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Mirrors > Home > MPE Home > Th. List > addnqf | Structured version Visualization version GIF version |
Description: Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
addnqf | ⊢ +Q :(Q × Q)⟶Q |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqerf 10927 | . . . 4 ⊢ [Q]:(N × N)⟶Q | |
2 | addpqf 10941 | . . . 4 ⊢ +pQ :((N × N) × (N × N))⟶(N × N) | |
3 | fco 6741 | . . . 4 ⊢ (([Q]:(N × N)⟶Q ∧ +pQ :((N × N) × (N × N))⟶(N × N)) → ([Q] ∘ +pQ ):((N × N) × (N × N))⟶Q) | |
4 | 1, 2, 3 | mp2an 690 | . . 3 ⊢ ([Q] ∘ +pQ ):((N × N) × (N × N))⟶Q |
5 | elpqn 10922 | . . . . 5 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N)) | |
6 | 5 | ssriv 3986 | . . . 4 ⊢ Q ⊆ (N × N) |
7 | xpss12 5691 | . . . 4 ⊢ ((Q ⊆ (N × N) ∧ Q ⊆ (N × N)) → (Q × Q) ⊆ ((N × N) × (N × N))) | |
8 | 6, 6, 7 | mp2an 690 | . . 3 ⊢ (Q × Q) ⊆ ((N × N) × (N × N)) |
9 | fssres 6757 | . . 3 ⊢ ((([Q] ∘ +pQ ):((N × N) × (N × N))⟶Q ∧ (Q × Q) ⊆ ((N × N) × (N × N))) → (([Q] ∘ +pQ ) ↾ (Q × Q)):(Q × Q)⟶Q) | |
10 | 4, 8, 9 | mp2an 690 | . 2 ⊢ (([Q] ∘ +pQ ) ↾ (Q × Q)):(Q × Q)⟶Q |
11 | df-plq 10911 | . . 3 ⊢ +Q = (([Q] ∘ +pQ ) ↾ (Q × Q)) | |
12 | 11 | feq1i 6708 | . 2 ⊢ ( +Q :(Q × Q)⟶Q ↔ (([Q] ∘ +pQ ) ↾ (Q × Q)):(Q × Q)⟶Q) |
13 | 10, 12 | mpbir 230 | 1 ⊢ +Q :(Q × Q)⟶Q |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3948 × cxp 5674 ↾ cres 5678 ∘ ccom 5680 ⟶wf 6539 Ncnpi 10841 +pQ cplpq 10845 Qcnq 10849 [Q]cerq 10851 +Q cplq 10852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-omul 8473 df-er 8705 df-ni 10869 df-pli 10870 df-mi 10871 df-lti 10872 df-plpq 10905 df-enq 10908 df-nq 10909 df-erq 10910 df-plq 10911 df-1nq 10913 |
This theorem is referenced by: addcomnq 10948 adderpq 10953 addassnq 10955 distrnq 10958 ltanq 10968 ltexnq 10972 nsmallnq 10974 ltbtwnnq 10975 prlem934 11030 ltaddpr 11031 ltexprlem2 11034 ltexprlem3 11035 ltexprlem4 11036 ltexprlem6 11038 ltexprlem7 11039 prlem936 11044 |
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