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Theorem axaddf 11174
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 11180. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 11223. (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 3702 . . . . . . . . 9 ∃*𝑧 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩
21mosubop 5515 . . . . . . . 8 ∃*𝑧𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
32mosubop 5515 . . . . . . 7 ∃*𝑧𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))
4 anass 467 . . . . . . . . . . 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 1843 . . . . . . . . . 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 1953 . . . . . . . . . 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 274 . . . . . . . . 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 1843 . . . . . . . 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 2537 . . . . . . 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 230 . . . . . 6 ∃*𝑧𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
1110moani 2542 . . . . 5 ∃*𝑧((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))
1211funoprab 7546 . . . 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 11155 . . . . 5 + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
1413funeqi 6577 . . . 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 230 . . 3 Fun +
1613dmeqi 5909 . . . . 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 7527 . . . . 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 4014 . . . 4 dom + ⊆ (ℂ × ℂ)
19 0ncn 11162 . . . . 5 ¬ ∅ ∈ ℂ
20 df-c 11150 . . . . . . 7 ℂ = (R × R)
21 oveq1 7431 . . . . . . . 8 (⟨𝑧, 𝑤⟩ = 𝑥 → (⟨𝑧, 𝑤⟩ + ⟨𝑣, 𝑢⟩) = (𝑥 + ⟨𝑣, 𝑢⟩))
2221eleq1d 2813 . . . . . . 7 (⟨𝑧, 𝑤⟩ = 𝑥 → ((⟨𝑧, 𝑤⟩ + ⟨𝑣, 𝑢⟩) ∈ (R × R) ↔ (𝑥 + ⟨𝑣, 𝑢⟩) ∈ (R × R)))
23 oveq2 7432 . . . . . . . 8 (⟨𝑣, 𝑢⟩ = 𝑦 → (𝑥 + ⟨𝑣, 𝑢⟩) = (𝑥 + 𝑦))
2423eleq1d 2813 . . . . . . 7 (⟨𝑣, 𝑢⟩ = 𝑦 → ((𝑥 + ⟨𝑣, 𝑢⟩) ∈ (R × R) ↔ (𝑥 + 𝑦) ∈ (R × R)))
25 addcnsr 11164 . . . . . . . 8 (((𝑧R𝑤R) ∧ (𝑣R𝑢R)) → (⟨𝑧, 𝑤⟩ + ⟨𝑣, 𝑢⟩) = ⟨(𝑧 +R 𝑣), (𝑤 +R 𝑢)⟩)
26 addclsr 11112 . . . . . . . . . . 11 ((𝑧R𝑣R) → (𝑧 +R 𝑣) ∈ R)
27 addclsr 11112 . . . . . . . . . . 11 ((𝑤R𝑢R) → (𝑤 +R 𝑢) ∈ R)
2826, 27anim12i 611 . . . . . . . . . 10 (((𝑧R𝑣R) ∧ (𝑤R𝑢R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R))
2928an4s 658 . . . . . . . . 9 (((𝑧R𝑤R) ∧ (𝑣R𝑢R)) → ((𝑧 +R 𝑣) ∈ R ∧ (𝑤 +R 𝑢) ∈ R))
30 opelxpi 5717 . . . . . . . . 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 2828 . . . . . . 7 (((𝑧R𝑤R) ∧ (𝑣R𝑢R)) → (⟨𝑧, 𝑤⟩ + ⟨𝑣, 𝑢⟩) ∈ (R × R))
3320, 22, 24, 322optocl 5775 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ (R × R))
3433, 20eleqtrrdi 2839 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
3519, 34oprssdm 7606 . . . 4 (ℂ × ℂ) ⊆ dom +
3618, 35eqssi 3996 . . 3 dom + = (ℂ × ℂ)
37 df-fn 6554 . . 3 ( + Fn (ℂ × ℂ) ↔ (Fun + ∧ dom + = (ℂ × ℂ)))
3815, 36, 37mpbir2an 709 . 2 + Fn (ℂ × ℂ)
3934rgen2 3193 . 2 𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 + 𝑦) ∈ ℂ
40 ffnov 7551 . 2 ( + :(ℂ × ℂ)⟶ℂ ↔ ( + Fn (ℂ × ℂ) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 + 𝑦) ∈ ℂ))
4138, 39, 40mpbir2an 709 1 + :(ℂ × ℂ)⟶ℂ
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wex 1773  wcel 2098  ∃*wmo 2527  wral 3057  cop 4636   × cxp 5678  dom cdm 5680  Fun wfun 6545   Fn wfn 6546  wf 6547  (class class class)co 7424  {coprab 7425  Rcnr 10894   +R cplr 10898  cc 11142   + caddc 11147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-inf2 9670
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-1o 8491  df-oadd 8495  df-omul 8496  df-er 8729  df-ec 8731  df-qs 8735  df-ni 10901  df-pli 10902  df-mi 10903  df-lti 10904  df-plpq 10937  df-mpq 10938  df-ltpq 10939  df-enq 10940  df-nq 10941  df-erq 10942  df-plq 10943  df-mq 10944  df-1nq 10945  df-rq 10946  df-ltnq 10947  df-np 11010  df-plp 11012  df-ltp 11014  df-enr 11084  df-nr 11085  df-plr 11086  df-c 11150  df-add 11155
This theorem is referenced by:  axaddcl  11180
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