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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 10873 | . . . 4 ⊢ [Q]:(N × N)⟶Q | |
2 | addpqf 10887 | . . . 4 ⊢ +pQ :((N × N) × (N × N))⟶(N × N) | |
3 | fco 6697 | . . . 4 ⊢ (([Q]:(N × N)⟶Q ∧ +pQ :((N × N) × (N × N))⟶(N × N)) → ([Q] ∘ +pQ ):((N × N) × (N × N))⟶Q) | |
4 | 1, 2, 3 | mp2an 691 | . . 3 ⊢ ([Q] ∘ +pQ ):((N × N) × (N × N))⟶Q |
5 | elpqn 10868 | . . . . 5 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N)) | |
6 | 5 | ssriv 3953 | . . . 4 ⊢ Q ⊆ (N × N) |
7 | xpss12 5653 | . . . 4 ⊢ ((Q ⊆ (N × N) ∧ Q ⊆ (N × N)) → (Q × Q) ⊆ ((N × N) × (N × N))) | |
8 | 6, 6, 7 | mp2an 691 | . . 3 ⊢ (Q × Q) ⊆ ((N × N) × (N × N)) |
9 | fssres 6713 | . . 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 691 | . 2 ⊢ (([Q] ∘ +pQ ) ↾ (Q × Q)):(Q × Q)⟶Q |
11 | df-plq 10857 | . . 3 ⊢ +Q = (([Q] ∘ +pQ ) ↾ (Q × Q)) | |
12 | 11 | feq1i 6664 | . 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 3915 × cxp 5636 ↾ cres 5640 ∘ ccom 5642 ⟶wf 6497 Ncnpi 10787 +pQ cplpq 10791 Qcnq 10795 [Q]cerq 10797 +Q cplq 10798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-omul 8422 df-er 8655 df-ni 10815 df-pli 10816 df-mi 10817 df-lti 10818 df-plpq 10851 df-enq 10854 df-nq 10855 df-erq 10856 df-plq 10857 df-1nq 10859 |
This theorem is referenced by: addcomnq 10894 adderpq 10899 addassnq 10901 distrnq 10904 ltanq 10914 ltexnq 10918 nsmallnq 10920 ltbtwnnq 10921 prlem934 10976 ltaddpr 10977 ltexprlem2 10980 ltexprlem3 10981 ltexprlem4 10982 ltexprlem6 10984 ltexprlem7 10985 prlem936 10990 |
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