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Theorem ax0id 7650
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 7692.

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.)

Assertion
Ref Expression
ax0id (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)

Proof of Theorem ax0id
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-c 7590 . 2 ℂ = (R × R)
2 oveq1 5747 . . 3 (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ + 0) = (𝐴 + 0))
3 id 19 . . 3 (⟨𝑥, 𝑦⟩ = 𝐴 → ⟨𝑥, 𝑦⟩ = 𝐴)
42, 3eqeq12d 2130 . 2 (⟨𝑥, 𝑦⟩ = 𝐴 → ((⟨𝑥, 𝑦⟩ + 0) = ⟨𝑥, 𝑦⟩ ↔ (𝐴 + 0) = 𝐴))
5 0r 7522 . . . 4 0RR
6 addcnsr 7606 . . . 4 (((𝑥R𝑦R) ∧ (0RR ∧ 0RR)) → (⟨𝑥, 𝑦⟩ + ⟨0R, 0R⟩) = ⟨(𝑥 +R 0R), (𝑦 +R 0R)⟩)
75, 5, 6mpanr12 433 . . 3 ((𝑥R𝑦R) → (⟨𝑥, 𝑦⟩ + ⟨0R, 0R⟩) = ⟨(𝑥 +R 0R), (𝑦 +R 0R)⟩)
8 df-0 7591 . . . . . 6 0 = ⟨0R, 0R
98eqcomi 2119 . . . . 5 ⟨0R, 0R⟩ = 0
109a1i 9 . . . 4 ((𝑥R𝑦R) → ⟨0R, 0R⟩ = 0)
1110oveq2d 5756 . . 3 ((𝑥R𝑦R) → (⟨𝑥, 𝑦⟩ + ⟨0R, 0R⟩) = (⟨𝑥, 𝑦⟩ + 0))
12 0idsr 7539 . . . . 5 (𝑥R → (𝑥 +R 0R) = 𝑥)
1312adantr 272 . . . 4 ((𝑥R𝑦R) → (𝑥 +R 0R) = 𝑥)
14 0idsr 7539 . . . . 5 (𝑦R → (𝑦 +R 0R) = 𝑦)
1514adantl 273 . . . 4 ((𝑥R𝑦R) → (𝑦 +R 0R) = 𝑦)
1613, 15opeq12d 3681 . . 3 ((𝑥R𝑦R) → ⟨(𝑥 +R 0R), (𝑦 +R 0R)⟩ = ⟨𝑥, 𝑦⟩)
177, 11, 163eqtr3d 2156 . 2 ((𝑥R𝑦R) → (⟨𝑥, 𝑦⟩ + 0) = ⟨𝑥, 𝑦⟩)
181, 4, 17optocl 4583 1 (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wcel 1463  cop 3498  (class class class)co 5740  Rcnr 7069  0Rc0r 7070   +R cplr 7073  cc 7582  0cc0 7584   + caddc 7587
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-i1p 7239  df-iplp 7240  df-enr 7498  df-nr 7499  df-plr 7500  df-0r 7503  df-c 7590  df-0 7591  df-add 7595
This theorem is referenced by: (None)
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