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Theorem axaddass 11187
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 11211 be used later. Instead, use addass 11233. (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 11173 . 2 ℂ = ((R × R) / E )
2 addcnsrec 11174 . 2 (((𝑥R𝑦R) ∧ (𝑧R𝑤R)) → ([⟨𝑥, 𝑦⟩] E + [⟨𝑧, 𝑤⟩] E ) = [⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩] E )
3 addcnsrec 11174 . 2 (((𝑧R𝑤R) ∧ (𝑣R𝑢R)) → ([⟨𝑧, 𝑤⟩] E + [⟨𝑣, 𝑢⟩] E ) = [⟨(𝑧 +R 𝑣), (𝑤 +R 𝑢)⟩] E )
4 addcnsrec 11174 . 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 11174 . 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 11114 . . . 4 ((𝑥R𝑧R) → (𝑥 +R 𝑧) ∈ R)
7 addclsr 11114 . . . 4 ((𝑦R𝑤R) → (𝑦 +R 𝑤) ∈ R)
86, 7anim12i 611 . . 3 (((𝑥R𝑧R) ∧ (𝑦R𝑤R)) → ((𝑥 +R 𝑧) ∈ R ∧ (𝑦 +R 𝑤) ∈ R))
98an4s 658 . 2 (((𝑥R𝑦R) ∧ (𝑧R𝑤R)) → ((𝑥 +R 𝑧) ∈ R ∧ (𝑦 +R 𝑤) ∈ R))
10 addclsr 11114 . . . 4 ((𝑧R𝑣R) → (𝑧 +R 𝑣) ∈ R)
11 addclsr 11114 . . . 4 ((𝑤R𝑢R) → (𝑤 +R 𝑢) ∈ R)
1210, 11anim12i 611 . . 3 (((𝑧R𝑣R) ∧ (𝑤R𝑢R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R))
1312an4s 658 . 2 (((𝑧R𝑤R) ∧ (𝑣R𝑢R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R))
14 addasssr 11119 . 2 ((𝑥 +R 𝑧) +R 𝑣) = (𝑥 +R (𝑧 +R 𝑣))
15 addasssr 11119 . 2 ((𝑦 +R 𝑤) +R 𝑢) = (𝑦 +R (𝑤 +R 𝑢))
161, 2, 3, 4, 5, 9, 13, 14, 15ecovass 8849 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098   E cep 5585  ccnv 5681  (class class class)co 7426  Rcnr 10896   +R cplr 10900  cc 11144   + caddc 11149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-oadd 8497  df-omul 8498  df-er 8731  df-ec 8733  df-qs 8737  df-ni 10903  df-pli 10904  df-mi 10905  df-lti 10906  df-plpq 10939  df-mpq 10940  df-ltpq 10941  df-enq 10942  df-nq 10943  df-erq 10944  df-plq 10945  df-mq 10946  df-1nq 10947  df-rq 10948  df-ltnq 10949  df-np 11012  df-plp 11014  df-ltp 11016  df-enr 11086  df-nr 11087  df-plr 11088  df-c 11152  df-add 11157
This theorem is referenced by: (None)
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