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Mirrors > Home > ILE Home > Th. List > ax0id | GIF version |
Description: 0
is an identity element for real addition. Axiom for real and
complex numbers, derived from set theory. This construction-dependent
theorem should not be referenced directly; instead, use ax-0id 7949.
In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on excluded middle and it is not known whether it is possible to prove this from the other axioms without excluded middle. (Contributed by Jim Kingdon, 16-Jan-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax0id | ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-c 7847 | . 2 ⊢ ℂ = (R × R) | |
2 | oveq1 5903 | . . 3 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (〈𝑥, 𝑦〉 + 0) = (𝐴 + 0)) | |
3 | id 19 | . . 3 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → 〈𝑥, 𝑦〉 = 𝐴) | |
4 | 2, 3 | eqeq12d 2204 | . 2 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → ((〈𝑥, 𝑦〉 + 0) = 〈𝑥, 𝑦〉 ↔ (𝐴 + 0) = 𝐴)) |
5 | 0r 7779 | . . . 4 ⊢ 0R ∈ R | |
6 | addcnsr 7863 | . . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (0R ∈ R ∧ 0R ∈ R)) → (〈𝑥, 𝑦〉 + 〈0R, 0R〉) = 〈(𝑥 +R 0R), (𝑦 +R 0R)〉) | |
7 | 5, 5, 6 | mpanr12 439 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 𝑦〉 + 〈0R, 0R〉) = 〈(𝑥 +R 0R), (𝑦 +R 0R)〉) |
8 | df-0 7848 | . . . . . 6 ⊢ 0 = 〈0R, 0R〉 | |
9 | 8 | eqcomi 2193 | . . . . 5 ⊢ 〈0R, 0R〉 = 0 |
10 | 9 | a1i 9 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → 〈0R, 0R〉 = 0) |
11 | 10 | oveq2d 5912 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 𝑦〉 + 〈0R, 0R〉) = (〈𝑥, 𝑦〉 + 0)) |
12 | 0idsr 7796 | . . . . 5 ⊢ (𝑥 ∈ R → (𝑥 +R 0R) = 𝑥) | |
13 | 12 | adantr 276 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 +R 0R) = 𝑥) |
14 | 0idsr 7796 | . . . . 5 ⊢ (𝑦 ∈ R → (𝑦 +R 0R) = 𝑦) | |
15 | 14 | adantl 277 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑦 +R 0R) = 𝑦) |
16 | 13, 15 | opeq12d 3801 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → 〈(𝑥 +R 0R), (𝑦 +R 0R)〉 = 〈𝑥, 𝑦〉) |
17 | 7, 11, 16 | 3eqtr3d 2230 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 𝑦〉 + 0) = 〈𝑥, 𝑦〉) |
18 | 1, 4, 17 | optocl 4720 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 〈cop 3610 (class class class)co 5896 Rcnr 7326 0Rc0r 7327 +R cplr 7330 ℂcc 7839 0cc0 7841 + caddc 7844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-eprel 4307 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-recs 6330 df-irdg 6395 df-1o 6441 df-2o 6442 df-oadd 6445 df-omul 6446 df-er 6559 df-ec 6561 df-qs 6565 df-ni 7333 df-pli 7334 df-mi 7335 df-lti 7336 df-plpq 7373 df-mpq 7374 df-enq 7376 df-nqqs 7377 df-plqqs 7378 df-mqqs 7379 df-1nqqs 7380 df-rq 7381 df-ltnqqs 7382 df-enq0 7453 df-nq0 7454 df-0nq0 7455 df-plq0 7456 df-mq0 7457 df-inp 7495 df-i1p 7496 df-iplp 7497 df-enr 7755 df-nr 7756 df-plr 7757 df-0r 7760 df-c 7847 df-0 7848 df-add 7852 |
This theorem is referenced by: (None) |
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