Theorem divfnzn 9623

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 9260	. . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
2	1	ad2antrr 488	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑥 ∈ ℂ)
3		nncn 8929	. . . . . . 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 8950	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 # 0)
7	6	ad2antlr 489	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑦 # 0)
8	2, 4, 5, 7	divmulapd 8771	. . . . 5 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝑥 / 𝑦) = 𝑧 ↔ (𝑦 · 𝑧) = 𝑥))
9	8	riotabidva 5849	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧) = (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
10		eqcom 2179	. . . . . . 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 5835	. . . . 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 8749	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → (𝑥 / 𝑦) ∈ ℂ)
17		reueq 2938	. . . . . . . 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 5847	. . . . . . 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 2255	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧) ∈ ℂ)
23	9, 22	eqeltrrd 2255	. . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ)
24	23	rgen2 2563	. 2 ⊢ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ
25		df-div 8632	. . . . 5 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
26	25	reseq1i 4905	. . . 4 ⊢ ( / ↾ (ℤ × ℕ)) = ((𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) ↾ (ℤ × ℕ))
27		zsscn 9263	. . . . 5 ⊢ ℤ ⊆ ℂ
28		nncn 8929	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
29		nnne0 8949	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0)
30		eldifsn 3721	. . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
31	28, 29, 30	sylanbrc 417	. . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℂ ∖ {0}))
32	31	ssriv 3161	. . . . 5 ⊢ ℕ ⊆ (ℂ ∖ {0})
33		resmpo 5975	. . . . 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 2198	. . 3 ⊢ ( / ↾ (ℤ × ℕ)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
36	35	fnmpo 6205	. 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 1353   ∈ wcel 2148   ≠ wne 2347  ∀wral 2455  ∃!wreu 2457   ∖ cdif 3128   ⊆ wss 3131  {csn 3594   class class class wbr 4005   × cxp 4626   ↾ cres 4630   Fn wfn 5213  ℩crio 5832  (class class class)co 5877   ∈ cmpo 5879  ℂcc 7811  0cc0 7813   · cmul 7818   # cap 8540   / cdiv 8631  ℕcn 8921  ℤcz 9255

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-z 9256

This theorem is referenced by:  elq  9624  qnnen  12434