Theorem axaddass 10015

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 10039 be used later. Instead, use addass 10061. (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.

Step	Hyp	Ref	Expression
1		dfcnqs 10001	. 2 ⊢ ℂ = ((R × R) / ◡ E )
2		addcnsrec 10002	. 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ([⟨𝑥, 𝑦⟩]◡ E + [⟨𝑧, 𝑤⟩]◡ E ) = [⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩]◡ E )
3		addcnsrec 10002	. 2 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ([⟨𝑧, 𝑤⟩]◡ E + [⟨𝑣, 𝑢⟩]◡ E ) = [⟨(𝑧 +R 𝑣), (𝑤 +R 𝑢)⟩]◡ E )
4		addcnsrec 10002	. 2 ⊢ ((((𝑥 +R 𝑧) ∈ R ∧ (𝑦 +R 𝑤) ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ([⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩]◡ E + [⟨𝑣, 𝑢⟩]◡ E ) = [⟨((𝑥 +R 𝑧) +R 𝑣), ((𝑦 +R 𝑤) +R 𝑢)⟩]◡ E )
5		addcnsrec 10002	. 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ((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 9942	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑧 ∈ R) → (𝑥 +R 𝑧) ∈ R)
7		addclsr 9942	. . . 4 ⊢ ((𝑦 ∈ R ∧ 𝑤 ∈ R) → (𝑦 +R 𝑤) ∈ R)
8	6, 7	anim12i 589	. . 3 ⊢ (((𝑥 ∈ R ∧ 𝑧 ∈ R) ∧ (𝑦 ∈ R ∧ 𝑤 ∈ R)) → ((𝑥 +R 𝑧) ∈ R ∧ (𝑦 +R 𝑤) ∈ R))
9	8	an4s 886	. 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ((𝑥 +R 𝑧) ∈ R ∧ (𝑦 +R 𝑤) ∈ R))
10		addclsr 9942	. . . 4 ⊢ ((𝑧 ∈ R ∧ 𝑣 ∈ R) → (𝑧 +R 𝑣) ∈ R)
11		addclsr 9942	. . . 4 ⊢ ((𝑤 ∈ R ∧ 𝑢 ∈ R) → (𝑤 +R 𝑢) ∈ R)
12	10, 11	anim12i 589	. . 3 ⊢ (((𝑧 ∈ R ∧ 𝑣 ∈ R) ∧ (𝑤 ∈ R ∧ 𝑢 ∈ R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R))
13	12	an4s 886	. 2 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R))
14		addasssr 9947	. 2 ⊢ ((𝑥 +R 𝑧) +R 𝑣) = (𝑥 +R (𝑧 +R 𝑣))
15		addasssr 9947	. 2 ⊢ ((𝑦 +R 𝑤) +R 𝑢) = (𝑦 +R (𝑤 +R 𝑢))
16	1, 2, 3, 4, 5, 9, 13, 14, 15	ecovass 7897	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   E cep 5057  ^◡ccnv 5142  (class class class)co 6690  Rcnr 9725   +_R cplr 9729  ℂcc 9972   + caddc 9977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ec 7789  df-qs 7793  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-rq 9777  df-ltnq 9778  df-np 9841  df-plp 9843  df-ltp 9845  df-enr 9915  df-nr 9916  df-plr 9917  df-c 9980  df-add 9985

This theorem is referenced by: (None)