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Theorem dmaddpq 7299
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.
StepHypRef Expression
1 dmoprab 5902 . . 3 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))}
2 df-plqqs 7269 . . . 4 +Q = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))}
32dmeqi 4787 . . 3 dom +Q = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))}
4 dmaddpqlem 7297 . . . . . . . . 9 (𝑥Q → ∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
5 dmaddpqlem 7297 . . . . . . . . 9 (𝑦Q → ∃𝑢𝑓 𝑦 = [⟨𝑢, 𝑓⟩] ~Q )
64, 5anim12i 336 . . . . . . . 8 ((𝑥Q𝑦Q) → (∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ ∃𝑢𝑓 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ))
7 ee4anv 1914 . . . . . . . 8 (∃𝑤𝑣𝑢𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ↔ (∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ ∃𝑢𝑓 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ))
86, 7sylibr 133 . . . . . . 7 ((𝑥Q𝑦Q) → ∃𝑤𝑣𝑢𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ))
9 enqex 7280 . . . . . . . . . . . . . 14 ~Q ∈ V
10 ecexg 6484 . . . . . . . . . . . . . 14 ( ~Q ∈ V → [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ∈ V)
119, 10ax-mp 5 . . . . . . . . . . . . 13 [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ∈ V
1211isseti 2720 . . . . . . . . . . . 12 𝑧 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q
13 ax-ia3 107 . . . . . . . . . . . . 13 ((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) → (𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q → ((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q )))
1413eximdv 1860 . . . . . . . . . . . 12 ((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) → (∃𝑧 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q → ∃𝑧((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q )))
1512, 14mpi 15 . . . . . . . . . . 11 ((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) → ∃𝑧((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))
16152eximi 1581 . . . . . . . . . 10 (∃𝑢𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) → ∃𝑢𝑓𝑧((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))
17 exrot3 1670 . . . . . . . . . 10 (∃𝑧𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ) ↔ ∃𝑢𝑓𝑧((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))
1816, 17sylibr 133 . . . . . . . . 9 (∃𝑢𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) → ∃𝑧𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))
19182eximi 1581 . . . . . . . 8 (∃𝑤𝑣𝑢𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) → ∃𝑤𝑣𝑧𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))
20 exrot3 1670 . . . . . . . 8 (∃𝑧𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ) ↔ ∃𝑤𝑣𝑧𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))
2119, 20sylibr 133 . . . . . . 7 (∃𝑤𝑣𝑢𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) → ∃𝑧𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))
228, 21syl 14 . . . . . 6 ((𝑥Q𝑦Q) → ∃𝑧𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))
2322pm4.71i 389 . . . . 5 ((𝑥Q𝑦Q) ↔ ((𝑥Q𝑦Q) ∧ ∃𝑧𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q )))
24 19.42v 1886 . . . . 5 (∃𝑧((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q )) ↔ ((𝑥Q𝑦Q) ∧ ∃𝑧𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q )))
2523, 24bitr4i 186 . . . 4 ((𝑥Q𝑦Q) ↔ ∃𝑧((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q )))
2625opabbii 4031 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))}
271, 3, 263eqtr4i 2188 . 2 dom +Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q)}
28 df-xp 4592 . 2 (Q × Q) = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q)}
2927, 28eqtr4i 2181 1 dom +Q = (Q × Q)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1335  wex 1472  wcel 2128  Vcvv 2712  cop 3563  {copab 4024   × cxp 4584  dom cdm 4586  (class class class)co 5824  {coprab 5825  [cec 6478   +pQ cplpq 7196   ~Q ceq 7199  Qcnq 7200   +Q cplq 7202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-iom 4550  df-xp 4592  df-cnv 4594  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-oprab 5828  df-ec 6482  df-qs 6486  df-ni 7224  df-enq 7267  df-nqqs 7268  df-plqqs 7269
This theorem is referenced by: (None)
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