elreal2 - Metamath Proof Explorer

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Theorem elreal2 11112

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 11105	. . 3 ⊢ ℝ = (R × {0_R})
2	1	eleq2i 2855	. 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0_R}))
3		xp1st 8014	. . . 4 ⊢ (𝐴 ∈ (R × {0_R}) → (1^st ‘𝐴) ∈ R)
4		1st2nd2 8021	. . . . 5 ⊢ (𝐴 ∈ (R × {0_R}) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
5		xp2nd 8015	. . . . . . 7 ⊢ (𝐴 ∈ (R × {0_R}) → (2^nd ‘𝐴) ∈ {0_R})
6		elsni 4606	. . . . . . 7 ⊢ ((2^nd ‘𝐴) ∈ {0_R} → (2^nd ‘𝐴) = 0_R)
7	5, 6	syl 18	. . . . . 6 ⊢ (𝐴 ∈ (R × {0_R}) → (2^nd ‘𝐴) = 0_R)
8	7	opeq2d 4845	. . . . 5 ⊢ (𝐴 ∈ (R × {0_R}) → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐴), 0_R⟩)
9	4, 8	eqtrd 2798	. . . 4 ⊢ (𝐴 ∈ (R × {0_R}) → 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩)
10	3, 9	jca 520	. . 3 ⊢ (𝐴 ∈ (R × {0_R}) → ((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩))
11		eleq1 2851	. . . . 5 ⊢ (𝐴 = ⟨(1^st ‘𝐴), 0_R⟩ → (𝐴 ∈ (R × {0_R}) ↔ ⟨(1^st ‘𝐴), 0_R⟩ ∈ (R × {0_R})))
12		0r 11060	. . . . . . . 8 ⊢ 0_R ∈ R
13	12	elexi 3477	. . . . . . 7 ⊢ 0_R ∈ V
14	13	snid 4628	. . . . . 6 ⊢ 0_R ∈ {0_R}
15		opelxp 5697	. . . . . 6 ⊢ (⟨(1^st ‘𝐴), 0_R⟩ ∈ (R × {0_R}) ↔ ((1^st ‘𝐴) ∈ R ∧ 0_R ∈ {0_R}))
16	14, 15	mpbiran2 722	. . . . 5 ⊢ (⟨(1^st ‘𝐴), 0_R⟩ ∈ (R × {0_R}) ↔ (1^st ‘𝐴) ∈ R)
17	11, 16	bitrdi 290	. . . 4 ⊢ (𝐴 = ⟨(1^st ‘𝐴), 0_R⟩ → (𝐴 ∈ (R × {0_R}) ↔ (1^st ‘𝐴) ∈ R))
18	17	biimparc 484	. . 3 ⊢ (((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩) → 𝐴 ∈ (R × {0_R}))
19	10, 18	impbii 212	. 2 ⊢ (𝐴 ∈ (R × {0_R}) ↔ ((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩))
20	2, 19	bitri 278	1 ⊢ (𝐴 ∈ ℝ ↔ ((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 {csn 4589 ⟨cop 4595 × cxp 5659 ‘cfv 6536 1^st c1st 7980 2^nd c2nd 7981 Rcnr 10845 0_Rc0r 10846 ℝcr 11094

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-omul 8454 df-er 8690 df-ec 8692 df-qs 8696 df-ni 10852 df-pli 10853 df-mi 10854 df-lti 10855 df-plpq 10888 df-mpq 10889 df-ltpq 10890 df-enq 10891 df-nq 10892 df-erq 10893 df-plq 10894 df-mq 10895 df-1nq 10896 df-rq 10897 df-ltnq 10898 df-np 10961 df-1p 10962 df-enr 11035 df-nr 11036 df-0r 11040 df-r 11105

This theorem is referenced by: ltresr2 11121 axrnegex 11142 axpre-sup 11149

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