Theorem divfnzn 9845

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 9474	. . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
2	1	ad2antrr 488	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑥 ∈ ℂ)
3		nncn 9141	. . . . . . 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 9162	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 # 0)
7	6	ad2antlr 489	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑦 # 0)
8	2, 4, 5, 7	divmulapd 8982	. . . . 5 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝑥 / 𝑦) = 𝑧 ↔ (𝑦 · 𝑧) = 𝑥))
9	8	riotabidva 5984	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧) = (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
10		eqcom 2231	. . . . . . 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 5968	. . . . 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 8960	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℕ) → (𝑥 / 𝑦) ∈ ℂ)
17		reueq 3003	. . . . . . . 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 5982	. . . . . . 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 2307	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ (𝑥 / 𝑦) = 𝑧) ∈ ℂ)
23	9, 22	eqeltrrd 2307	. . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ)
24	23	rgen2 2616	. 2 ⊢ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ ℂ
25		df-div 8843	. . . . 5 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
26	25	reseq1i 5007	. . . 4 ⊢ ( / ↾ (ℤ × ℕ)) = ((𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) ↾ (ℤ × ℕ))
27		zsscn 9477	. . . . 5 ⊢ ℤ ⊆ ℂ
28		nncn 9141	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
29		nnne0 9161	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0)
30		eldifsn 3798	. . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
31	28, 29, 30	sylanbrc 417	. . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℂ ∖ {0}))
32	31	ssriv 3229	. . . . 5 ⊢ ℕ ⊆ (ℂ ∖ {0})
33		resmpo 6114	. . . . 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 2250	. . 3 ⊢ ( / ↾ (ℤ × ℕ)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
36	35	fnmpo 6362	. 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 1395   ∈ wcel 2200   ≠ wne 2400  ∀wral 2508  ∃!wreu 2510   ∖ cdif 3195   ⊆ wss 3198  {csn 3667   class class class wbr 4086   × cxp 4721   ↾ cres 4725   Fn wfn 5319  ℩crio 5965  (class class class)co 6013   ∈ cmpo 6015  ℂcc 8020  0cc0 8022   · cmul 8027   # cap 8751   / cdiv 8842  ℕcn 9133  ℤcz 9469

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-po 4391  df-iso 4392  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-z 9470

This theorem is referenced by:  elq  9846  qnnen  13042