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Theorem dmmulpq 7396
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.
StepHypRef Expression
1 dmoprab 5971 . . 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-mqqs 7366 . . . 4 ·Q = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q ))}
32dmeqi 4842 . . 3 dom ·Q = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q ))}
4 dmaddpqlem 7393 . . . . . . . . 9 (𝑥Q → ∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
5 dmaddpqlem 7393 . . . . . . . . 9 (𝑦Q → ∃𝑢𝑓 𝑦 = [⟨𝑢, 𝑓⟩] ~Q )
64, 5anim12i 338 . . . . . . . 8 ((𝑥Q𝑦Q) → (∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ ∃𝑢𝑓 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ))
7 ee4anv 1945 . . . . . . . 8 (∃𝑤𝑣𝑢𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ↔ (∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ ∃𝑢𝑓 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ))
86, 7sylibr 134 . . . . . . 7 ((𝑥Q𝑦Q) → ∃𝑤𝑣𝑢𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ))
9 enqex 7376 . . . . . . . . . . . . . 14 ~Q ∈ V
10 ecexg 6556 . . . . . . . . . . . . . 14 ( ~Q ∈ V → [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q ∈ V)
119, 10ax-mp 5 . . . . . . . . . . . . 13 [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q ∈ V
1211isseti 2759 . . . . . . . . . . . 12 𝑧 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q
13 ax-ia3 108 . . . . . . . . . . . . 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 1890 . . . . . . . . . . . 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 1611 . . . . . . . . . 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 1700 . . . . . . . . . 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 134 . . . . . . . . 9 (∃𝑢𝑓(𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) → ∃𝑧𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q ))
19182eximi 1611 . . . . . . . 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 1700 . . . . . . . 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 134 . . . . . . 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 391 . . . . 5 ((𝑥Q𝑦Q) ↔ ((𝑥Q𝑦Q) ∧ ∃𝑧𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q )))
24 19.42v 1917 . . . . 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 187 . . . 4 ((𝑥Q𝑦Q) ↔ ∃𝑧((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q )))
2625opabbii 4084 . . 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 2219 . 2 dom ·Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q)}
28 df-xp 4646 . 2 (Q × Q) = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q)}
2927, 28eqtr4i 2212 1 dom ·Q = (Q × Q)
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1363  wex 1502  wcel 2159  Vcvv 2751  cop 3609  {copab 4077   × cxp 4638  dom cdm 4640  (class class class)co 5890  {coprab 5891  [cec 6550   ·pQ cmpq 7293   ~Q ceq 7295  Qcnq 7296   ·Q cmq 7299
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-iinf 4601
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-br 4018  df-opab 4079  df-iom 4604  df-xp 4646  df-cnv 4648  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-oprab 5894  df-ec 6554  df-qs 6558  df-ni 7320  df-enq 7363  df-nqqs 7364  df-mqqs 7366
This theorem is referenced by: (None)
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