1st ‘𝐴) ∈ R)
4		1st2nd2 6319	. . . . 5 ⊢ (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
5		xp2nd 6310	. . . . . . 7 ⊢ (𝐴 ∈ (R × {0R}) → (2nd ‘𝐴) ∈ {0R})
6		elsni 3684	. . . . . . 7 ⊢ ((2nd ‘𝐴) ∈ {0R} → (2nd ‘𝐴) = 0R)
7	5, 6	syl 14	. . . . . 6 ⊢ (𝐴 ∈ (R × {0R}) → (2nd ‘𝐴) = 0R)
8	7	opeq2d 3863	. . . . 5 ⊢ (𝐴 ∈ (R × {0R}) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨(1st ‘𝐴), 0R⟩)
9	4, 8	eqtrd 2262	. . . 4 ⊢ (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st ‘𝐴), 0R⟩)
10	3, 9	jca 306	. . 3 ⊢ (𝐴 ∈ (R × {0R}) → ((1st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1st ‘𝐴), 0R⟩))
11		eleq1 2292	. . . . 5 ⊢ (𝐴 = ⟨(1st ‘𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ ⟨(1st ‘𝐴), 0R⟩ ∈ (R × {0R})))
12		0r 7933	. . . . . . . 8 ⊢ 0R ∈ R
13	12	elexi 2812	. . . . . . 7 ⊢ 0R ∈ V
14	13	snid 3697	. . . . . 6 ⊢ 0R ∈ {0R}
15		opelxp 4748	. . . . . 6 ⊢ (⟨(1st ‘𝐴), 0R⟩ ∈ (R × {0R}) ↔ ((1st ‘𝐴) ∈ R ∧ 0R ∈ {0R}))
16	14, 15	mpbiran2 947	. . . . 5 ⊢ (⟨(1st ‘𝐴), 0R⟩ ∈ (R × {0R}) ↔ (1st ‘𝐴) ∈ R)
17	11, 16	bitrdi 196	. . . 4 ⊢ (𝐴 = ⟨(1st ‘𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ (1st ‘𝐴) ∈ R))
18	17	biimparc 299	. . 3 ⊢ (((1st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1st ‘𝐴), 0R⟩) → 𝐴 ∈ (R × {0R}))
19	10, 18	impbii 126	. 2 ⊢ (𝐴 ∈ (R × {0R}) ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1st ‘𝐴), 0R⟩))
20	2, 19	bitri 184	1 ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1st ‘𝐴), 0R⟩))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  {csn 3666  ⟨cop 3669   × cxp 4716  ‘cfv 5317  1^st c1st 6282  2^nd c2nd 6283  Rcnr 7480  0_Rc0r 7481  ℝcr 7994

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4379  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-1o 6560  df-oadd 6564  df-omul 6565  df-er 6678  df-ec 6680  df-qs 6684  df-ni 7487  df-pli 7488  df-mi 7489  df-lti 7490  df-plpq 7527  df-mpq 7528  df-enq 7530  df-nqqs 7531  df-plqqs 7532  df-mqqs 7533  df-1nqqs 7534  df-rq 7535  df-ltnqqs 7536  df-inp 7649  df-i1p 7650  df-enr 7909  df-nr 7910  df-0r 7914  df-r 8005

This theorem is referenced by:  ltresr2  8023  axrnegex  8062  axpre-suploclemres  8084