![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > elreal2 | GIF version |
Description: Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) |
Ref | Expression |
---|---|
elreal2 | ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-r 6899 | . . 3 ⊢ ℝ = (R × {0R}) | |
2 | 1 | eleq2i 2104 | . 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R})) |
3 | xp1st 5792 | . . . 4 ⊢ (𝐴 ∈ (R × {0R}) → (1st ‘𝐴) ∈ R) | |
4 | 1st2nd2 5801 | . . . . 5 ⊢ (𝐴 ∈ (R × {0R}) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
5 | xp2nd 5793 | . . . . . . 7 ⊢ (𝐴 ∈ (R × {0R}) → (2nd ‘𝐴) ∈ {0R}) | |
6 | elsni 3393 | . . . . . . 7 ⊢ ((2nd ‘𝐴) ∈ {0R} → (2nd ‘𝐴) = 0R) | |
7 | 5, 6 | syl 14 | . . . . . 6 ⊢ (𝐴 ∈ (R × {0R}) → (2nd ‘𝐴) = 0R) |
8 | 7 | opeq2d 3556 | . . . . 5 ⊢ (𝐴 ∈ (R × {0R}) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈(1st ‘𝐴), 0R〉) |
9 | 4, 8 | eqtrd 2072 | . . . 4 ⊢ (𝐴 ∈ (R × {0R}) → 𝐴 = 〈(1st ‘𝐴), 0R〉) |
10 | 3, 9 | jca 290 | . . 3 ⊢ (𝐴 ∈ (R × {0R}) → ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) |
11 | eleq1 2100 | . . . . 5 ⊢ (𝐴 = 〈(1st ‘𝐴), 0R〉 → (𝐴 ∈ (R × {0R}) ↔ 〈(1st ‘𝐴), 0R〉 ∈ (R × {0R}))) | |
12 | 0r 6835 | . . . . . . . 8 ⊢ 0R ∈ R | |
13 | 12 | elexi 2567 | . . . . . . 7 ⊢ 0R ∈ V |
14 | 13 | snid 3402 | . . . . . 6 ⊢ 0R ∈ {0R} |
15 | opelxp 4374 | . . . . . 6 ⊢ (〈(1st ‘𝐴), 0R〉 ∈ (R × {0R}) ↔ ((1st ‘𝐴) ∈ R ∧ 0R ∈ {0R})) | |
16 | 14, 15 | mpbiran2 848 | . . . . 5 ⊢ (〈(1st ‘𝐴), 0R〉 ∈ (R × {0R}) ↔ (1st ‘𝐴) ∈ R) |
17 | 11, 16 | syl6bb 185 | . . . 4 ⊢ (𝐴 = 〈(1st ‘𝐴), 0R〉 → (𝐴 ∈ (R × {0R}) ↔ (1st ‘𝐴) ∈ R)) |
18 | 17 | biimparc 283 | . . 3 ⊢ (((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉) → 𝐴 ∈ (R × {0R})) |
19 | 10, 18 | impbii 117 | . 2 ⊢ (𝐴 ∈ (R × {0R}) ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) |
20 | 2, 19 | bitri 173 | 1 ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 {csn 3375 〈cop 3378 × cxp 4343 ‘cfv 4902 1st c1st 5765 2nd c2nd 5766 Rcnr 6395 0Rc0r 6396 ℝcr 6888 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-inp 6564 df-i1p 6565 df-enr 6811 df-nr 6812 df-0r 6816 df-r 6899 |
This theorem is referenced by: ltresr2 6916 axrnegex 6953 |
Copyright terms: Public domain | W3C validator |