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Theorem addnqf 9730
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
StepHypRef Expression
1 nqerf 9712 . . . 4 [Q]:(N × N)⟶Q
2 addpqf 9726 . . . 4 +pQ :((N × N) × (N × N))⟶(N × N)
3 fco 6025 . . . 4 (([Q]:(N × N)⟶Q ∧ +pQ :((N × N) × (N × N))⟶(N × N)) → ([Q] ∘ +pQ ):((N × N) × (N × N))⟶Q)
41, 2, 3mp2an 707 . . 3 ([Q] ∘ +pQ ):((N × N) × (N × N))⟶Q
5 elpqn 9707 . . . . 5 (𝑥Q𝑥 ∈ (N × N))
65ssriv 3592 . . . 4 Q ⊆ (N × N)
7 xpss12 5196 . . . 4 ((Q ⊆ (N × N) ∧ Q ⊆ (N × N)) → (Q × Q) ⊆ ((N × N) × (N × N)))
86, 6, 7mp2an 707 . . 3 (Q × Q) ⊆ ((N × N) × (N × N))
9 fssres 6037 . . 3 ((([Q] ∘ +pQ ):((N × N) × (N × N))⟶Q ∧ (Q × Q) ⊆ ((N × N) × (N × N))) → (([Q] ∘ +pQ ) ↾ (Q × Q)):(Q × Q)⟶Q)
104, 8, 9mp2an 707 . 2 (([Q] ∘ +pQ ) ↾ (Q × Q)):(Q × Q)⟶Q
11 df-plq 9696 . . 3 +Q = (([Q] ∘ +pQ ) ↾ (Q × Q))
1211feq1i 6003 . 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 3560   × cxp 5082  cres 5086  ccom 5088  wf 5853  Ncnpi 9626   +pQ cplpq 9630  Qcnq 9634  [Q]cerq 9636   +Q cplq 9637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-omul 7525  df-er 7702  df-ni 9654  df-pli 9655  df-mi 9656  df-lti 9657  df-plpq 9690  df-enq 9693  df-nq 9694  df-erq 9695  df-plq 9696  df-1nq 9698
This theorem is referenced by:  addcomnq  9733  adderpq  9738  addassnq  9740  distrnq  9743  ltanq  9753  ltexnq  9757  nsmallnq  9759  ltbtwnnq  9760  prlem934  9815  ltaddpr  9816  ltexprlem2  9819  ltexprlem3  9820  ltexprlem4  9821  ltexprlem6  9823  ltexprlem7  9824  prlem936  9829
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