Theorem dmmulpq 7440

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 5999	. . 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 7410	. . . 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 4863	. . 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 7437	. . . . . . . . 9 ⊢ (𝑥 ∈ Q → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
5		dmaddpqlem 7437	. . . . . . . . 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 1950	. . . . . . . 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 7420	. . . . . . . . . . . . . 14 ⊢ ~Q ∈ V
10		ecexg 6591	. . . . . . . . . . . . . 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 2768	. . . . . . . . . . . 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 1891	. . . . . . . . . . . 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 1612	. . . . . . . . . 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 1701	. . . . . . . . . 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 1612	. . . . . . . 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 1701	. . . . . . . 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 391	. . . . 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 1918	. . . . 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 4096	. . 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 2224	. 2 ⊢ dom ·Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)}
28		df-xp 4665	. 2 ⊢ (Q × Q) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)}
29	27, 28	eqtr4i 2217	1 ⊢ dom ·Q = (Q × Q)

Syntax hints:   ∧ wa 104   = wceq 1364  ∃wex 1503   ∈ wcel 2164  Vcvv 2760  ⟨cop 3621  {copab 4089   × cxp 4657  dom cdm 4659  (class class class)co 5918  {coprab 5919  [cec 6585   ·_pQ cmpq 7337   ~_Q ceq 7339  Qcnq 7340   ·_Q cmq 7343

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-iom 4623  df-xp 4665  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-oprab 5922  df-ec 6589  df-qs 6593  df-ni 7364  df-enq 7407  df-nqqs 7408  df-mqqs 7410

This theorem is referenced by: (None)