elreal2 - Metamath Proof Explorer

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Theorem elreal2 10819

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

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

**Proof of Theorem elreal2**
Step	Hyp	Ref	Expression
1		df-r 10812	. . 3 ⊢ ℝ = (R × {0_R})
2	1	eleq2i 2830	. 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0_R}))
3		xp1st 7836	. . . 4 ⊢ (𝐴 ∈ (R × {0_R}) → (1^st ‘𝐴) ∈ R)
4		1st2nd2 7843	. . . . 5 ⊢ (𝐴 ∈ (R × {0_R}) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
5		xp2nd 7837	. . . . . . 7 ⊢ (𝐴 ∈ (R × {0_R}) → (2^nd ‘𝐴) ∈ {0_R})
6		elsni 4575	. . . . . . 7 ⊢ ((2^nd ‘𝐴) ∈ {0_R} → (2^nd ‘𝐴) = 0_R)
7	5, 6	syl 17	. . . . . 6 ⊢ (𝐴 ∈ (R × {0_R}) → (2^nd ‘𝐴) = 0_R)
8	7	opeq2d 4808	. . . . 5 ⊢ (𝐴 ∈ (R × {0_R}) → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐴), 0_R⟩)
9	4, 8	eqtrd 2778	. . . 4 ⊢ (𝐴 ∈ (R × {0_R}) → 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩)
10	3, 9	jca 511	. . 3 ⊢ (𝐴 ∈ (R × {0_R}) → ((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩))
11		eleq1 2826	. . . . 5 ⊢ (𝐴 = ⟨(1^st ‘𝐴), 0_R⟩ → (𝐴 ∈ (R × {0_R}) ↔ ⟨(1^st ‘𝐴), 0_R⟩ ∈ (R × {0_R})))
12		0r 10767	. . . . . . . 8 ⊢ 0_R ∈ R
13	12	elexi 3441	. . . . . . 7 ⊢ 0_R ∈ V
14	13	snid 4594	. . . . . 6 ⊢ 0_R ∈ {0_R}
15		opelxp 5616	. . . . . 6 ⊢ (⟨(1^st ‘𝐴), 0_R⟩ ∈ (R × {0_R}) ↔ ((1^st ‘𝐴) ∈ R ∧ 0_R ∈ {0_R}))
16	14, 15	mpbiran2 706	. . . . 5 ⊢ (⟨(1^st ‘𝐴), 0_R⟩ ∈ (R × {0_R}) ↔ (1^st ‘𝐴) ∈ R)
17	11, 16	bitrdi 286	. . . 4 ⊢ (𝐴 = ⟨(1^st ‘𝐴), 0_R⟩ → (𝐴 ∈ (R × {0_R}) ↔ (1^st ‘𝐴) ∈ R))
18	17	biimparc 479	. . 3 ⊢ (((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩) → 𝐴 ∈ (R × {0_R}))
19	10, 18	impbii 208	. 2 ⊢ (𝐴 ∈ (R × {0_R}) ↔ ((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩))
20	2, 19	bitri 274	1 ⊢ (𝐴 ∈ ℝ ↔ ((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 {csn 4558 ⟨cop 4564 × cxp 5578 ‘cfv 6418 1^st c1st 7802 2^nd c2nd 7803 Rcnr 10552 0_Rc0r 10553 ℝcr 10801

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-oadd 8271 df-omul 8272 df-er 8456 df-ec 8458 df-qs 8462 df-ni 10559 df-pli 10560 df-mi 10561 df-lti 10562 df-plpq 10595 df-mpq 10596 df-ltpq 10597 df-enq 10598 df-nq 10599 df-erq 10600 df-plq 10601 df-mq 10602 df-1nq 10603 df-rq 10604 df-ltnq 10605 df-np 10668 df-1p 10669 df-enr 10742 df-nr 10743 df-0r 10747 df-r 10812

This theorem is referenced by: ltresr2 10828 axrnegex 10849 axpre-sup 10856

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