Theorem axaddcom 8133

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 8175 be used later. Instead, use addcom 8359.

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 8081	. 2 ⊢ ℂ = (R × R)
2		oveq1 6035	. . 3 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = (𝐴 + ⟨𝑧, 𝑤⟩))
3		oveq2 6036	. . 3 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩) = (⟨𝑧, 𝑤⟩ + 𝐴))
4	2, 3	eqeq12d 2246	. 2 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → ((⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩) ↔ (𝐴 + ⟨𝑧, 𝑤⟩) = (⟨𝑧, 𝑤⟩ + 𝐴)))
5		oveq2 6036	. . 3 ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → (𝐴 + ⟨𝑧, 𝑤⟩) = (𝐴 + 𝐵))
6		oveq1 6035	. . 3 ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → (⟨𝑧, 𝑤⟩ + 𝐴) = (𝐵 + 𝐴))
7	5, 6	eqeq12d 2246	. 2 ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → ((𝐴 + ⟨𝑧, 𝑤⟩) = (⟨𝑧, 𝑤⟩ + 𝐴) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴)))
8		addcomsrg 8018	. . . . 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 8018	. . . . 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 3875	. . 3 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩ = ⟨(𝑧 +R 𝑥), (𝑤 +R 𝑦)⟩)
13		addcnsr 8097	. . 3 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = ⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩)
14		addcnsr 8097	. . . 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 2274	. 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = (⟨𝑧, 𝑤⟩ + ⟨𝑥, 𝑦⟩))
17	1, 4, 7, 16	2optocl 4809	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202  ⟨cop 3676  (class class class)co 6028  Rcnr 7560   +_R cplr 7564  ℂcc 8073   + caddc 8078

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616  df-enq0 7687  df-nq0 7688  df-0nq0 7689  df-plq0 7690  df-mq0 7691  df-inp 7729  df-iplp 7731  df-enr 7989  df-nr 7990  df-plr 7991  df-c 8081  df-add 8086

This theorem is referenced by: (None)