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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 11156 | . . 3 ⊢ ℝ = (R × {0R}) | |
2 | 1 | eleq2i 2821 | . 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R})) |
3 | xp1st 8031 | . . . 4 ⊢ (𝐴 ∈ (R × {0R}) → (1st ‘𝐴) ∈ R) | |
4 | 1st2nd2 8038 | . . . . 5 ⊢ (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩) | |
5 | xp2nd 8032 | . . . . . . 7 ⊢ (𝐴 ∈ (R × {0R}) → (2nd ‘𝐴) ∈ {0R}) | |
6 | elsni 4649 | . . . . . . 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 2768 | . . . 4 ⊢ (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st ‘𝐴), 0R⟩) |
10 | 3, 9 | jca 510 | . . 3 ⊢ (𝐴 ∈ (R × {0R}) → ((1st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1st ‘𝐴), 0R⟩)) |
11 | eleq1 2817 | . . . . 5 ⊢ (𝐴 = ⟨(1st ‘𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ ⟨(1st ‘𝐴), 0R⟩ ∈ (R × {0R}))) | |
12 | 0r 11111 | . . . . . . . 8 ⊢ 0R ∈ R | |
13 | 12 | elexi 3493 | . . . . . . 7 ⊢ 0R ∈ V |
14 | 13 | snid 4669 | . . . . . 6 ⊢ 0R ∈ {0R} |
15 | opelxp 5718 | . . . . . 6 ⊢ (⟨(1st ‘𝐴), 0R⟩ ∈ (R × {0R}) ↔ ((1st ‘𝐴) ∈ R ∧ 0R ∈ {0R})) | |
16 | 14, 15 | mpbiran2 708 | . . . . 5 ⊢ (⟨(1st ‘𝐴), 0R⟩ ∈ (R × {0R}) ↔ (1st ‘𝐴) ∈ R) |
17 | 11, 16 | bitrdi 286 | . . . 4 ⊢ (𝐴 = ⟨(1st ‘𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ (1st ‘𝐴) ∈ R)) |
18 | 17 | biimparc 478 | . . 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 274 | 1 ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1st ‘𝐴), 0R⟩)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {csn 4632 ⟨cop 4638 × cxp 5680 ‘cfv 6553 1st c1st 7997 2nd c2nd 7998 Rcnr 10896 0Rc0r 10897 ℝcr 11145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-oadd 8497 df-omul 8498 df-er 8731 df-ec 8733 df-qs 8737 df-ni 10903 df-pli 10904 df-mi 10905 df-lti 10906 df-plpq 10939 df-mpq 10940 df-ltpq 10941 df-enq 10942 df-nq 10943 df-erq 10944 df-plq 10945 df-mq 10946 df-1nq 10947 df-rq 10948 df-ltnq 10949 df-np 11012 df-1p 11013 df-enr 11086 df-nr 11087 df-0r 11091 df-r 11156 |
This theorem is referenced by: ltresr2 11172 axrnegex 11193 axpre-sup 11200 |
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