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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 10598 be used later. Instead, in most cases use readdcl 10620. (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 10553 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 10553 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | oveq1 7163 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 + 〈𝑦, 0R〉) = (𝐴 + 〈𝑦, 0R〉)) | |
4 | 3 | eleq1d 2897 | . 2 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((〈𝑥, 0R〉 + 〈𝑦, 0R〉) ∈ ℝ ↔ (𝐴 + 〈𝑦, 0R〉) ∈ ℝ)) |
5 | oveq2 7164 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 + 〈𝑦, 0R〉) = (𝐴 + 𝐵)) | |
6 | 5 | eleq1d 2897 | . 2 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((𝐴 + 〈𝑦, 0R〉) ∈ ℝ ↔ (𝐴 + 𝐵) ∈ ℝ)) |
7 | addresr 10560 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 + 〈𝑦, 0R〉) = 〈(𝑥 +R 𝑦), 0R〉) | |
8 | addclsr 10505 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 +R 𝑦) ∈ R) | |
9 | opelreal 10552 | . . . 4 ⊢ (〈(𝑥 +R 𝑦), 0R〉 ∈ ℝ ↔ (𝑥 +R 𝑦) ∈ R) | |
10 | 8, 9 | sylibr 236 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → 〈(𝑥 +R 𝑦), 0R〉 ∈ ℝ) |
11 | 7, 10 | eqeltrd 2913 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 + 〈𝑦, 0R〉) ∈ ℝ) |
12 | 1, 2, 4, 6, 11 | 2gencl 3535 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4573 (class class class)co 7156 Rcnr 10287 0Rc0r 10288 +R cplr 10291 ℝcr 10536 + caddc 10540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-omul 8107 df-er 8289 df-ec 8291 df-qs 8295 df-ni 10294 df-pli 10295 df-mi 10296 df-lti 10297 df-plpq 10330 df-mpq 10331 df-ltpq 10332 df-enq 10333 df-nq 10334 df-erq 10335 df-plq 10336 df-mq 10337 df-1nq 10338 df-rq 10339 df-ltnq 10340 df-np 10403 df-1p 10404 df-plp 10405 df-ltp 10407 df-enr 10477 df-nr 10478 df-plr 10479 df-0r 10482 df-c 10543 df-r 10547 df-add 10548 |
This theorem is referenced by: (None) |
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