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Mirrors > Home > MPE Home > Th. List > elreal2 | Structured version Visualization version GIF version |
Description: Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elreal2 | ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-r 10536 | . . 3 ⊢ ℝ = (R × {0R}) | |
2 | 1 | eleq2i 2881 | . 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R})) |
3 | xp1st 7703 | . . . 4 ⊢ (𝐴 ∈ (R × {0R}) → (1st ‘𝐴) ∈ R) | |
4 | 1st2nd2 7710 | . . . . 5 ⊢ (𝐴 ∈ (R × {0R}) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
5 | xp2nd 7704 | . . . . . . 7 ⊢ (𝐴 ∈ (R × {0R}) → (2nd ‘𝐴) ∈ {0R}) | |
6 | elsni 4542 | . . . . . . 7 ⊢ ((2nd ‘𝐴) ∈ {0R} → (2nd ‘𝐴) = 0R) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ (R × {0R}) → (2nd ‘𝐴) = 0R) |
8 | 7 | opeq2d 4772 | . . . . 5 ⊢ (𝐴 ∈ (R × {0R}) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈(1st ‘𝐴), 0R〉) |
9 | 4, 8 | eqtrd 2833 | . . . 4 ⊢ (𝐴 ∈ (R × {0R}) → 𝐴 = 〈(1st ‘𝐴), 0R〉) |
10 | 3, 9 | jca 515 | . . 3 ⊢ (𝐴 ∈ (R × {0R}) → ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) |
11 | eleq1 2877 | . . . . 5 ⊢ (𝐴 = 〈(1st ‘𝐴), 0R〉 → (𝐴 ∈ (R × {0R}) ↔ 〈(1st ‘𝐴), 0R〉 ∈ (R × {0R}))) | |
12 | 0r 10491 | . . . . . . . 8 ⊢ 0R ∈ R | |
13 | 12 | elexi 3460 | . . . . . . 7 ⊢ 0R ∈ V |
14 | 13 | snid 4561 | . . . . . 6 ⊢ 0R ∈ {0R} |
15 | opelxp 5555 | . . . . . 6 ⊢ (〈(1st ‘𝐴), 0R〉 ∈ (R × {0R}) ↔ ((1st ‘𝐴) ∈ R ∧ 0R ∈ {0R})) | |
16 | 14, 15 | mpbiran2 709 | . . . . 5 ⊢ (〈(1st ‘𝐴), 0R〉 ∈ (R × {0R}) ↔ (1st ‘𝐴) ∈ R) |
17 | 11, 16 | syl6bb 290 | . . . 4 ⊢ (𝐴 = 〈(1st ‘𝐴), 0R〉 → (𝐴 ∈ (R × {0R}) ↔ (1st ‘𝐴) ∈ R)) |
18 | 17 | biimparc 483 | . . 3 ⊢ (((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉) → 𝐴 ∈ (R × {0R})) |
19 | 10, 18 | impbii 212 | . 2 ⊢ (𝐴 ∈ (R × {0R}) ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) |
20 | 2, 19 | bitri 278 | 1 ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {csn 4525 〈cop 4531 × cxp 5517 ‘cfv 6324 1st c1st 7669 2nd c2nd 7670 Rcnr 10276 0Rc0r 10277 ℝcr 10525 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-omul 8090 df-er 8272 df-ec 8274 df-qs 8278 df-ni 10283 df-pli 10284 df-mi 10285 df-lti 10286 df-plpq 10319 df-mpq 10320 df-ltpq 10321 df-enq 10322 df-nq 10323 df-erq 10324 df-plq 10325 df-mq 10326 df-1nq 10327 df-rq 10328 df-ltnq 10329 df-np 10392 df-1p 10393 df-enr 10466 df-nr 10467 df-0r 10471 df-r 10536 |
This theorem is referenced by: ltresr2 10552 axrnegex 10573 axpre-sup 10580 |
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