Theorem elreal2 7973

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 7965	. . 3 ⊢ ℝ = (R × {0R})
2	1	eleq2i 2273	. 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R}))
3		xp1st 6269	. . . 4 ⊢ (𝐴 ∈ (R × {0R}) → (1st ‘𝐴) ∈ R)
4		1st2nd2 6279	. . . . 5 ⊢ (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
5		xp2nd 6270	. . . . . . 7 ⊢ (𝐴 ∈ (R × {0R}) → (2nd ‘𝐴) ∈ {0R})
6		elsni 3656	. . . . . . 7 ⊢ ((2nd ‘𝐴) ∈ {0R} → (2nd ‘𝐴) = 0R)
7	5, 6	syl 14	. . . . . 6 ⊢ (𝐴 ∈ (R × {0R}) → (2nd ‘𝐴) = 0R)
8	7	opeq2d 3835	. . . . 5 ⊢ (𝐴 ∈ (R × {0R}) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨(1st ‘𝐴), 0R⟩)
9	4, 8	eqtrd 2239	. . . 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 2269	. . . . 5 ⊢ (𝐴 = ⟨(1st ‘𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ ⟨(1st ‘𝐴), 0R⟩ ∈ (R × {0R})))
12		0r 7893	. . . . . . . 8 ⊢ 0R ∈ R
13	12	elexi 2786	. . . . . . 7 ⊢ 0R ∈ V
14	13	snid 3669	. . . . . 6 ⊢ 0R ∈ {0R}
15		opelxp 4718	. . . . . 6 ⊢ (⟨(1st ‘𝐴), 0R⟩ ∈ (R × {0R}) ↔ ((1st ‘𝐴) ∈ R ∧ 0R ∈ {0R}))
16	14, 15	mpbiran2 944	. . . . 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 1373   ∈ wcel 2177  {csn 3638  ⟨cop 3641   × cxp 4686  ‘cfv 5285  1^st c1st 6242  2^nd c2nd 6243  Rcnr 7440  0_Rc0r 7441  ℝcr 7954

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-eprel 4349  df-id 4353  df-po 4356  df-iso 4357  df-iord 4426  df-on 4428  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-irdg 6474  df-1o 6520  df-oadd 6524  df-omul 6525  df-er 6638  df-ec 6640  df-qs 6644  df-ni 7447  df-pli 7448  df-mi 7449  df-lti 7450  df-plpq 7487  df-mpq 7488  df-enq 7490  df-nqqs 7491  df-plqqs 7492  df-mqqs 7493  df-1nqqs 7494  df-rq 7495  df-ltnqqs 7496  df-inp 7609  df-i1p 7610  df-enr 7869  df-nr 7870  df-0r 7874  df-r 7965

This theorem is referenced by:  ltresr2  7983  axrnegex  8022  axpre-suploclemres  8044