Theorem axaddcom 7954

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 7996 be used later. Instead, use addcom 8180.

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.

Step	Hyp	Ref	Expression
1		df-c 7902	. 2 ⊢ ℂ = (R × R)
2		oveq1 5932	. . 3 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = (𝐴 + ⟨𝑧, 𝑤⟩))
3		oveq2 5933	. . 3 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩) = (⟨𝑧, 𝑤⟩ + 𝐴))
4	2, 3	eqeq12d 2211	. 2 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → ((⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩) ↔ (𝐴 + ⟨𝑧, 𝑤⟩) = (⟨𝑧, 𝑤⟩ + 𝐴)))
5		oveq2 5933	. . 3 ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → (𝐴 + ⟨𝑧, 𝑤⟩) = (𝐴 + 𝐵))
6		oveq1 5932	. . 3 ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → (⟨𝑧, 𝑤⟩ + 𝐴) = (𝐵 + 𝐴))
7	5, 6	eqeq12d 2211	. 2 ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → ((𝐴 + ⟨𝑧, 𝑤⟩) = (⟨𝑧, 𝑤⟩ + 𝐴) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴)))
8		addcomsrg 7839	. . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑧 ∈ R) → (𝑥 +R 𝑧) = (𝑧 +R 𝑥))
9	8	ad2ant2r 509	. . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (𝑥 +R 𝑧) = (𝑧 +R 𝑥))
10		addcomsrg 7839	. . . . 5 ⊢ ((𝑦 ∈ R ∧ 𝑤 ∈ R) → (𝑦 +R 𝑤) = (𝑤 +R 𝑦))
11	10	ad2ant2l 508	. . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (𝑦 +R 𝑤) = (𝑤 +R 𝑦))
12	9, 11	opeq12d 3817	. . 3 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩ = ⟨(𝑧 +R 𝑥), (𝑤 +R 𝑦)⟩)
13		addcnsr 7918	. . 3 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = ⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩)
14		addcnsr 7918	. . . 4 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩) = ⟨(𝑧 +R 𝑥), (𝑤 +R 𝑦)⟩)
15	14	ancoms 268	. . 3 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩) = ⟨(𝑧 +R 𝑥), (𝑤 +R 𝑦)⟩)
16	12, 13, 15	3eqtr4d 2239	. 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩))
17	1, 4, 7, 16	2optocl 4741	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ⟨cop 3626  (class class class)co 5925  Rcnr 7381   +_R cplr 7385  ℂcc 7894   + caddc 7899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-iplp 7552  df-enr 7810  df-nr 7811  df-plr 7812  df-c 7902  df-add 7907

This theorem is referenced by: (None)