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Theorem dmaddpq 7341
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 5934 . . 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 7311 . . . 4 +Q = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))}
32dmeqi 4812 . . 3 dom +Q = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))}
4 dmaddpqlem 7339 . . . . . . . . 9 (𝑥Q → ∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
5 dmaddpqlem 7339 . . . . . . . . 9 (𝑦Q → ∃𝑢𝑓 𝑦 = [⟨𝑢, 𝑓⟩] ~Q )
64, 5anim12i 336 . . . . . . . 8 ((𝑥Q𝑦Q) → (∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ ∃𝑢𝑓 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ))
7 ee4anv 1927 . . . . . . . 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 7322 . . . . . . . . . . . . . 14 ~Q ∈ V
10 ecexg 6517 . . . . . . . . . . . . . 14 ( ~Q ∈ V → [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ∈ V)
119, 10ax-mp 5 . . . . . . . . . . . . 13 [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ∈ V
1211isseti 2738 . . . . . . . . . . . 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 1873 . . . . . . . . . . . 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 1594 . . . . . . . . . 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 1683 . . . . . . . . . 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 1594 . . . . . . . 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 1683 . . . . . . . 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 1899 . . . . 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 4056 . . 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 2201 . 2 dom +Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q)}
28 df-xp 4617 . 2 (Q × Q) = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q)}
2927, 28eqtr4i 2194 1 dom +Q = (Q × Q)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1348  wex 1485  wcel 2141  Vcvv 2730  cop 3586  {copab 4049   × cxp 4609  dom cdm 4611  (class class class)co 5853  {coprab 5854  [cec 6511   +pQ cplpq 7238   ~Q ceq 7241  Qcnq 7242   +Q cplq 7244
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-iom 4575  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-oprab 5857  df-ec 6515  df-qs 6519  df-ni 7266  df-enq 7309  df-nqqs 7310  df-plqqs 7311
This theorem is referenced by: (None)
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