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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 7696.
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 7594 | . 2 ⊢ ℂ = (R × R) | |
2 | oveq1 5749 | . . 3 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (〈𝑥, 𝑦〉 + 0) = (𝐴 + 0)) | |
3 | id 19 | . . 3 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → 〈𝑥, 𝑦〉 = 𝐴) | |
4 | 2, 3 | eqeq12d 2132 | . 2 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → ((〈𝑥, 𝑦〉 + 0) = 〈𝑥, 𝑦〉 ↔ (𝐴 + 0) = 𝐴)) |
5 | 0r 7526 | . . . 4 ⊢ 0R ∈ R | |
6 | addcnsr 7610 | . . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (0R ∈ R ∧ 0R ∈ R)) → (〈𝑥, 𝑦〉 + 〈0R, 0R〉) = 〈(𝑥 +R 0R), (𝑦 +R 0R)〉) | |
7 | 5, 5, 6 | mpanr12 435 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 𝑦〉 + 〈0R, 0R〉) = 〈(𝑥 +R 0R), (𝑦 +R 0R)〉) |
8 | df-0 7595 | . . . . . 6 ⊢ 0 = 〈0R, 0R〉 | |
9 | 8 | eqcomi 2121 | . . . . 5 ⊢ 〈0R, 0R〉 = 0 |
10 | 9 | a1i 9 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → 〈0R, 0R〉 = 0) |
11 | 10 | oveq2d 5758 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 𝑦〉 + 〈0R, 0R〉) = (〈𝑥, 𝑦〉 + 0)) |
12 | 0idsr 7543 | . . . . 5 ⊢ (𝑥 ∈ R → (𝑥 +R 0R) = 𝑥) | |
13 | 12 | adantr 274 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 +R 0R) = 𝑥) |
14 | 0idsr 7543 | . . . . 5 ⊢ (𝑦 ∈ R → (𝑦 +R 0R) = 𝑦) | |
15 | 14 | adantl 275 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑦 +R 0R) = 𝑦) |
16 | 13, 15 | opeq12d 3683 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → 〈(𝑥 +R 0R), (𝑦 +R 0R)〉 = 〈𝑥, 𝑦〉) |
17 | 7, 11, 16 | 3eqtr3d 2158 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 𝑦〉 + 0) = 〈𝑥, 𝑦〉) |
18 | 1, 4, 17 | optocl 4585 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 〈cop 3500 (class class class)co 5742 Rcnr 7073 0Rc0r 7074 +R cplr 7077 ℂcc 7586 0cc0 7588 + caddc 7591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-eprel 4181 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-2o 6282 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-pli 7081 df-mi 7082 df-lti 7083 df-plpq 7120 df-mpq 7121 df-enq 7123 df-nqqs 7124 df-plqqs 7125 df-mqqs 7126 df-1nqqs 7127 df-rq 7128 df-ltnqqs 7129 df-enq0 7200 df-nq0 7201 df-0nq0 7202 df-plq0 7203 df-mq0 7204 df-inp 7242 df-i1p 7243 df-iplp 7244 df-enr 7502 df-nr 7503 df-plr 7504 df-0r 7507 df-c 7594 df-0 7595 df-add 7599 |
This theorem is referenced by: (None) |
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