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Theorem addnqf 9649
Description: Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
Assertion
Ref Expression
addnqf +Q :(Q × Q)⟶Q

Proof of Theorem addnqf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nqerf 9631 . . . 4 [Q]:(N × N)⟶Q
2 addpqf 9645 . . . 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-plq 9615 . . 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 cplpq 9549  Qcnq 9553  [Q]cerq 9555   +Q cplq 9556
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-pli 9574  df-mi 9575  df-lti 9576  df-plpq 9609  df-enq 9612  df-nq 9613  df-erq 9614  df-plq 9615  df-1nq 9617
This theorem is referenced by:  addcomnq  9652  adderpq  9657  addassnq  9659  distrnq  9662  ltanq  9672  ltexnq  9676  nsmallnq  9678  ltbtwnnq  9679  prlem934  9734  ltaddpr  9735  ltexprlem2  9738  ltexprlem3  9739  ltexprlem4  9740  ltexprlem6  9742  ltexprlem7  9743  prlem936  9748
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