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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 11068 | . . 3 ⊢ ℝ = (R × {0R}) | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R})) |
3 | xp1st 7958 | . . . 4 ⊢ (𝐴 ∈ (R × {0R}) → (1st ‘𝐴) ∈ R) | |
4 | 1st2nd2 7965 | . . . . 5 ⊢ (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩) | |
5 | xp2nd 7959 | . . . . . . 7 ⊢ (𝐴 ∈ (R × {0R}) → (2nd ‘𝐴) ∈ {0R}) | |
6 | elsni 4608 | . . . . . . 7 ⊢ ((2nd ‘𝐴) ∈ {0R} → (2nd ‘𝐴) = 0R) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ (R × {0R}) → (2nd ‘𝐴) = 0R) |
8 | 7 | opeq2d 4842 | . . . . 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 513 | . . 3 ⊢ (𝐴 ∈ (R × {0R}) → ((1st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1st ‘𝐴), 0R⟩)) |
11 | eleq1 2826 | . . . . 5 ⊢ (𝐴 = ⟨(1st ‘𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ ⟨(1st ‘𝐴), 0R⟩ ∈ (R × {0R}))) | |
12 | 0r 11023 | . . . . . . . 8 ⊢ 0R ∈ R | |
13 | 12 | elexi 3467 | . . . . . . 7 ⊢ 0R ∈ V |
14 | 13 | snid 4627 | . . . . . 6 ⊢ 0R ∈ {0R} |
15 | opelxp 5674 | . . . . . 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 | bitrdi 287 | . . . 4 ⊢ (𝐴 = ⟨(1st ‘𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ (1st ‘𝐴) ∈ R)) |
18 | 17 | biimparc 481 | . . 3 ⊢ (((1st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1st ‘𝐴), 0R⟩) → 𝐴 ∈ (R × {0R})) |
19 | 10, 18 | impbii 208 | . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4591 ⟨cop 4597 × cxp 5636 ‘cfv 6501 1st c1st 7924 2nd c2nd 7925 Rcnr 10808 0Rc0r 10809 ℝcr 11057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-omul 8422 df-er 8655 df-ec 8657 df-qs 8661 df-ni 10815 df-pli 10816 df-mi 10817 df-lti 10818 df-plpq 10851 df-mpq 10852 df-ltpq 10853 df-enq 10854 df-nq 10855 df-erq 10856 df-plq 10857 df-mq 10858 df-1nq 10859 df-rq 10860 df-ltnq 10861 df-np 10924 df-1p 10925 df-enr 10998 df-nr 10999 df-0r 11003 df-r 11068 |
This theorem is referenced by: ltresr2 11084 axrnegex 11105 axpre-sup 11112 |
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