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Theorem dmmulpq 6364
 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 5527 . . 3 dom {⟨⟨x, y⟩, z⟩ ∣ ((x Q y Q) wvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q ))} = {⟨x, y⟩ ∣ z((x Q y Q) wvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q ))}
2 df-mqqs 6334 . . . 4 ·Q = {⟨⟨x, y⟩, z⟩ ∣ ((x Q y Q) wvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q ))}
32dmeqi 4479 . . 3 dom ·Q = dom {⟨⟨x, y⟩, z⟩ ∣ ((x Q y Q) wvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q ))}
4 dmaddpqlem 6361 . . . . . . . . 9 (x Qwv x = [⟨w, v⟩] ~Q )
5 dmaddpqlem 6361 . . . . . . . . 9 (y Quf y = [⟨u, f⟩] ~Q )
64, 5anim12i 321 . . . . . . . 8 ((x Q y Q) → (wv x = [⟨w, v⟩] ~Q uf y = [⟨u, f⟩] ~Q ))
7 ee4anv 1806 . . . . . . . 8 (wvuf(x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) ↔ (wv x = [⟨w, v⟩] ~Q uf y = [⟨u, f⟩] ~Q ))
86, 7sylibr 137 . . . . . . 7 ((x Q y Q) → wvuf(x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ))
9 enqex 6344 . . . . . . . . . . . . . 14 ~Q V
10 ecexg 6046 . . . . . . . . . . . . . 14 ( ~Q V → [(⟨w, v⟩ ·pQu, f⟩)] ~Q V)
119, 10ax-mp 7 . . . . . . . . . . . . 13 [(⟨w, v⟩ ·pQu, f⟩)] ~Q V
1211isseti 2557 . . . . . . . . . . . 12 z z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q
13 ax-ia3 101 . . . . . . . . . . . . 13 ((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) → (z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q → ((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q )))
1413eximdv 1757 . . . . . . . . . . . 12 ((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) → (z z = [(⟨w, v⟩ ·pQu, f⟩)] ~Qz((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q )))
1512, 14mpi 15 . . . . . . . . . . 11 ((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) → z((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q ))
16152eximi 1489 . . . . . . . . . 10 (uf(x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) → ufz((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q ))
17 exrot3 1577 . . . . . . . . . 10 (zuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q ) ↔ ufz((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q ))
1816, 17sylibr 137 . . . . . . . . 9 (uf(x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) → zuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q ))
19182eximi 1489 . . . . . . . 8 (wvuf(x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) → wvzuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q ))
20 exrot3 1577 . . . . . . . 8 (zwvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q ) ↔ wvzuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q ))
2119, 20sylibr 137 . . . . . . 7 (wvuf(x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) → zwvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q ))
228, 21syl 14 . . . . . 6 ((x Q y Q) → zwvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q ))
2322pm4.71i 371 . . . . 5 ((x Q y Q) ↔ ((x Q y Q) zwvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q )))
24 19.42v 1783 . . . . 5 (z((x Q y Q) wvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q )) ↔ ((x Q y Q) zwvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q )))
2523, 24bitr4i 176 . . . 4 ((x Q y Q) ↔ z((x Q y Q) wvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q )))
2625opabbii 3815 . . 3 {⟨x, y⟩ ∣ (x Q y Q)} = {⟨x, y⟩ ∣ z((x Q y Q) wvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q ))}
271, 3, 263eqtr4i 2067 . 2 dom ·Q = {⟨x, y⟩ ∣ (x Q y Q)}
28 df-xp 4294 . 2 (Q × Q) = {⟨x, y⟩ ∣ (x Q y Q)}
2927, 28eqtr4i 2060 1 dom ·Q = (Q × Q)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370  {copab 3808   × cxp 4286  dom cdm 4288  (class class class)co 5455  {coprab 5456  [cec 6040   ·pQ cmpq 6261   ~Q ceq 6263  Qcnq 6264   ·Q cmq 6267 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-opab 3810  df-iom 4257  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-oprab 5459  df-ec 6044  df-qs 6048  df-ni 6288  df-enq 6331  df-nqqs 6332  df-mqqs 6334 This theorem is referenced by: (None)
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