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Theorem dmmulpq 7181
 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 5845 . . 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 7151 . . . 4 ·Q = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q ))}
32dmeqi 4735 . . 3 dom ·Q = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q ))}
4 dmaddpqlem 7178 . . . . . . . . 9 (𝑥Q → ∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
5 dmaddpqlem 7178 . . . . . . . . 9 (𝑦Q → ∃𝑢𝑓 𝑦 = [⟨𝑢, 𝑓⟩] ~Q )
64, 5anim12i 336 . . . . . . . 8 ((𝑥Q𝑦Q) → (∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ ∃𝑢𝑓 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ))
7 ee4anv 1904 . . . . . . . 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 7161 . . . . . . . . . . . . . 14 ~Q ∈ V
10 ecexg 6426 . . . . . . . . . . . . . 14 ( ~Q ∈ V → [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q ∈ V)
119, 10ax-mp 5 . . . . . . . . . . . . 13 [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q ∈ V
1211isseti 2689 . . . . . . . . . . . 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 1852 . . . . . . . . . . . 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 1580 . . . . . . . . . 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 1668 . . . . . . . . . 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 1580 . . . . . . . 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 1668 . . . . . . . 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 388 . . . . 5 ((𝑥Q𝑦Q) ↔ ((𝑥Q𝑦Q) ∧ ∃𝑧𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q )))
24 19.42v 1878 . . . . 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 3990 . . 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 2168 . 2 dom ·Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q)}
28 df-xp 4540 . 2 (Q × Q) = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q)}
2927, 28eqtr4i 2161 1 dom ·Q = (Q × Q)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2681  ⟨cop 3525  {copab 3983   × cxp 4532  dom cdm 4534  (class class class)co 5767  {coprab 5768  [cec 6420   ·pQ cmpq 7078   ~Q ceq 7080  Qcnq 7081   ·Q cmq 7084 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-iinf 4497 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-iom 4500  df-xp 4540  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-oprab 5771  df-ec 6424  df-qs 6428  df-ni 7105  df-enq 7148  df-nqqs 7149  df-mqqs 7151 This theorem is referenced by: (None)
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