Theorem axaddf 11081

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 11087. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 11130. (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 3665	. . . . . . . . 9 ⊢ ∃*𝑧 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩
2	1	mosubop 5468	. . . . . . . 8 ⊢ ∃*𝑧∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
3	2	mosubop 5468	. . . . . . 7 ⊢ ∃*𝑧∃𝑤∃𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))
4		anass 469	. . . . . . . . . . 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 1851	. . . . . . . . . 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 1961	. . . . . . . . . 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 274	. . . . . . . . 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 1851	. . . . . . . 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 2546	. . . . . . 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 230	. . . . . 6 ⊢ ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
11	10	moani 2551	. . . . 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 7478	. . . 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 11062	. . . . 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 6522	. . . 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 230	. . 3 ⊢ Fun +
16	13	dmeqi 5860	. . . . 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 7459	. . . . 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 3978	. . . 4 ⊢ dom + ⊆ (ℂ × ℂ)
19		0ncn 11069	. . . . 5 ⊢ ¬ ∅ ∈ ℂ
20		df-c 11057	. . . . . . 7 ⊢ ℂ = (R × R)
21		oveq1 7364	. . . . . . . 8 ⊢ (⟨𝑧, 𝑤⟩ = 𝑥 → (⟨𝑧, 𝑤⟩ + ⟨𝑣, 𝑢⟩) = (𝑥 + ⟨𝑣, 𝑢⟩))
22	21	eleq1d 2822	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = 𝑥 → ((⟨𝑧, 𝑤⟩ + ⟨𝑣, 𝑢⟩) ∈ (R × R) ↔ (𝑥 + ⟨𝑣, 𝑢⟩) ∈ (R × R)))
23		oveq2 7365	. . . . . . . 8 ⊢ (⟨𝑣, 𝑢⟩ = 𝑦 → (𝑥 + ⟨𝑣, 𝑢⟩) = (𝑥 + 𝑦))
24	23	eleq1d 2822	. . . . . . 7 ⊢ (⟨𝑣, 𝑢⟩ = 𝑦 → ((𝑥 + ⟨𝑣, 𝑢⟩) ∈ (R × R) ↔ (𝑥 + 𝑦) ∈ (R × R)))
25		addcnsr 11071	. . . . . . . 8 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → (⟨𝑧, 𝑤⟩ + ⟨𝑣, 𝑢⟩) = ⟨(𝑧 +R 𝑣), (𝑤 +R 𝑢)⟩)
26		addclsr 11019	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ R ∧ 𝑣 ∈ R) → (𝑧 +R 𝑣) ∈ R)
27		addclsr 11019	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ R ∧ 𝑢 ∈ R) → (𝑤 +R 𝑢) ∈ R)
28	26, 27	anim12i 613	. . . . . . . . . 10 ⊢ (((𝑧 ∈ R ∧ 𝑣 ∈ R) ∧ (𝑤 ∈ R ∧ 𝑢 ∈ R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R))
29	28	an4s 658	. . . . . . . . 9 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R))
30		opelxpi 5670	. . . . . . . . 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 2838	. . . . . . 7 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑣 ∈ R ∧ 𝑢 ∈ R)) → (⟨𝑧, 𝑤⟩ + ⟨𝑣, 𝑢⟩) ∈ (R × R))
33	20, 22, 24, 32	2optocl 5727	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ (R × R))
34	33, 20	eleqtrrdi 2849	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
35	19, 34	oprssdm 7535	. . . 4 ⊢ (ℂ × ℂ) ⊆ dom +
36	18, 35	eqssi 3960	. . 3 ⊢ dom + = (ℂ × ℂ)
37		df-fn 6499	. . 3 ⊢ ( + Fn (ℂ × ℂ) ↔ (Fun + ∧ dom + = (ℂ × ℂ)))
38	15, 36, 37	mpbir2an 709	. 2 ⊢ + Fn (ℂ × ℂ)
39	34	rgen2 3194	. 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 + 𝑦) ∈ ℂ
40		ffnov 7483	. 2 ⊢ ( + :(ℂ × ℂ)⟶ℂ ↔ ( + Fn (ℂ × ℂ) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 + 𝑦) ∈ ℂ))
41	38, 39, 40	mpbir2an 709	1 ⊢ + :(ℂ × ℂ)⟶ℂ

Syntax hints:   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∃*wmo 2536  ∀wral 3064  ⟨cop 4592   × cxp 5631  dom cdm 5633  Fun wfun 6490   Fn wfn 6491  ⟶wf 6492  (class class class)co 7357  {coprab 7358  Rcnr 10801   +_R cplr 10805  ℂcc 11049   + caddc 11054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-er 8648  df-ec 8650  df-qs 8654  df-ni 10808  df-pli 10809  df-mi 10810  df-lti 10811  df-plpq 10844  df-mpq 10845  df-ltpq 10846  df-enq 10847  df-nq 10848  df-erq 10849  df-plq 10850  df-mq 10851  df-1nq 10852  df-rq 10853  df-ltnq 10854  df-np 10917  df-plp 10919  df-ltp 10921  df-enr 10991  df-nr 10992  df-plr 10993  df-c 11057  df-add 11062

This theorem is referenced by:  axaddcl  11087