Theorem elreal2 8028

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 8020	. . 3 ⊢ ℝ = (R × {0R})
2	1	eleq2i 2296	. 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R}))
3		xp1st 6317	. . . 4 ⊢ (𝐴 ∈ (R × {0R}) → (1st ‘𝐴) ∈ R)
4		1st2nd2 6327	. . . . 5 ⊢ (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
5		xp2nd 6318	. . . . . . 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 3864	. . . . 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 7948	. . . . . . . 8 ⊢ 0R ∈ R
13	12	elexi 2812	. . . . . . 7 ⊢ 0R ∈ V
14	13	snid 3697	. . . . . 6 ⊢ 0R ∈ {0R}
15		opelxp 4749	. . . . . 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 4717  ‘cfv 5318  1^st c1st 6290  2^nd c2nd 6291  Rcnr 7495  0_Rc0r 7496  ℝcr 8009

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

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 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7502  df-pli 7503  df-mi 7504  df-lti 7505  df-plpq 7542  df-mpq 7543  df-enq 7545  df-nqqs 7546  df-plqqs 7547  df-mqqs 7548  df-1nqqs 7549  df-rq 7550  df-ltnqqs 7551  df-inp 7664  df-i1p 7665  df-enr 7924  df-nr 7925  df-0r 7929  df-r 8020

This theorem is referenced by:  ltresr2  8038  axrnegex  8077  axpre-suploclemres  8099