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Theorem divfnzn 9436
 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 ℕ 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 9079 . . . . . . 7 (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
21ad2antrr 480 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑥 ∈ ℂ)
3 nncn 8748 . . . . . . 7 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
43ad2antlr 481 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑦 ∈ ℂ)
5 simpr 109 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
6 nnap0 8769 . . . . . . 7 (𝑦 ∈ ℕ → 𝑦 # 0)
76ad2antlr 481 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑦 # 0)
82, 4, 5, 7divmulapd 8592 . . . . 5 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝑥 / 𝑦) = 𝑧 ↔ (𝑦 · 𝑧) = 𝑥))
98riotabidva 5750 . . . 4 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧) = (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
10 eqcom 2142 . . . . . . 7 (𝑧 = (𝑥 / 𝑦) ↔ (𝑥 / 𝑦) = 𝑧)
1110a1i 9 . . . . . 6 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 = (𝑥 / 𝑦) ↔ (𝑥 / 𝑦) = 𝑧))
1211riotabidv 5736 . . . . 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 275 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℂ)
156adantl 275 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → 𝑦 # 0)
1613, 14, 15divclapd 8570 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → (𝑥 / 𝑦) ∈ ℂ)
17 reueq 2884 . . . . . . . 8 ((𝑥 / 𝑦) ∈ ℂ ↔ ∃!𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦))
1816, 17sylib 121 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → ∃!𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦))
19 riotacl 5748 . . . . . . 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 281 . . . . 5 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦)) ∈ ℂ)
2212, 21eqeltrrd 2218 . . . 4 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧) ∈ ℂ)
239, 22eqeltrrd 2218 . . 3 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ)
2423rgen2 2519 . 2 𝑥 ∈ ℤ ∀𝑦 ∈ ℕ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ
25 df-div 8453 . . . . 5 / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
2625reseq1i 4819 . . . 4 ( / ↾ (ℤ × ℕ)) = ((𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) ↾ (ℤ × ℕ))
27 zsscn 9082 . . . . 5 ℤ ⊆ ℂ
28 nncn 8748 . . . . . . 7 (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
29 nnne0 8768 . . . . . . 7 (𝑥 ∈ ℕ → 𝑥 ≠ 0)
30 eldifsn 3654 . . . . . . 7 (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
3128, 29, 30sylanbrc 414 . . . . . 6 (𝑥 ∈ ℕ → 𝑥 ∈ (ℂ ∖ {0}))
3231ssriv 3102 . . . . 5 ℕ ⊆ (ℂ ∖ {0})
33 resmpo 5873 . . . . 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 423 . . . 4 ((𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) ↾ (ℤ × ℕ)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
3526, 34eqtri 2161 . . 3 ( / ↾ (ℤ × ℕ)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
3635fnmpo 6104 . 2 (∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ → ( / ↾ (ℤ × ℕ)) Fn (ℤ × ℕ))
3724, 36ax-mp 5 1 ( / ↾ (ℤ × ℕ)) Fn (ℤ × ℕ)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481   ≠ wne 2309  ∀wral 2417  ∃!wreu 2419   ∖ cdif 3069   ⊆ wss 3072  {csn 3528   class class class wbr 3933   × cxp 4541   ↾ cres 4545   Fn wfn 5122  ℩crio 5733  (class class class)co 5778   ∈ cmpo 5780  ℂcc 7638  0cc0 7640   · cmul 7645   # cap 8363   / cdiv 8452  ℕcn 8740  ℤcz 9074 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-cnex 7731  ax-resscn 7732  ax-1cn 7733  ax-1re 7734  ax-icn 7735  ax-addcl 7736  ax-addrcl 7737  ax-mulcl 7738  ax-mulrcl 7739  ax-addcom 7740  ax-mulcom 7741  ax-addass 7742  ax-mulass 7743  ax-distr 7744  ax-i2m1 7745  ax-0lt1 7746  ax-1rid 7747  ax-0id 7748  ax-rnegex 7749  ax-precex 7750  ax-cnre 7751  ax-pre-ltirr 7752  ax-pre-ltwlin 7753  ax-pre-lttrn 7754  ax-pre-apti 7755  ax-pre-ltadd 7756  ax-pre-mulgt0 7757  ax-pre-mulext 7758 This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-id 4219  df-po 4222  df-iso 4223  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-fv 5135  df-riota 5734  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-pnf 7822  df-mnf 7823  df-xr 7824  df-ltxr 7825  df-le 7826  df-sub 7955  df-neg 7956  df-reap 8357  df-ap 8364  df-div 8453  df-inn 8741  df-z 9075 This theorem is referenced by:  elq  9437  qnnen  11971
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