Theorem divfnzn 9741

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 9376	. . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
2	1	ad2antrr 488	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑥 ∈ ℂ)
3		nncn 9043	. . . . . . 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 9064	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 # 0)
7	6	ad2antlr 489	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑦 # 0)
8	2, 4, 5, 7	divmulapd 8884	. . . . 5 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝑥 / 𝑦) = 𝑧 ↔ (𝑦 · 𝑧) = 𝑥))
9	8	riotabidva 5915	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧) = (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
10		eqcom 2206	. . . . . . 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 5900	. . . . 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 8862	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → (𝑥 / 𝑦) ∈ ℂ)
17		reueq 2971	. . . . . . . 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 5913	. . . . . . 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 2282	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧) ∈ ℂ)
23	9, 22	eqeltrrd 2282	. . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ)
24	23	rgen2 2591	. 2 ⊢ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ
25		df-div 8745	. . . . 5 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
26	25	reseq1i 4954	. . . 4 ⊢ ( / ↾ (ℤ × ℕ)) = ((𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) ↾ (ℤ × ℕ))
27		zsscn 9379	. . . . 5 ⊢ ℤ ⊆ ℂ
28		nncn 9043	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
29		nnne0 9063	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0)
30		eldifsn 3759	. . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
31	28, 29, 30	sylanbrc 417	. . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℂ ∖ {0}))
32	31	ssriv 3196	. . . . 5 ⊢ ℕ ⊆ (ℂ ∖ {0})
33		resmpo 6042	. . . . 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 2225	. . 3 ⊢ ( / ↾ (ℤ × ℕ)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
36	35	fnmpo 6287	. 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 1372   ∈ wcel 2175   ≠ wne 2375  ∀wral 2483  ∃!wreu 2485   ∖ cdif 3162   ⊆ wss 3165  {csn 3632   class class class wbr 4043   × cxp 4672   ↾ cres 4676   Fn wfn 5265  ℩crio 5897  (class class class)co 5943   ∈ cmpo 5945  ℂcc 7922  0cc0 7924   · cmul 7929   # cap 8653   / cdiv 8744  ℕcn 9035  ℤcz 9371

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-po 4342  df-iso 4343  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-z 9372

This theorem is referenced by:  elq  9742  qnnen  12773