Theorem dmmulpq 7747

Description: Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)

Assertion

Ref	Expression
dmmulpq	⊢ dom ·Q = (Q × Q)

Proof of Theorem dmmulpq

Dummy variables 𝑥 𝑦 𝑧 𝑣 𝑤 𝑢 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmoprab 6169	. . 3 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ))}
2		df-mqqs 7717	. . . 4 ⊢ ·Q = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ))}
3	2	dmeqi 4982	. . 3 ⊢ dom ·Q = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ))}
4		dmaddpqlem 7744	. . . . . . . . 9 ⊢ (𝑥 ∈ Q → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
5		dmaddpqlem 7744	. . . . . . . . 9 ⊢ (𝑦 ∈ Q → ∃𝑢∃𝑓 𝑦 = [⟨𝑢, 𝑓⟩] ~Q )
6	4, 5	anim12i 338	. . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ ∃𝑢∃𝑓 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ))
7		ee4anv 1994	. . . . . . . 8 ⊢ (∃𝑤∃𝑣∃𝑢∃𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ↔ (∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ ∃𝑢∃𝑓 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ))
8	6, 7	sylibr 134	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ∃𝑤∃𝑣∃𝑢∃𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ))
9		enqex 7727	. . . . . . . . . . . . . 14 ⊢ ~Q ∈ V
10		ecexg 6811	. . . . . . . . . . . . . 14 ⊢ ( ~Q ∈ V → [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ∈ V)
11	9, 10	ax-mp 5	. . . . . . . . . . . . 13 ⊢ [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ∈ V
12	11	isseti 2830	. . . . . . . . . . . 12 ⊢ ∃𝑧 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q
13		ax-ia3 108	. . . . . . . . . . . . 13 ⊢ ((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) → (𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q → ((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q )))
14	13	eximdv 1933	. . . . . . . . . . . 12 ⊢ ((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) → (∃𝑧 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q → ∃𝑧((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q )))
15	12, 14	mpi 15	. . . . . . . . . . 11 ⊢ ((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) → ∃𝑧((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ))
16	15	2eximi 1654	. . . . . . . . . 10 ⊢ (∃𝑢∃𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) → ∃𝑢∃𝑓∃𝑧((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ))
17		exrot3 1742	. . . . . . . . . 10 ⊢ (∃𝑧∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ) ↔ ∃𝑢∃𝑓∃𝑧((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ))
18	16, 17	sylibr 134	. . . . . . . . 9 ⊢ (∃𝑢∃𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) → ∃𝑧∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ))
19	18	2eximi 1654	. . . . . . . 8 ⊢ (∃𝑤∃𝑣∃𝑢∃𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) → ∃𝑤∃𝑣∃𝑧∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ))
20		exrot3 1742	. . . . . . . 8 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ) ↔ ∃𝑤∃𝑣∃𝑧∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ))
21	19, 20	sylibr 134	. . . . . . 7 ⊢ (∃𝑤∃𝑣∃𝑢∃𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) → ∃𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ))
22	8, 21	syl 14	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ∃𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ))
23	22	pm4.71i 395	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ↔ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q )))
24		19.42v 1962	. . . . 5 ⊢ (∃𝑧((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q )) ↔ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q )))
25	23, 24	bitr4i 187	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ↔ ∃𝑧((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q )))
26	25	opabbii 4198	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ))}
27	1, 3, 26	3eqtr4i 2269	. 2 ⊢ dom ·Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)}
28		df-xp 4780	. 2 ⊢ (Q × Q) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)}
29	27, 28	eqtr4i 2262	1 ⊢ dom ·Q = (Q × Q)

This proof depends on syntax axioms:   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821  ⟨cop 3712  {copab 4191   × cxp 4772  dom cdm 4774  (class class class)co 6085  {coprab 6086  [cec 6805   ·_pQ cmpq 7644   ~_Q ceq 7646  Qcnq 7647   ·_Q cmq 7650

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-iom 4738  df-xp 4780  df-cnv 4782  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-oprab 6089  df-ec 6809  df-qs 6813  df-ni 7671  df-enq 7714  df-nqqs 7715  df-mqqs 7717

This theorem is used by: (None)