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Theorem mulnqf 9809
Description: Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
Assertion
Ref Expression
mulnqf ·Q :(Q × Q)⟶Q

Proof of Theorem mulnqf
StepHypRef Expression
1 nqerf 9790 . . . 4 [Q]:(N × N)⟶Q
2 mulpqf 9806 . . . 4 ·pQ :((N × N) × (N × N))⟶(N × N)
3 fco 6096 . . . 4 (([Q]:(N × N)⟶Q ∧ ·pQ :((N × N) × (N × N))⟶(N × N)) → ([Q] ∘ ·pQ ):((N × N) × (N × N))⟶Q)
41, 2, 3mp2an 708 . . 3 ([Q] ∘ ·pQ ):((N × N) × (N × N))⟶Q
5 elpqn 9785 . . . . 5 (𝑥Q𝑥 ∈ (N × N))
65ssriv 3640 . . . 4 Q ⊆ (N × N)
7 xpss12 5158 . . . 4 ((Q ⊆ (N × N) ∧ Q ⊆ (N × N)) → (Q × Q) ⊆ ((N × N) × (N × N)))
86, 6, 7mp2an 708 . . 3 (Q × Q) ⊆ ((N × N) × (N × N))
9 fssres 6108 . . 3 ((([Q] ∘ ·pQ ):((N × N) × (N × N))⟶Q ∧ (Q × Q) ⊆ ((N × N) × (N × N))) → (([Q] ∘ ·pQ ) ↾ (Q × Q)):(Q × Q)⟶Q)
104, 8, 9mp2an 708 . 2 (([Q] ∘ ·pQ ) ↾ (Q × Q)):(Q × Q)⟶Q
11 df-mq 9775 . . 3 ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q))
1211feq1i 6074 . 2 ( ·Q :(Q × Q)⟶Q ↔ (([Q] ∘ ·pQ ) ↾ (Q × Q)):(Q × Q)⟶Q)
1310, 12mpbir 221 1 ·Q :(Q × Q)⟶Q
Colors of variables: wff setvar class
Syntax hints:  wss 3607   × cxp 5141  cres 5145  ccom 5147  wf 5922  Ncnpi 9704   ·pQ cmpq 9709  Qcnq 9712  [Q]cerq 9714   ·Q cmq 9716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ni 9732  df-mi 9734  df-lti 9735  df-mpq 9769  df-enq 9771  df-nq 9772  df-erq 9773  df-mq 9775  df-1nq 9776
This theorem is referenced by:  mulcomnq  9813  mulerpq  9817  mulassnq  9819  distrnq  9821  recmulnq  9824  recclnq  9826  dmrecnq  9828  ltmnq  9832  prlem936  9907
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