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 Description: Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
Assertion
Ref Expression
dmaddpq dom +Q = (Q × Q)

Dummy variables 𝑥 𝑦 𝑧 𝑣 𝑤 𝑢 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmoprab 5585 . . 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 6447 . . . 4 +Q = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))}
32dmeqi 4536 . . 3 dom +Q = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))}
4 dmaddpqlem 6475 . . . . . . . . 9 (𝑥Q → ∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
5 dmaddpqlem 6475 . . . . . . . . 9 (𝑦Q → ∃𝑢𝑓 𝑦 = [⟨𝑢, 𝑓⟩] ~Q )
64, 5anim12i 321 . . . . . . . 8 ((𝑥Q𝑦Q) → (∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ ∃𝑢𝑓 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ))
7 ee4anv 1809 . . . . . . . 8 (∃𝑤𝑣𝑢𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ↔ (∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ ∃𝑢𝑓 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ))
86, 7sylibr 137 . . . . . . 7 ((𝑥Q𝑦Q) → ∃𝑤𝑣𝑢𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ))
9 enqex 6458 . . . . . . . . . . . . . 14 ~Q ∈ V
10 ecexg 6110 . . . . . . . . . . . . . 14 ( ~Q ∈ V → [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ∈ V)
119, 10ax-mp 7 . . . . . . . . . . . . 13 [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ∈ V
1211isseti 2563 . . . . . . . . . . . 12 𝑧 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q
13 ax-ia3 101 . . . . . . . . . . . . 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 1760 . . . . . . . . . . . 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 1492 . . . . . . . . . 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 1580 . . . . . . . . . 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 137 . . . . . . . . 9 (∃𝑢𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) → ∃𝑧𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))
19182eximi 1492 . . . . . . . 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 1580 . . . . . . . 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 137 . . . . . . 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 371 . . . . 5 ((𝑥Q𝑦Q) ↔ ((𝑥Q𝑦Q) ∧ ∃𝑧𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q )))
24 19.42v 1786 . . . . 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 176 . . . 4 ((𝑥Q𝑦Q) ↔ ∃𝑧((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q )))
2625opabbii 3824 . . 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 2070 . 2 dom +Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q)}
28 df-xp 4351 . 2 (Q × Q) = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q)}
2927, 28eqtr4i 2063 1 dom +Q = (Q × Q)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557  ⟨cop 3378  {copab 3817   × cxp 4343  dom cdm 4345  (class class class)co 5512  {coprab 5513  [cec 6104   +pQ cplpq 6374   ~Q ceq 6377  Qcnq 6378   +Q cplq 6380 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-iom 4314  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-oprab 5516  df-ec 6108  df-qs 6112  df-ni 6402  df-enq 6445  df-nqqs 6446  df-plqqs 6447 This theorem is referenced by: (None)
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