Theorem divfnzn 9695

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 9331	. . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
2	1	ad2antrr 488	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑥 ∈ ℂ)
3		nncn 8998	. . . . . . 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 9019	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 # 0)
7	6	ad2antlr 489	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑦 # 0)
8	2, 4, 5, 7	divmulapd 8839	. . . . 5 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝑥 / 𝑦) = 𝑧 ↔ (𝑦 · 𝑧) = 𝑥))
9	8	riotabidva 5894	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧) = (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
10		eqcom 2198	. . . . . . 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 5879	. . . . 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 8817	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → (𝑥 / 𝑦) ∈ ℂ)
17		reueq 2963	. . . . . . . 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 5892	. . . . . . 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 2274	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧) ∈ ℂ)
23	9, 22	eqeltrrd 2274	. . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ)
24	23	rgen2 2583	. 2 ⊢ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ
25		df-div 8700	. . . . 5 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
26	25	reseq1i 4942	. . . 4 ⊢ ( / ↾ (ℤ × ℕ)) = ((𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) ↾ (ℤ × ℕ))
27		zsscn 9334	. . . . 5 ⊢ ℤ ⊆ ℂ
28		nncn 8998	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
29		nnne0 9018	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0)
30		eldifsn 3749	. . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
31	28, 29, 30	sylanbrc 417	. . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℂ ∖ {0}))
32	31	ssriv 3187	. . . . 5 ⊢ ℕ ⊆ (ℂ ∖ {0})
33		resmpo 6020	. . . . 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 2217	. . 3 ⊢ ( / ↾ (ℤ × ℕ)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
36	35	fnmpo 6260	. 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 2167   ≠ wne 2367  ∀wral 2475  ∃!wreu 2477   ∖ cdif 3154   ⊆ wss 3157  {csn 3622   class class class wbr 4033   × cxp 4661   ↾ cres 4665   Fn wfn 5253  ℩crio 5876  (class class class)co 5922   ∈ cmpo 5924  ℂcc 7877  0cc0 7879   · cmul 7884   # cap 8608   / cdiv 8699  ℕcn 8990  ℤcz 9326

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-z 9327

This theorem is referenced by:  elq  9696  qnnen  12648