Theorem axaddf 9966

Description: Addition is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axaddcl 9972. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 10015. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)

Assertion

Ref	Expression
axaddf	⊢ + :(ℂ × ℂ)⟶ℂ

Proof of Theorem axaddf

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		moeq 3382	. . . . . . . . 9 ⊢ ∃*𝑧 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩
2	1	mosubop 4973	. . . . . . . 8 ⊢ ∃*𝑧∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
3	2	mosubop 4973	. . . . . . 7 ⊢ ∃*𝑧∃𝑤∃𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))
4		anass 681	. . . . . . . . . . 11 ⊢ (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)))
5	4	2exbii 1775	. . . . . . . . . 10 ⊢ (∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩) ↔ ∃𝑢∃𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)))
6		19.42vv 1920	. . . . . . . . . 10 ⊢ (∃𝑢∃𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)))
7	5, 6	bitri 264	. . . . . . . . 9 ⊢ (∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)))
8	7	2exbii 1775	. . . . . . . 8 ⊢ (∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩) ↔ ∃𝑤∃𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)))
9	8	mobii 2493	. . . . . . 7 ⊢ (∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩) ↔ ∃*𝑧∃𝑤∃𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)))
10	3, 9	mpbir 221	. . . . . 6 ⊢ ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
11	10	moani 2525	. . . . 5 ⊢ ∃*𝑧((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))
12	11	funoprab 6760	. . . 4 ⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
13		df-add 9947	. . . . 5 ⊢ + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
14	13	funeqi 5909	. . . 4 ⊢ (Fun + ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))})
15	12, 14	mpbir 221	. . 3 ⊢ Fun +
16	13	dmeqi 5325	. . . . 5 ⊢ dom + = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
17		dmoprabss 6742	. . . . 5 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))} ⊆ (ℂ × ℂ)
18	16, 17	eqsstri 3635	. . . 4 ⊢ dom + ⊆ (ℂ × ℂ)
19		0ncn 9954	. . . . 5 ⊢ ¬ ∅ ∈ ℂ
20		df-c 9942	. . . . . . 7 ⊢ ℂ = (R × R)
21		oveq1 6657	. . . . . . . 8 ⊢ (⟨𝑧, 𝑤⟩ = 𝑥 → (⟨𝑧, 𝑤⟩ + ⟨𝑣, 𝑢⟩) = (𝑥 + ⟨𝑣, 𝑢⟩))
22	21	eleq1d 2686	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = 𝑥 → ((⟨𝑧, 𝑤⟩ + ⟨𝑣, 𝑢⟩) ∈ (R × R) ↔ (𝑥 + ⟨𝑣, 𝑢⟩) ∈ (R × R)))
23		oveq2 6658	. . . . . . . 8 ⊢ (⟨𝑣, 𝑢⟩ = 𝑦 → (𝑥 + ⟨𝑣, 𝑢⟩) = (𝑥 + 𝑦))
24	23	eleq1d 2686	. . . . . . 7 ⊢ (⟨𝑣, 𝑢⟩ = 𝑦 → ((𝑥 + ⟨𝑣, 𝑢⟩) ∈ (R × R) ↔ (𝑥 + 𝑦) ∈ (R × R)))
25		addcnsr 9956	. . . . . . . 8 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → (⟨𝑧, 𝑤⟩ + ⟨𝑣, 𝑢⟩) = ⟨(𝑧 +R 𝑣), (𝑤 +R 𝑢)⟩)
26		addclsr 9904	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ R ∧ 𝑣 ∈ R) → (𝑧 +R 𝑣) ∈ R)
27		addclsr 9904	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ R ∧ 𝑢 ∈ R) → (𝑤 +R 𝑢) ∈ R)
28	26, 27	anim12i 590	. . . . . . . . . 10 ⊢ (((𝑧 ∈ R ∧ 𝑣 ∈ R) ∧ (𝑤 ∈ R ∧ 𝑢 ∈ R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R))
29	28	an4s 869	. . . . . . . . 9 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R))
30		opelxpi 5148	. . . . . . . . 9 ⊢ (((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R) → ⟨(𝑧 +R 𝑣), (𝑤 +R 𝑢)⟩ ∈ (R × R))
31	29, 30	syl 17	. . . . . . . 8 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ⟨(𝑧 +R 𝑣), (𝑤 +R 𝑢)⟩ ∈ (R × R))
32	25, 31	eqeltrd 2701	. . . . . . 7 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → (⟨𝑧, 𝑤⟩ + ⟨𝑣, 𝑢⟩) ∈ (R × R))
33	20, 22, 24, 32	2optocl 5196	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ (R × R))
34	33, 20	syl6eleqr 2712	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
35	19, 34	oprssdm 6815	. . . 4 ⊢ (ℂ × ℂ) ⊆ dom +
36	18, 35	eqssi 3619	. . 3 ⊢ dom + = (ℂ × ℂ)
37		df-fn 5891	. . 3 ⊢ ( + Fn (ℂ × ℂ) ↔ (Fun + ∧ dom + = (ℂ × ℂ)))
38	15, 36, 37	mpbir2an 955	. 2 ⊢ + Fn (ℂ × ℂ)
39	34	rgen2a 2977	. 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 + 𝑦) ∈ ℂ
40		ffnov 6764	. 2 ⊢ ( + :(ℂ × ℂ)⟶ℂ ↔ ( + Fn (ℂ × ℂ) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 + 𝑦) ∈ ℂ))
41	38, 39, 40	mpbir2an 955	1 ⊢ + :(ℂ × ℂ)⟶ℂ

Syntax hints:   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃*wmo 2471  ∀wral 2912  ⟨cop 4183   × cxp 5112  dom cdm 5114  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  (class class class)co 6650  {coprab 6651  Rcnr 9687   +_R cplr 9691  ℂcc 9934   + caddc 9939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-plp 9805  df-ltp 9807  df-enr 9877  df-nr 9878  df-plr 9879  df-c 9942  df-add 9947

This theorem is referenced by:  axaddcl  9972