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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 11163 | . . 3 ⊢ ℝ = (R × {0R}) | |
2 | 1 | eleq2i 2831 | . 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R})) |
3 | xp1st 8045 | . . . 4 ⊢ (𝐴 ∈ (R × {0R}) → (1st ‘𝐴) ∈ R) | |
4 | 1st2nd2 8052 | . . . . 5 ⊢ (𝐴 ∈ (R × {0R}) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
5 | xp2nd 8046 | . . . . . . 7 ⊢ (𝐴 ∈ (R × {0R}) → (2nd ‘𝐴) ∈ {0R}) | |
6 | elsni 4648 | . . . . . . 7 ⊢ ((2nd ‘𝐴) ∈ {0R} → (2nd ‘𝐴) = 0R) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ (R × {0R}) → (2nd ‘𝐴) = 0R) |
8 | 7 | opeq2d 4885 | . . . . 5 ⊢ (𝐴 ∈ (R × {0R}) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈(1st ‘𝐴), 0R〉) |
9 | 4, 8 | eqtrd 2775 | . . . 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 2827 | . . . . 5 ⊢ (𝐴 = 〈(1st ‘𝐴), 0R〉 → (𝐴 ∈ (R × {0R}) ↔ 〈(1st ‘𝐴), 0R〉 ∈ (R × {0R}))) | |
12 | 0r 11118 | . . . . . . . 8 ⊢ 0R ∈ R | |
13 | 12 | elexi 3501 | . . . . . . 7 ⊢ 0R ∈ V |
14 | 13 | snid 4667 | . . . . . 6 ⊢ 0R ∈ {0R} |
15 | opelxp 5725 | . . . . . 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 1537 ∈ wcel 2106 {csn 4631 〈cop 4637 × cxp 5687 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 Rcnr 10903 0Rc0r 10904 ℝcr 11152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-omul 8510 df-er 8744 df-ec 8746 df-qs 8750 df-ni 10910 df-pli 10911 df-mi 10912 df-lti 10913 df-plpq 10946 df-mpq 10947 df-ltpq 10948 df-enq 10949 df-nq 10950 df-erq 10951 df-plq 10952 df-mq 10953 df-1nq 10954 df-rq 10955 df-ltnq 10956 df-np 11019 df-1p 11020 df-enr 11093 df-nr 11094 df-0r 11098 df-r 11163 |
This theorem is referenced by: ltresr2 11179 axrnegex 11200 axpre-sup 11207 |
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