Theorem dmaddpq 7642

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

Assertion

Ref	Expression
dmaddpq	⊢ dom +Q = (Q × Q)

Proof of Theorem dmaddpq

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

Step	Hyp	Ref	Expression
1		dmoprab 6112	. . 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-plqqs 7612	. . . 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 4938	. . 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 7640	. . . . . . . . 9 ⊢ (𝑥 ∈ Q → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
5		dmaddpqlem 7640	. . . . . . . . 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 1987	. . . . . . . 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 7623	. . . . . . . . . . . . . 14 ⊢ ~Q ∈ V
10		ecexg 6749	. . . . . . . . . . . . . 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 2812	. . . . . . . . . . . 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 1928	. . . . . . . . . . . 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 1650	. . . . . . . . . 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 1738	. . . . . . . . . 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 1650	. . . . . . . 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 1738	. . . . . . . 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 1955	. . . . 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 4161	. . 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 2262	. 2 ⊢ dom +Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)}
28		df-xp 4737	. 2 ⊢ (Q × Q) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)}
29	27, 28	eqtr4i 2255	1 ⊢ dom +Q = (Q × Q)

Syntax hints:   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2202  Vcvv 2803  ⟨cop 3676  {copab 4154   × cxp 4729  dom cdm 4731  (class class class)co 6028  {coprab 6029  [cec 6743   +_pQ cplpq 7539   ~_Q ceq 7542  Qcnq 7543   +_Q cplq 7545

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-iom 4695  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-oprab 6032  df-ec 6747  df-qs 6751  df-ni 7567  df-enq 7610  df-nqqs 7611  df-plqqs 7612

This theorem is referenced by: (None)