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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 7084.
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 6987 | . 2 ⊢ ℂ = (R × R) | |
2 | oveq1 5539 | . . 3 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (〈𝑥, 𝑦〉 + 0) = (𝐴 + 0)) | |
3 | id 19 | . . 3 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → 〈𝑥, 𝑦〉 = 𝐴) | |
4 | 2, 3 | eqeq12d 2095 | . 2 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → ((〈𝑥, 𝑦〉 + 0) = 〈𝑥, 𝑦〉 ↔ (𝐴 + 0) = 𝐴)) |
5 | 0r 6927 | . . . 4 ⊢ 0R ∈ R | |
6 | addcnsr 7002 | . . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (0R ∈ R ∧ 0R ∈ R)) → (〈𝑥, 𝑦〉 + 〈0R, 0R〉) = 〈(𝑥 +R 0R), (𝑦 +R 0R)〉) | |
7 | 5, 5, 6 | mpanr12 429 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 𝑦〉 + 〈0R, 0R〉) = 〈(𝑥 +R 0R), (𝑦 +R 0R)〉) |
8 | df-0 6988 | . . . . . 6 ⊢ 0 = 〈0R, 0R〉 | |
9 | 8 | eqcomi 2085 | . . . . 5 ⊢ 〈0R, 0R〉 = 0 |
10 | 9 | a1i 9 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → 〈0R, 0R〉 = 0) |
11 | 10 | oveq2d 5548 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 𝑦〉 + 〈0R, 0R〉) = (〈𝑥, 𝑦〉 + 0)) |
12 | 0idsr 6944 | . . . . 5 ⊢ (𝑥 ∈ R → (𝑥 +R 0R) = 𝑥) | |
13 | 12 | adantr 270 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 +R 0R) = 𝑥) |
14 | 0idsr 6944 | . . . . 5 ⊢ (𝑦 ∈ R → (𝑦 +R 0R) = 𝑦) | |
15 | 14 | adantl 271 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑦 +R 0R) = 𝑦) |
16 | 13, 15 | opeq12d 3578 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → 〈(𝑥 +R 0R), (𝑦 +R 0R)〉 = 〈𝑥, 𝑦〉) |
17 | 7, 11, 16 | 3eqtr3d 2121 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 𝑦〉 + 0) = 〈𝑥, 𝑦〉) |
18 | 1, 4, 17 | optocl 4434 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 〈cop 3401 (class class class)co 5532 Rcnr 6487 0Rc0r 6488 +R cplr 6491 ℂcc 6979 0cc0 6981 + caddc 6984 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-i1p 6657 df-iplp 6658 df-enr 6903 df-nr 6904 df-plr 6905 df-0r 6908 df-c 6987 df-0 6988 df-add 6992 |
This theorem is referenced by: (None) |
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