MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axaddf Structured version   Visualization version   GIF version

Theorem axaddf 10158
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 10164. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 10207. (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.
StepHypRef Expression
1 moeq 3523 . . . . . . . . 9 ∃*𝑧 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩
21mosubop 5121 . . . . . . . 8 ∃*𝑧𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
32mosubop 5121 . . . . . . 7 ∃*𝑧𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))
4 anass 684 . . . . . . . . . . 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)⟩)))
542exbii 1924 . . . . . . . . . 10 (∃𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩) ↔ ∃𝑢𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)))
6 19.42vv 2032 . . . . . . . . . 10 (∃𝑢𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)))
75, 6bitri 264 . . . . . . . . 9 (∃𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)))
872exbii 1924 . . . . . . . 8 (∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩) ↔ ∃𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)))
98mobii 2630 . . . . . . 7 (∃*𝑧𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩) ↔ ∃*𝑧𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)))
103, 9mpbir 221 . . . . . 6 ∃*𝑧𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
1110moani 2663 . . . . 5 ∃*𝑧((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))
1211funoprab 6925 . . . 4 Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
13 df-add 10139 . . . . 5 + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
1413funeqi 6070 . . . 4 (Fun + ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))})
1512, 14mpbir 221 . . 3 Fun +
1613dmeqi 5480 . . . . 5 dom + = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
17 dmoprabss 6907 . . . . 5 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))} ⊆ (ℂ × ℂ)
1816, 17eqsstri 3776 . . . 4 dom + ⊆ (ℂ × ℂ)
19 0ncn 10146 . . . . 5 ¬ ∅ ∈ ℂ
20 df-c 10134 . . . . . . 7 ℂ = (R × R)
21 oveq1 6820 . . . . . . . 8 (⟨𝑧, 𝑤⟩ = 𝑥 → (⟨𝑧, 𝑤⟩ + ⟨𝑣, 𝑢⟩) = (𝑥 + ⟨𝑣, 𝑢⟩))
2221eleq1d 2824 . . . . . . 7 (⟨𝑧, 𝑤⟩ = 𝑥 → ((⟨𝑧, 𝑤⟩ + ⟨𝑣, 𝑢⟩) ∈ (R × R) ↔ (𝑥 + ⟨𝑣, 𝑢⟩) ∈ (R × R)))
23 oveq2 6821 . . . . . . . 8 (⟨𝑣, 𝑢⟩ = 𝑦 → (𝑥 + ⟨𝑣, 𝑢⟩) = (𝑥 + 𝑦))
2423eleq1d 2824 . . . . . . 7 (⟨𝑣, 𝑢⟩ = 𝑦 → ((𝑥 + ⟨𝑣, 𝑢⟩) ∈ (R × R) ↔ (𝑥 + 𝑦) ∈ (R × R)))
25 addcnsr 10148 . . . . . . . 8 (((𝑧R𝑤R) ∧ (𝑣R𝑢R)) → (⟨𝑧, 𝑤⟩ + ⟨𝑣, 𝑢⟩) = ⟨(𝑧 +R 𝑣), (𝑤 +R 𝑢)⟩)
26 addclsr 10096 . . . . . . . . . . 11 ((𝑧R𝑣R) → (𝑧 +R 𝑣) ∈ R)
27 addclsr 10096 . . . . . . . . . . 11 ((𝑤R𝑢R) → (𝑤 +R 𝑢) ∈ R)
2826, 27anim12i 591 . . . . . . . . . 10 (((𝑧R𝑣R) ∧ (𝑤R𝑢R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R))
2928an4s 904 . . . . . . . . 9 (((𝑧R𝑤R) ∧ (𝑣R𝑢R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R))
30 opelxpi 5305 . . . . . . . . 9 (((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R) → ⟨(𝑧 +R 𝑣), (𝑤 +R 𝑢)⟩ ∈ (R × R))
3129, 30syl 17 . . . . . . . 8 (((𝑧R𝑤R) ∧ (𝑣R𝑢R)) → ⟨(𝑧 +R 𝑣), (𝑤 +R 𝑢)⟩ ∈ (R × R))
3225, 31eqeltrd 2839 . . . . . . 7 (((𝑧R𝑤R) ∧ (𝑣R𝑢R)) → (⟨𝑧, 𝑤⟩ + ⟨𝑣, 𝑢⟩) ∈ (R × R))
3320, 22, 24, 322optocl 5353 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ (R × R))
3433, 20syl6eleqr 2850 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
3519, 34oprssdm 6980 . . . 4 (ℂ × ℂ) ⊆ dom +
3618, 35eqssi 3760 . . 3 dom + = (ℂ × ℂ)
37 df-fn 6052 . . 3 ( + Fn (ℂ × ℂ) ↔ (Fun + ∧ dom + = (ℂ × ℂ)))
3815, 36, 37mpbir2an 993 . 2 + Fn (ℂ × ℂ)
3934rgen2a 3115 . 2 𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 + 𝑦) ∈ ℂ
40 ffnov 6929 . 2 ( + :(ℂ × ℂ)⟶ℂ ↔ ( + Fn (ℂ × ℂ) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 + 𝑦) ∈ ℂ))
4138, 39, 40mpbir2an 993 1 + :(ℂ × ℂ)⟶ℂ
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wex 1853  wcel 2139  ∃*wmo 2608  wral 3050  cop 4327   × cxp 5264  dom cdm 5266  Fun wfun 6043   Fn wfn 6044  wf 6045  (class class class)co 6813  {coprab 6814  Rcnr 9879   +R cplr 9883  cc 10126   + caddc 10131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-omul 7734  df-er 7911  df-ec 7913  df-qs 7917  df-ni 9886  df-pli 9887  df-mi 9888  df-lti 9889  df-plpq 9922  df-mpq 9923  df-ltpq 9924  df-enq 9925  df-nq 9926  df-erq 9927  df-plq 9928  df-mq 9929  df-1nq 9930  df-rq 9931  df-ltnq 9932  df-np 9995  df-plp 9997  df-ltp 9999  df-enr 10069  df-nr 10070  df-plr 10071  df-c 10134  df-add 10139
This theorem is referenced by:  axaddcl  10164
  Copyright terms: Public domain W3C validator