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Theorem divfnzn 9205
Description: Division restricted to ℤ × ℕ is a function. Given excluded middle, it would be easy to prove this for ℂ × (ℂ ∖ {0}). The key difference is that an element of is apart from zero, whereas being an element of ℂ ∖ {0} implies being not equal to zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
Assertion
Ref Expression
divfnzn ( / ↾ (ℤ × ℕ)) Fn (ℤ × ℕ)

Proof of Theorem divfnzn
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zcn 8853 . . . . . . 7 (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
21ad2antrr 473 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑥 ∈ ℂ)
3 nncn 8528 . . . . . . 7 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
43ad2antlr 474 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑦 ∈ ℂ)
5 simpr 109 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
6 nnap0 8549 . . . . . . 7 (𝑦 ∈ ℕ → 𝑦 # 0)
76ad2antlr 474 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑦 # 0)
82, 4, 5, 7divmulapd 8376 . . . . 5 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝑥 / 𝑦) = 𝑧 ↔ (𝑦 · 𝑧) = 𝑥))
98riotabidva 5662 . . . 4 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧) = (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
10 eqcom 2097 . . . . . . 7 (𝑧 = (𝑥 / 𝑦) ↔ (𝑥 / 𝑦) = 𝑧)
1110a1i 9 . . . . . 6 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 = (𝑥 / 𝑦) ↔ (𝑥 / 𝑦) = 𝑧))
1211riotabidv 5648 . . . . 5 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦)) = (𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧))
13 simpl 108 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → 𝑥 ∈ ℂ)
143adantl 272 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℂ)
156adantl 272 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → 𝑦 # 0)
1613, 14, 15divclapd 8354 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → (𝑥 / 𝑦) ∈ ℂ)
17 reueq 2828 . . . . . . . 8 ((𝑥 / 𝑦) ∈ ℂ ↔ ∃!𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦))
1816, 17sylib 121 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → ∃!𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦))
19 riotacl 5660 . . . . . . 7 (∃!𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦) → (𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦)) ∈ ℂ)
2018, 19syl 14 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦)) ∈ ℂ)
211, 20sylan 278 . . . . 5 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦)) ∈ ℂ)
2212, 21eqeltrrd 2172 . . . 4 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧) ∈ ℂ)
239, 22eqeltrrd 2172 . . 3 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ)
2423rgen2 2471 . 2 𝑥 ∈ ℤ ∀𝑦 ∈ ℕ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ
25 df-div 8237 . . . . 5 / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
2625reseq1i 4741 . . . 4 ( / ↾ (ℤ × ℕ)) = ((𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) ↾ (ℤ × ℕ))
27 zsscn 8856 . . . . 5 ℤ ⊆ ℂ
28 nncn 8528 . . . . . . 7 (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
29 nnne0 8548 . . . . . . 7 (𝑥 ∈ ℕ → 𝑥 ≠ 0)
30 eldifsn 3589 . . . . . . 7 (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
3128, 29, 30sylanbrc 409 . . . . . 6 (𝑥 ∈ ℕ → 𝑥 ∈ (ℂ ∖ {0}))
3231ssriv 3043 . . . . 5 ℕ ⊆ (ℂ ∖ {0})
33 resmpt2 5781 . . . . 5 ((ℤ ⊆ ℂ ∧ ℕ ⊆ (ℂ ∖ {0})) → ((𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) ↾ (ℤ × ℕ)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)))
3427, 32, 33mp2an 418 . . . 4 ((𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) ↾ (ℤ × ℕ)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
3526, 34eqtri 2115 . . 3 ( / ↾ (ℤ × ℕ)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
3635fnmpt2 6010 . 2 (∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ → ( / ↾ (ℤ × ℕ)) Fn (ℤ × ℕ))
3724, 36ax-mp 7 1 ( / ↾ (ℤ × ℕ)) Fn (ℤ × ℕ)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1296  wcel 1445  wne 2262  wral 2370  ∃!wreu 2372  cdif 3010  wss 3013  {csn 3466   class class class wbr 3867   × cxp 4465  cres 4469   Fn wfn 5044  crio 5645  (class class class)co 5690  cmpt2 5692  cc 7445  0cc0 7447   · cmul 7452   # cap 8155   / cdiv 8236  cn 8520  cz 8848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560
This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-po 4147  df-iso 4148  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-z 8849
This theorem is referenced by:  elq  9206
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