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 Description: Addition commutes. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcom 6984 be used later. Instead, use addcom 7150. In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on real number trichotomy and it is not known whether it is possible to prove this from the other axioms without it. (Contributed by Jim Kingdon, 17-Jan-2020.) (New usage is discouraged.)
Assertion
Ref Expression
axaddcom ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Proof of Theorem axaddcom
Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-c 6895 . 2 ℂ = (R × R)
2 oveq1 5519 . . 3 (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = (𝐴 + ⟨𝑧, 𝑤⟩))
3 oveq2 5520 . . 3 (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩) = (⟨𝑧, 𝑤⟩ + 𝐴))
42, 3eqeq12d 2054 . 2 (⟨𝑥, 𝑦⟩ = 𝐴 → ((⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩) ↔ (𝐴 + ⟨𝑧, 𝑤⟩) = (⟨𝑧, 𝑤⟩ + 𝐴)))
5 oveq2 5520 . . 3 (⟨𝑧, 𝑤⟩ = 𝐵 → (𝐴 + ⟨𝑧, 𝑤⟩) = (𝐴 + 𝐵))
6 oveq1 5519 . . 3 (⟨𝑧, 𝑤⟩ = 𝐵 → (⟨𝑧, 𝑤⟩ + 𝐴) = (𝐵 + 𝐴))
75, 6eqeq12d 2054 . 2 (⟨𝑧, 𝑤⟩ = 𝐵 → ((𝐴 + ⟨𝑧, 𝑤⟩) = (⟨𝑧, 𝑤⟩ + 𝐴) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴)))
8 addcomsrg 6840 . . . . 5 ((𝑥R𝑧R) → (𝑥 +R 𝑧) = (𝑧 +R 𝑥))
98ad2ant2r 478 . . . 4 (((𝑥R𝑦R) ∧ (𝑧R𝑤R)) → (𝑥 +R 𝑧) = (𝑧 +R 𝑥))
10 addcomsrg 6840 . . . . 5 ((𝑦R𝑤R) → (𝑦 +R 𝑤) = (𝑤 +R 𝑦))
1110ad2ant2l 477 . . . 4 (((𝑥R𝑦R) ∧ (𝑧R𝑤R)) → (𝑦 +R 𝑤) = (𝑤 +R 𝑦))
129, 11opeq12d 3557 . . 3 (((𝑥R𝑦R) ∧ (𝑧R𝑤R)) → ⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩ = ⟨(𝑧 +R 𝑥), (𝑤 +R 𝑦)⟩)
13 addcnsr 6910 . . 3 (((𝑥R𝑦R) ∧ (𝑧R𝑤R)) → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = ⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩)
14 addcnsr 6910 . . . 4 (((𝑧R𝑤R) ∧ (𝑥R𝑦R)) → (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩) = ⟨(𝑧 +R 𝑥), (𝑤 +R 𝑦)⟩)
1514ancoms 255 . . 3 (((𝑥R𝑦R) ∧ (𝑧R𝑤R)) → (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩) = ⟨(𝑧 +R 𝑥), (𝑤 +R 𝑦)⟩)
1612, 13, 153eqtr4d 2082 . 2 (((𝑥R𝑦R) ∧ (𝑧R𝑤R)) → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩))
171, 4, 7, 162optocl 4417 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ⟨cop 3378  (class class class)co 5512  Rcnr 6395   +R cplr 6399  ℂcc 6887   + caddc 6892 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566  df-enr 6811  df-nr 6812  df-plr 6813  df-c 6895  df-add 6900 This theorem is referenced by: (None)
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