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Mirrors > Home > ILE Home > Th. List > axaddrcl | GIF version |
Description: Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addrcl 7636 be used later. Instead, in most cases use readdcl 7664. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axaddrcl | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal 7557 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 7557 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | oveq1 5733 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 + 〈𝑦, 0R〉) = (𝐴 + 〈𝑦, 0R〉)) | |
4 | 3 | eleq1d 2181 | . 2 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((〈𝑥, 0R〉 + 〈𝑦, 0R〉) ∈ ℝ ↔ (𝐴 + 〈𝑦, 0R〉) ∈ ℝ)) |
5 | oveq2 5734 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 + 〈𝑦, 0R〉) = (𝐴 + 𝐵)) | |
6 | 5 | eleq1d 2181 | . 2 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((𝐴 + 〈𝑦, 0R〉) ∈ ℝ ↔ (𝐴 + 𝐵) ∈ ℝ)) |
7 | addresr 7566 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 + 〈𝑦, 0R〉) = 〈(𝑥 +R 𝑦), 0R〉) | |
8 | addclsr 7490 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 +R 𝑦) ∈ R) | |
9 | opelreal 7556 | . . . 4 ⊢ (〈(𝑥 +R 𝑦), 0R〉 ∈ ℝ ↔ (𝑥 +R 𝑦) ∈ R) | |
10 | 8, 9 | sylibr 133 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → 〈(𝑥 +R 𝑦), 0R〉 ∈ ℝ) |
11 | 7, 10 | eqeltrd 2189 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 + 〈𝑦, 0R〉) ∈ ℝ) |
12 | 1, 2, 4, 6, 11 | 2gencl 2688 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1312 ∈ wcel 1461 〈cop 3494 (class class class)co 5726 Rcnr 7047 0Rc0r 7048 +R cplr 7051 ℝcr 7540 + caddc 7544 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-iinf 4460 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-tr 3985 df-eprel 4169 df-id 4173 df-po 4176 df-iso 4177 df-iord 4246 df-on 4248 df-suc 4251 df-iom 4463 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-recs 6154 df-irdg 6219 df-1o 6265 df-2o 6266 df-oadd 6269 df-omul 6270 df-er 6381 df-ec 6383 df-qs 6387 df-ni 7054 df-pli 7055 df-mi 7056 df-lti 7057 df-plpq 7094 df-mpq 7095 df-enq 7097 df-nqqs 7098 df-plqqs 7099 df-mqqs 7100 df-1nqqs 7101 df-rq 7102 df-ltnqqs 7103 df-enq0 7174 df-nq0 7175 df-0nq0 7176 df-plq0 7177 df-mq0 7178 df-inp 7216 df-i1p 7217 df-iplp 7218 df-enr 7463 df-nr 7464 df-plr 7465 df-0r 7468 df-c 7547 df-r 7551 df-add 7552 |
This theorem is referenced by: peano5nnnn 7621 |
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