Theorem divfnzn 9686

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.

Step	Hyp	Ref	Expression
1		zcn 9322	. . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
2	1	ad2antrr 488	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑥 ∈ ℂ)
3		nncn 8990	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
4	3	ad2antlr 489	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑦 ∈ ℂ)
5		simpr 110	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
6		nnap0 9011	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 # 0)
7	6	ad2antlr 489	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑦 # 0)
8	2, 4, 5, 7	divmulapd 8831	. . . . 5 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝑥 / 𝑦) = 𝑧 ↔ (𝑦 · 𝑧) = 𝑥))
9	8	riotabidva 5890	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧) = (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
10		eqcom 2195	. . . . . . 7 ⊢ (𝑧 = (𝑥 / 𝑦) ↔ (𝑥 / 𝑦) = 𝑧)
11	10	a1i 9	. . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑧 = (𝑥 / 𝑦) ↔ (𝑥 / 𝑦) = 𝑧))
12	11	riotabidv 5875	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦)) = (℩𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧))
13		simpl 109	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → 𝑥 ∈ ℂ)
14	3	adantl 277	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℂ)
15	6	adantl 277	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → 𝑦 # 0)
16	13, 14, 15	divclapd 8809	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → (𝑥 / 𝑦) ∈ ℂ)
17		reueq 2959	. . . . . . . 8 ⊢ ((𝑥 / 𝑦) ∈ ℂ ↔ ∃!𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦))
18	16, 17	sylib 122	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → ∃!𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦))
19		riotacl 5888	. . . . . . 7 ⊢ (∃!𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦) → (℩𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦)) ∈ ℂ)
20	18, 19	syl 14	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦)) ∈ ℂ)
21	1, 20	sylan 283	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ 𝑧 = (𝑥 / 𝑦)) ∈ ℂ)
22	12, 21	eqeltrrd 2271	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧) ∈ ℂ)
23	9, 22	eqeltrrd 2271	. . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ)
24	23	rgen2 2580	. 2 ⊢ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ
25		df-div 8692	. . . . 5 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
26	25	reseq1i 4938	. . . 4 ⊢ ( / ↾ (ℤ × ℕ)) = ((𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) ↾ (ℤ × ℕ))
27		zsscn 9325	. . . . 5 ⊢ ℤ ⊆ ℂ
28		nncn 8990	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
29		nnne0 9010	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0)
30		eldifsn 3745	. . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
31	28, 29, 30	sylanbrc 417	. . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℂ ∖ {0}))
32	31	ssriv 3183	. . . . 5 ⊢ ℕ ⊆ (ℂ ∖ {0})
33		resmpo 6016	. . . . 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)))
34	27, 32, 33	mp2an 426	. . . 4 ⊢ ((𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) ↾ (ℤ × ℕ)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
35	26, 34	eqtri 2214	. . 3 ⊢ ( / ↾ (ℤ × ℕ)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
36	35	fnmpo 6255	. 2 ⊢ (∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ → ( / ↾ (ℤ × ℕ)) Fn (ℤ × ℕ))
37	24, 36	ax-mp 5	1 ⊢ ( / ↾ (ℤ × ℕ)) Fn (ℤ × ℕ)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164   ≠ wne 2364  ∀wral 2472  ∃!wreu 2474   ∖ cdif 3150   ⊆ wss 3153  {csn 3618   class class class wbr 4029   × cxp 4657   ↾ cres 4661   Fn wfn 5249  ℩crio 5872  (class class class)co 5918   ∈ cmpo 5920  ℂcc 7870  0cc0 7872   · cmul 7877   # cap 8600   / cdiv 8691  ℕcn 8982  ℤcz 9317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-po 4327  df-iso 4328  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-z 9318

This theorem is referenced by:  elq  9687  qnnen  12588