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Theorem ax0id 6742
 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 6771. 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 6697 . 2 ℂ = (R × R)
2 oveq1 5462 . . 3 (⟨x, y⟩ = A → (⟨x, y⟩ + 0) = (A + 0))
3 id 19 . . 3 (⟨x, y⟩ = A → ⟨x, y⟩ = A)
42, 3eqeq12d 2051 . 2 (⟨x, y⟩ = A → ((⟨x, y⟩ + 0) = ⟨x, y⟩ ↔ (A + 0) = A))
5 0r 6658 . . . 4 0R R
6 addcnsr 6711 . . . 4 (((x R y R) (0R R 0R R)) → (⟨x, y⟩ + ⟨0R, 0R⟩) = ⟨(x +R 0R), (y +R 0R)⟩)
75, 5, 6mpanr12 415 . . 3 ((x R y R) → (⟨x, y⟩ + ⟨0R, 0R⟩) = ⟨(x +R 0R), (y +R 0R)⟩)
8 df-0 6698 . . . . . 6 0 = ⟨0R, 0R
98eqcomi 2041 . . . . 5 ⟨0R, 0R⟩ = 0
109a1i 9 . . . 4 ((x R y R) → ⟨0R, 0R⟩ = 0)
1110oveq2d 5471 . . 3 ((x R y R) → (⟨x, y⟩ + ⟨0R, 0R⟩) = (⟨x, y⟩ + 0))
12 0idsr 6675 . . . . 5 (x R → (x +R 0R) = x)
1312adantr 261 . . . 4 ((x R y R) → (x +R 0R) = x)
14 0idsr 6675 . . . . 5 (y R → (y +R 0R) = y)
1514adantl 262 . . . 4 ((x R y R) → (y +R 0R) = y)
1613, 15opeq12d 3548 . . 3 ((x R y R) → ⟨(x +R 0R), (y +R 0R)⟩ = ⟨x, y⟩)
177, 11, 163eqtr3d 2077 . 2 ((x R y R) → (⟨x, y⟩ + 0) = ⟨x, y⟩)
181, 4, 17optocl 4359 1 (A ℂ → (A + 0) = A)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  ⟨cop 3370  (class class class)co 5455  Rcnr 6281  0Rc0r 6282   +R cplr 6285  ℂcc 6689  0cc0 6691   + caddc 6694 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254 This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-i1p 6449  df-iplp 6450  df-enr 6634  df-nr 6635  df-plr 6636  df-0r 6639  df-c 6697  df-0 6698  df-add 6702 This theorem is referenced by: (None)
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