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Theorem divfnzn 9263
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 8911 . . . . . . 7 (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
21ad2antrr 475 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑥 ∈ ℂ)
3 nncn 8586 . . . . . . 7 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
43ad2antlr 476 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑦 ∈ ℂ)
5 simpr 109 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
6 nnap0 8607 . . . . . . 7 (𝑦 ∈ ℕ → 𝑦 # 0)
76ad2antlr 476 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑦 # 0)
82, 4, 5, 7divmulapd 8433 . . . . 5 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝑥 / 𝑦) = 𝑧 ↔ (𝑦 · 𝑧) = 𝑥))
98riotabidva 5678 . . . 4 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧) = (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
10 eqcom 2102 . . . . . . 7 (𝑧 = (𝑥 / 𝑦) ↔ (𝑥 / 𝑦) = 𝑧)
1110a1i 9 . . . . . 6 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 = (𝑥 / 𝑦) ↔ (𝑥 / 𝑦) = 𝑧))
1211riotabidv 5664 . . . . 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 273 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℂ)
156adantl 273 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → 𝑦 # 0)
1613, 14, 15divclapd 8411 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → (𝑥 / 𝑦) ∈ ℂ)
17 reueq 2836 . . . . . . . 8 ((𝑥 / 𝑦) ∈ ℂ ↔ ∃!𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦))
1816, 17sylib 121 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → ∃!𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦))
19 riotacl 5676 . . . . . . 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 279 . . . . 5 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦)) ∈ ℂ)
2212, 21eqeltrrd 2177 . . . 4 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧) ∈ ℂ)
239, 22eqeltrrd 2177 . . 3 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ)
2423rgen2 2477 . 2 𝑥 ∈ ℤ ∀𝑦 ∈ ℕ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ
25 df-div 8294 . . . . 5 / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
2625reseq1i 4751 . . . 4 ( / ↾ (ℤ × ℕ)) = ((𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) ↾ (ℤ × ℕ))
27 zsscn 8914 . . . . 5 ℤ ⊆ ℂ
28 nncn 8586 . . . . . . 7 (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
29 nnne0 8606 . . . . . . 7 (𝑥 ∈ ℕ → 𝑥 ≠ 0)
30 eldifsn 3597 . . . . . . 7 (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
3128, 29, 30sylanbrc 411 . . . . . 6 (𝑥 ∈ ℕ → 𝑥 ∈ (ℂ ∖ {0}))
3231ssriv 3051 . . . . 5 ℕ ⊆ (ℂ ∖ {0})
33 resmpo 5801 . . . . 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 420 . . . 4 ((𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) ↾ (ℤ × ℕ)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
3526, 34eqtri 2120 . . 3 ( / ↾ (ℤ × ℕ)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
3635fnmpo 6030 . 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 1299  wcel 1448  wne 2267  wral 2375  ∃!wreu 2377  cdif 3018  wss 3021  {csn 3474   class class class wbr 3875   × cxp 4475  cres 4479   Fn wfn 5054  crio 5661  (class class class)co 5706  cmpo 5708  cc 7498  0cc0 7500   · cmul 7505   # cap 8209   / cdiv 8293  cn 8578  cz 8906
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 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613
This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-po 4156  df-iso 4157  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-z 8907
This theorem is referenced by:  elq  9264  qnnen  11736
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