Theorem axaddcom 7168

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 7208 be used later. Instead, use addcom 7382.

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 7119	. 2 ⊢ ℂ = (R × R)
2		oveq1 5571	. . 3 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = (𝐴 + ⟨𝑧, 𝑤⟩))
3		oveq2 5572	. . 3 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩) = (⟨𝑧, 𝑤⟩ + 𝐴))
4	2, 3	eqeq12d 2097	. 2 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → ((⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩) ↔ (𝐴 + ⟨𝑧, 𝑤⟩) = (⟨𝑧, 𝑤⟩ + 𝐴)))
5		oveq2 5572	. . 3 ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → (𝐴 + ⟨𝑧, 𝑤⟩) = (𝐴 + 𝐵))
6		oveq1 5571	. . 3 ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → (⟨𝑧, 𝑤⟩ + 𝐴) = (𝐵 + 𝐴))
7	5, 6	eqeq12d 2097	. 2 ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → ((𝐴 + ⟨𝑧, 𝑤⟩) = (⟨𝑧, 𝑤⟩ + 𝐴) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴)))
8		addcomsrg 7064	. . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑧 ∈ R) → (𝑥 +R 𝑧) = (𝑧 +R 𝑥))
9	8	ad2ant2r 493	. . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (𝑥 +R 𝑧) = (𝑧 +R 𝑥))
10		addcomsrg 7064	. . . . 5 ⊢ ((𝑦 ∈ R ∧ 𝑤 ∈ R) → (𝑦 +R 𝑤) = (𝑤 +R 𝑦))
11	10	ad2ant2l 492	. . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (𝑦 +R 𝑤) = (𝑤 +R 𝑦))
12	9, 11	opeq12d 3598	. . 3 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩ = ⟨(𝑧 +R 𝑥), (𝑤 +R 𝑦)⟩)
13		addcnsr 7134	. . 3 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = ⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩)
14		addcnsr 7134	. . . 4 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩) = ⟨(𝑧 +R 𝑥), (𝑤 +R 𝑦)⟩)
15	14	ancoms 264	. . 3 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩) = ⟨(𝑧 +R 𝑥), (𝑤 +R 𝑦)⟩)
16	12, 13, 15	3eqtr4d 2125	. 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩))
17	1, 4, 7, 16	2optocl 4463	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1285   ∈ wcel 1434  ⟨cop 3419  (class class class)co 5564  Rcnr 6619   +_R cplr 6623  ℂcc 7111   + caddc 7116

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-eprel 4072  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-1o 6086  df-2o 6087  df-oadd 6090  df-omul 6091  df-er 6194  df-ec 6196  df-qs 6200  df-ni 6626  df-pli 6627  df-mi 6628  df-lti 6629  df-plpq 6666  df-mpq 6667  df-enq 6669  df-nqqs 6670  df-plqqs 6671  df-mqqs 6672  df-1nqqs 6673  df-rq 6674  df-ltnqqs 6675  df-enq0 6746  df-nq0 6747  df-0nq0 6748  df-plq0 6749  df-mq0 6750  df-inp 6788  df-iplp 6790  df-enr 7035  df-nr 7036  df-plr 7037  df-c 7119  df-add 7124

This theorem is referenced by: (None)