![]() |
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 10948 | . . . 4 ⊢ [Q]:(N × N)⟶Q | |
2 | mulpqf 10964 | . . . 4 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N) | |
3 | fco 6742 | . . . 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 10943 | . . . . 5 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N)) | |
6 | 5 | ssriv 3983 | . . . 4 ⊢ Q ⊆ (N × N) |
7 | xpss12 5688 | . . . 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 6758 | . . 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 10933 | . . 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 3945 × cxp 5671 ↾ cres 5675 ∘ ccom 5677 ⟶wf 6539 Ncnpi 10862 ·pQ cmpq 10867 Qcnq 10870 [Q]cerq 10872 ·Q cmq 10874 |
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 5294 ax-nul 5301 ax-pr 5424 ax-un 7735 |
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 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-oadd 8485 df-omul 8486 df-er 8719 df-ni 10890 df-mi 10892 df-lti 10893 df-mpq 10927 df-enq 10929 df-nq 10930 df-erq 10931 df-mq 10933 df-1nq 10934 |
This theorem is referenced by: mulcomnq 10971 mulerpq 10975 mulassnq 10977 distrnq 10979 recmulnq 10982 recclnq 10984 dmrecnq 10986 ltmnq 10990 prlem936 11065 |
Copyright terms: Public domain | W3C validator |