Theorem axaddass 8082

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 for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addass 8124 be used later. Instead, use addass 8152. (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 8051	. 2 ⊢ ℂ = ((R × R) / ◡ E )
2		addcnsrec 8052	. 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ([⟨𝑥, 𝑦⟩]◡ E + [⟨𝑧, 𝑤⟩]◡ E ) = [⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩]◡ E )
3		addcnsrec 8052	. 2 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ([⟨𝑧, 𝑤⟩]◡ E + [⟨𝑣, 𝑢⟩]◡ E ) = [⟨(𝑧 +R 𝑣), (𝑤 +R 𝑢)⟩]◡ E )
4		addcnsrec 8052	. 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 8052	. 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 7963	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑧 ∈ R) → (𝑥 +R 𝑧) ∈ R)
7		addclsr 7963	. . . 4 ⊢ ((𝑦 ∈ R ∧ 𝑤 ∈ R) → (𝑦 +R 𝑤) ∈ R)
8	6, 7	anim12i 338	. . 3 ⊢ (((𝑥 ∈ R ∧ 𝑧 ∈ R) ∧ (𝑦 ∈ R ∧ 𝑤 ∈ R)) → ((𝑥 +R 𝑧) ∈ R ∧ (𝑦 +R 𝑤) ∈ R))
9	8	an4s 590	. 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ((𝑥 +R 𝑧) ∈ R ∧ (𝑦 +R 𝑤) ∈ R))
10		addclsr 7963	. . . 4 ⊢ ((𝑧 ∈ R ∧ 𝑣 ∈ R) → (𝑧 +R 𝑣) ∈ R)
11		addclsr 7963	. . . 4 ⊢ ((𝑤 ∈ R ∧ 𝑢 ∈ R) → (𝑤 +R 𝑢) ∈ R)
12	10, 11	anim12i 338	. . 3 ⊢ (((𝑧 ∈ R ∧ 𝑣 ∈ R) ∧ (𝑤 ∈ R ∧ 𝑢 ∈ R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R))
13	12	an4s 590	. 2 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R))
14		addasssrg 7966	. . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑧 ∈ R ∧ 𝑣 ∈ R) → ((𝑥 +R 𝑧) +R 𝑣) = (𝑥 +R (𝑧 +R 𝑣)))
15	14	3adant3r 1259	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑧 ∈ R ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ((𝑥 +R 𝑧) +R 𝑣) = (𝑥 +R (𝑧 +R 𝑣)))
16	15	3adant2r 1257	. . 3 ⊢ ((𝑥 ∈ R ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ((𝑥 +R 𝑧) +R 𝑣) = (𝑥 +R (𝑧 +R 𝑣)))
17	16	3adant1r 1255	. 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ((𝑥 +R 𝑧) +R 𝑣) = (𝑥 +R (𝑧 +R 𝑣)))
18		addasssrg 7966	. . . . 5 ⊢ ((𝑦 ∈ R ∧ 𝑤 ∈ R ∧ 𝑢 ∈ R) → ((𝑦 +R 𝑤) +R 𝑢) = (𝑦 +R (𝑤 +R 𝑢)))
19	18	3adant3l 1258	. . . 4 ⊢ ((𝑦 ∈ R ∧ 𝑤 ∈ R ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ((𝑦 +R 𝑤) +R 𝑢) = (𝑦 +R (𝑤 +R 𝑢)))
20	19	3adant2l 1256	. . 3 ⊢ ((𝑦 ∈ R ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ((𝑦 +R 𝑤) +R 𝑢) = (𝑦 +R (𝑤 +R 𝑢)))
21	20	3adant1l 1254	. 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ((𝑦 +R 𝑤) +R 𝑢) = (𝑦 +R (𝑤 +R 𝑢)))
22	1, 2, 3, 4, 5, 9, 13, 17, 21	ecoviass 6809	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200   E cep 4382  ^◡ccnv 4722  (class class class)co 6013  Rcnr 7507   +_R cplr 7511  ℂcc 8020   + caddc 8025

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-enq0 7634  df-nq0 7635  df-0nq0 7636  df-plq0 7637  df-mq0 7638  df-inp 7676  df-iplp 7678  df-enr 7936  df-nr 7937  df-plr 7938  df-c 8028  df-add 8033

This theorem is referenced by: (None)