elreal2 - Metamath Proof Explorer

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Theorem elreal2 11157

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 11150	. . 3 ⊢ ℝ = (R × {0_R})
2	1	eleq2i 2817	. 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0_R}))
3		xp1st 8026	. . . 4 ⊢ (𝐴 ∈ (R × {0_R}) → (1^st ‘𝐴) ∈ R)
4		1st2nd2 8033	. . . . 5 ⊢ (𝐴 ∈ (R × {0_R}) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
5		xp2nd 8027	. . . . . . 7 ⊢ (𝐴 ∈ (R × {0_R}) → (2^nd ‘𝐴) ∈ {0_R})
6		elsni 4647	. . . . . . 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 4882	. . . . 5 ⊢ (𝐴 ∈ (R × {0_R}) → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐴), 0_R⟩)
9	4, 8	eqtrd 2765	. . . 4 ⊢ (𝐴 ∈ (R × {0_R}) → 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩)
10	3, 9	jca 510	. . 3 ⊢ (𝐴 ∈ (R × {0_R}) → ((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩))
11		eleq1 2813	. . . . 5 ⊢ (𝐴 = ⟨(1^st ‘𝐴), 0_R⟩ → (𝐴 ∈ (R × {0_R}) ↔ ⟨(1^st ‘𝐴), 0_R⟩ ∈ (R × {0_R})))
12		0r 11105	. . . . . . . 8 ⊢ 0_R ∈ R
13	12	elexi 3482	. . . . . . 7 ⊢ 0_R ∈ V
14	13	snid 4666	. . . . . 6 ⊢ 0_R ∈ {0_R}
15		opelxp 5714	. . . . . 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 478	. . 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 394 = wceq 1533 ∈ wcel 2098 {csn 4630 ⟨cop 4636 × cxp 5676 ‘cfv 6549 1^st c1st 7992 2^nd c2nd 7993 Rcnr 10890 0_Rc0r 10891 ℝcr 11139

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-omul 8492 df-er 8725 df-ec 8727 df-qs 8731 df-ni 10897 df-pli 10898 df-mi 10899 df-lti 10900 df-plpq 10933 df-mpq 10934 df-ltpq 10935 df-enq 10936 df-nq 10937 df-erq 10938 df-plq 10939 df-mq 10940 df-1nq 10941 df-rq 10942 df-ltnq 10943 df-np 11006 df-1p 11007 df-enr 11080 df-nr 11081 df-0r 11085 df-r 11150

This theorem is referenced by: ltresr2 11166 axrnegex 11187 axpre-sup 11194

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