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Mirrors > Home > MPE Home > Th. List > axaddrcl | Structured version Visualization version GIF version |
Description: Closure law for addition in the real subfield of complex numbers. Axiom 5 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addrcl 11166 be used later. Instead, in most cases use readdcl 11188. (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 11121 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 11121 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | oveq1 7408 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 + 〈𝑦, 0R〉) = (𝐴 + 〈𝑦, 0R〉)) | |
4 | 3 | eleq1d 2810 | . 2 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((〈𝑥, 0R〉 + 〈𝑦, 0R〉) ∈ ℝ ↔ (𝐴 + 〈𝑦, 0R〉) ∈ ℝ)) |
5 | oveq2 7409 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 + 〈𝑦, 0R〉) = (𝐴 + 𝐵)) | |
6 | 5 | eleq1d 2810 | . 2 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((𝐴 + 〈𝑦, 0R〉) ∈ ℝ ↔ (𝐴 + 𝐵) ∈ ℝ)) |
7 | addresr 11128 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 + 〈𝑦, 0R〉) = 〈(𝑥 +R 𝑦), 0R〉) | |
8 | addclsr 11073 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 +R 𝑦) ∈ R) | |
9 | opelreal 11120 | . . . 4 ⊢ (〈(𝑥 +R 𝑦), 0R〉 ∈ ℝ ↔ (𝑥 +R 𝑦) ∈ R) | |
10 | 8, 9 | sylibr 233 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → 〈(𝑥 +R 𝑦), 0R〉 ∈ ℝ) |
11 | 7, 10 | eqeltrd 2825 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 + 〈𝑦, 0R〉) ∈ ℝ) |
12 | 1, 2, 4, 6, 11 | 2gencl 3509 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 〈cop 4626 (class class class)co 7401 Rcnr 10855 0Rc0r 10856 +R cplr 10859 ℝcr 11104 + caddc 11108 |
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 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-oadd 8465 df-omul 8466 df-er 8698 df-ec 8700 df-qs 8704 df-ni 10862 df-pli 10863 df-mi 10864 df-lti 10865 df-plpq 10898 df-mpq 10899 df-ltpq 10900 df-enq 10901 df-nq 10902 df-erq 10903 df-plq 10904 df-mq 10905 df-1nq 10906 df-rq 10907 df-ltnq 10908 df-np 10971 df-1p 10972 df-plp 10973 df-ltp 10975 df-enr 11045 df-nr 11046 df-plr 11047 df-0r 11050 df-c 11111 df-r 11115 df-add 11116 |
This theorem is referenced by: (None) |
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