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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 11165 | . . 3 ⊢ ℝ = (R × {0R}) | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R})) |
| 3 | xp1st 8046 | . . . 4 ⊢ (𝐴 ∈ (R × {0R}) → (1st ‘𝐴) ∈ R) | |
| 4 | 1st2nd2 8053 | . . . . 5 ⊢ (𝐴 ∈ (R × {0R}) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
| 5 | xp2nd 8047 | . . . . . . 7 ⊢ (𝐴 ∈ (R × {0R}) → (2nd ‘𝐴) ∈ {0R}) | |
| 6 | elsni 4643 | . . . . . . 7 ⊢ ((2nd ‘𝐴) ∈ {0R} → (2nd ‘𝐴) = 0R) | |
| 7 | 5, 6 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ (R × {0R}) → (2nd ‘𝐴) = 0R) |
| 8 | 7 | opeq2d 4880 | . . . . 5 ⊢ (𝐴 ∈ (R × {0R}) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈(1st ‘𝐴), 0R〉) |
| 9 | 4, 8 | eqtrd 2777 | . . . 4 ⊢ (𝐴 ∈ (R × {0R}) → 𝐴 = 〈(1st ‘𝐴), 0R〉) |
| 10 | 3, 9 | jca 511 | . . 3 ⊢ (𝐴 ∈ (R × {0R}) → ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) |
| 11 | eleq1 2829 | . . . . 5 ⊢ (𝐴 = 〈(1st ‘𝐴), 0R〉 → (𝐴 ∈ (R × {0R}) ↔ 〈(1st ‘𝐴), 0R〉 ∈ (R × {0R}))) | |
| 12 | 0r 11120 | . . . . . . . 8 ⊢ 0R ∈ R | |
| 13 | 12 | elexi 3503 | . . . . . . 7 ⊢ 0R ∈ V |
| 14 | 13 | snid 4662 | . . . . . 6 ⊢ 0R ∈ {0R} |
| 15 | opelxp 5721 | . . . . . 6 ⊢ (〈(1st ‘𝐴), 0R〉 ∈ (R × {0R}) ↔ ((1st ‘𝐴) ∈ R ∧ 0R ∈ {0R})) | |
| 16 | 14, 15 | mpbiran2 710 | . . . . 5 ⊢ (〈(1st ‘𝐴), 0R〉 ∈ (R × {0R}) ↔ (1st ‘𝐴) ∈ R) |
| 17 | 11, 16 | bitrdi 287 | . . . 4 ⊢ (𝐴 = 〈(1st ‘𝐴), 0R〉 → (𝐴 ∈ (R × {0R}) ↔ (1st ‘𝐴) ∈ R)) |
| 18 | 17 | biimparc 479 | . . 3 ⊢ (((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉) → 𝐴 ∈ (R × {0R})) |
| 19 | 10, 18 | impbii 209 | . 2 ⊢ (𝐴 ∈ (R × {0R}) ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) |
| 20 | 2, 19 | bitri 275 | 1 ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4626 〈cop 4632 × cxp 5683 ‘cfv 6561 1st c1st 8012 2nd c2nd 8013 Rcnr 10905 0Rc0r 10906 ℝcr 11154 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-omul 8511 df-er 8745 df-ec 8747 df-qs 8751 df-ni 10912 df-pli 10913 df-mi 10914 df-lti 10915 df-plpq 10948 df-mpq 10949 df-ltpq 10950 df-enq 10951 df-nq 10952 df-erq 10953 df-plq 10954 df-mq 10955 df-1nq 10956 df-rq 10957 df-ltnq 10958 df-np 11021 df-1p 11022 df-enr 11095 df-nr 11096 df-0r 11100 df-r 11165 |
| This theorem is referenced by: ltresr2 11181 axrnegex 11202 axpre-sup 11209 |
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