elreal2 - Metamath Proof Explorer

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Theorem elreal2 11123

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 11116	. . 3 ⊢ ℝ = (R × {0_R})
2	1	eleq2i 2825	. 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0_R}))
3		xp1st 8003	. . . 4 ⊢ (𝐴 ∈ (R × {0_R}) → (1^st ‘𝐴) ∈ R)
4		1st2nd2 8010	. . . . 5 ⊢ (𝐴 ∈ (R × {0_R}) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
5		xp2nd 8004	. . . . . . 7 ⊢ (𝐴 ∈ (R × {0_R}) → (2^nd ‘𝐴) ∈ {0_R})
6		elsni 4644	. . . . . . 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 4879	. . . . 5 ⊢ (𝐴 ∈ (R × {0_R}) → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐴), 0_R⟩)
9	4, 8	eqtrd 2772	. . . 4 ⊢ (𝐴 ∈ (R × {0_R}) → 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩)
10	3, 9	jca 512	. . 3 ⊢ (𝐴 ∈ (R × {0_R}) → ((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩))
11		eleq1 2821	. . . . 5 ⊢ (𝐴 = ⟨(1^st ‘𝐴), 0_R⟩ → (𝐴 ∈ (R × {0_R}) ↔ ⟨(1^st ‘𝐴), 0_R⟩ ∈ (R × {0_R})))
12		0r 11071	. . . . . . . 8 ⊢ 0_R ∈ R
13	12	elexi 3493	. . . . . . 7 ⊢ 0_R ∈ V
14	13	snid 4663	. . . . . 6 ⊢ 0_R ∈ {0_R}
15		opelxp 5711	. . . . . 6 ⊢ (⟨(1^st ‘𝐴), 0_R⟩ ∈ (R × {0_R}) ↔ ((1^st ‘𝐴) ∈ R ∧ 0_R ∈ {0_R}))
16	14, 15	mpbiran2 708	. . . . 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 480	. . 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 396 = wceq 1541 ∈ wcel 2106 {csn 4627 ⟨cop 4633 × cxp 5673 ‘cfv 6540 1^st c1st 7969 2^nd c2nd 7970 Rcnr 10856 0_Rc0r 10857 ℝcr 11105

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-omul 8467 df-er 8699 df-ec 8701 df-qs 8705 df-ni 10863 df-pli 10864 df-mi 10865 df-lti 10866 df-plpq 10899 df-mpq 10900 df-ltpq 10901 df-enq 10902 df-nq 10903 df-erq 10904 df-plq 10905 df-mq 10906 df-1nq 10907 df-rq 10908 df-ltnq 10909 df-np 10972 df-1p 10973 df-enr 11046 df-nr 11047 df-0r 11051 df-r 11116

This theorem is referenced by: ltresr2 11132 axrnegex 11153 axpre-sup 11160

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