![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mulnqf | Structured version Visualization version GIF version |
Description: Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulnqf | ⊢ ·Q :(Q × Q)⟶Q |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqerf 10873 | . . . 4 ⊢ [Q]:(N × N)⟶Q | |
2 | mulpqf 10889 | . . . 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-mq 10858 | . . 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 cmpq 10792 Qcnq 10795 [Q]cerq 10797 ·Q cmq 10799 |
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-mi 10817 df-lti 10818 df-mpq 10852 df-enq 10854 df-nq 10855 df-erq 10856 df-mq 10858 df-1nq 10859 |
This theorem is referenced by: mulcomnq 10896 mulerpq 10900 mulassnq 10902 distrnq 10904 recmulnq 10907 recclnq 10909 dmrecnq 10911 ltmnq 10915 prlem936 10990 |
Copyright terms: Public domain | W3C validator |