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Theorem axaddass 10376
Description: Addition of complex numbers is associative. This theorem transfers the associative laws for the real and imaginary signed real components of complex number pairs, to complex number addition itself. Axiom 9 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addass 10400 be used later. Instead, use addass 10422. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
Assertion
Ref Expression
axaddass ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Proof of Theorem axaddass
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfcnqs 10362 . 2 ℂ = ((R × R) / E )
2 addcnsrec 10363 . 2 (((𝑥R𝑦R) ∧ (𝑧R𝑤R)) → ([⟨𝑥, 𝑦⟩] E + [⟨𝑧, 𝑤⟩] E ) = [⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩] E )
3 addcnsrec 10363 . 2 (((𝑧R𝑤R) ∧ (𝑣R𝑢R)) → ([⟨𝑧, 𝑤⟩] E + [⟨𝑣, 𝑢⟩] E ) = [⟨(𝑧 +R 𝑣), (𝑤 +R 𝑢)⟩] E )
4 addcnsrec 10363 . 2 ((((𝑥 +R 𝑧) ∈ R ∧ (𝑦 +R 𝑤) ∈ R) ∧ (𝑣R𝑢R)) → ([⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩] E + [⟨𝑣, 𝑢⟩] E ) = [⟨((𝑥 +R 𝑧) +R 𝑣), ((𝑦 +R 𝑤) +R 𝑢)⟩] E )
5 addcnsrec 10363 . 2 (((𝑥R𝑦R) ∧ ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R)) → ([⟨𝑥, 𝑦⟩] E + [⟨(𝑧 +R 𝑣), (𝑤 +R 𝑢)⟩] E ) = [⟨(𝑥 +R (𝑧 +R 𝑣)), (𝑦 +R (𝑤 +R 𝑢))⟩] E )
6 addclsr 10303 . . . 4 ((𝑥R𝑧R) → (𝑥 +R 𝑧) ∈ R)
7 addclsr 10303 . . . 4 ((𝑦R𝑤R) → (𝑦 +R 𝑤) ∈ R)
86, 7anim12i 603 . . 3 (((𝑥R𝑧R) ∧ (𝑦R𝑤R)) → ((𝑥 +R 𝑧) ∈ R ∧ (𝑦 +R 𝑤) ∈ R))
98an4s 647 . 2 (((𝑥R𝑦R) ∧ (𝑧R𝑤R)) → ((𝑥 +R 𝑧) ∈ R ∧ (𝑦 +R 𝑤) ∈ R))
10 addclsr 10303 . . . 4 ((𝑧R𝑣R) → (𝑧 +R 𝑣) ∈ R)
11 addclsr 10303 . . . 4 ((𝑤R𝑢R) → (𝑤 +R 𝑢) ∈ R)
1210, 11anim12i 603 . . 3 (((𝑧R𝑣R) ∧ (𝑤R𝑢R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R))
1312an4s 647 . 2 (((𝑧R𝑤R) ∧ (𝑣R𝑢R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R))
14 addasssr 10308 . 2 ((𝑥 +R 𝑧) +R 𝑣) = (𝑥 +R (𝑧 +R 𝑣))
15 addasssr 10308 . 2 ((𝑦 +R 𝑤) +R 𝑢) = (𝑦 +R (𝑤 +R 𝑢))
161, 2, 3, 4, 5, 9, 13, 14, 15ecovass 8204 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wcel 2050   E cep 5316  ccnv 5406  (class class class)co 6976  Rcnr 10085   +R cplr 10089  cc 10333   + caddc 10338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-inf2 8898
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-oadd 7909  df-omul 7910  df-er 8089  df-ec 8091  df-qs 8095  df-ni 10092  df-pli 10093  df-mi 10094  df-lti 10095  df-plpq 10128  df-mpq 10129  df-ltpq 10130  df-enq 10131  df-nq 10132  df-erq 10133  df-plq 10134  df-mq 10135  df-1nq 10136  df-rq 10137  df-ltnq 10138  df-np 10201  df-plp 10203  df-ltp 10205  df-enr 10275  df-nr 10276  df-plr 10277  df-c 10341  df-add 10346
This theorem is referenced by: (None)
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