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Theorem mulnqf 9650
 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
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nqerf 9631 . . . 4 [Q]:(N × N)⟶Q
2 mulpqf 9647 . . . 4 ·pQ :((N × N) × (N × N))⟶(N × N)
3 fco 5971 . . . 4 (([Q]:(N × N)⟶Q ∧ ·pQ :((N × N) × (N × N))⟶(N × N)) → ([Q] ∘ ·pQ ):((N × N) × (N × N))⟶Q)
41, 2, 3mp2an 704 . . 3 ([Q] ∘ ·pQ ):((N × N) × (N × N))⟶Q
5 elpqn 9626 . . . . 5 (𝑥Q𝑥 ∈ (N × N))
65ssriv 3572 . . . 4 Q ⊆ (N × N)
7 xpss12 5148 . . . 4 ((Q ⊆ (N × N) ∧ Q ⊆ (N × N)) → (Q × Q) ⊆ ((N × N) × (N × N)))
86, 6, 7mp2an 704 . . 3 (Q × Q) ⊆ ((N × N) × (N × N))
9 fssres 5983 . . 3 ((([Q] ∘ ·pQ ):((N × N) × (N × N))⟶Q ∧ (Q × Q) ⊆ ((N × N) × (N × N))) → (([Q] ∘ ·pQ ) ↾ (Q × Q)):(Q × Q)⟶Q)
104, 8, 9mp2an 704 . 2 (([Q] ∘ ·pQ ) ↾ (Q × Q)):(Q × Q)⟶Q
11 df-mq 9616 . . 3 ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q))
1211feq1i 5949 . 2 ( ·Q :(Q × Q)⟶Q ↔ (([Q] ∘ ·pQ ) ↾ (Q × Q)):(Q × Q)⟶Q)
1310, 12mpbir 220 1 ·Q :(Q × Q)⟶Q
 Colors of variables: wff setvar class Syntax hints:   ⊆ wss 3540   × cxp 5036   ↾ cres 5040   ∘ ccom 5042  ⟶wf 5800  Ncnpi 9545   ·pQ cmpq 9550  Qcnq 9553  [Q]cerq 9555   ·Q cmq 9557 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ni 9573  df-mi 9575  df-lti 9576  df-mpq 9610  df-enq 9612  df-nq 9613  df-erq 9614  df-mq 9616  df-1nq 9617 This theorem is referenced by:  mulcomnq  9654  mulerpq  9658  mulassnq  9660  distrnq  9662  recmulnq  9665  recclnq  9667  dmrecnq  9669  ltmnq  9673  prlem936  9748
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