![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 10954 | . . . 4 ⊢ [Q]:(N × N)⟶Q | |
2 | addpqf 10968 | . . . 4 ⊢ +pQ :((N × N) × (N × N))⟶(N × N) | |
3 | fco 6747 | . . . 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 10949 | . . . . 5 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N)) | |
6 | 5 | ssriv 3984 | . . . 4 ⊢ Q ⊆ (N × N) |
7 | xpss12 5693 | . . . 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 6763 | . . 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 10938 | . . 3 ⊢ +Q = (([Q] ∘ +pQ ) ↾ (Q × Q)) | |
12 | 11 | feq1i 6713 | . 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 3947 × cxp 5676 ↾ cres 5680 ∘ ccom 5682 ⟶wf 6544 Ncnpi 10868 +pQ cplpq 10872 Qcnq 10876 [Q]cerq 10878 +Q cplq 10879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-omul 8492 df-er 8725 df-ni 10896 df-pli 10897 df-mi 10898 df-lti 10899 df-plpq 10932 df-enq 10935 df-nq 10936 df-erq 10937 df-plq 10938 df-1nq 10940 |
This theorem is referenced by: addcomnq 10975 adderpq 10980 addassnq 10982 distrnq 10985 ltanq 10995 ltexnq 10999 nsmallnq 11001 ltbtwnnq 11002 prlem934 11057 ltaddpr 11058 ltexprlem2 11061 ltexprlem3 11062 ltexprlem4 11063 ltexprlem6 11065 ltexprlem7 11066 prlem936 11071 |
Copyright terms: Public domain | W3C validator |