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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 9790 | . . . 4 ⊢ [Q]:(N × N)⟶Q | |
2 | mulpqf 9806 | . . . 4 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N) | |
3 | fco 6096 | . . . 4 ⊢ (([Q]:(N × N)⟶Q ∧ ·pQ :((N × N) × (N × N))⟶(N × N)) → ([Q] ∘ ·pQ ):((N × N) × (N × N))⟶Q) | |
4 | 1, 2, 3 | mp2an 708 | . . 3 ⊢ ([Q] ∘ ·pQ ):((N × N) × (N × N))⟶Q |
5 | elpqn 9785 | . . . . 5 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N)) | |
6 | 5 | ssriv 3640 | . . . 4 ⊢ Q ⊆ (N × N) |
7 | xpss12 5158 | . . . 4 ⊢ ((Q ⊆ (N × N) ∧ Q ⊆ (N × N)) → (Q × Q) ⊆ ((N × N) × (N × N))) | |
8 | 6, 6, 7 | mp2an 708 | . . 3 ⊢ (Q × Q) ⊆ ((N × N) × (N × N)) |
9 | fssres 6108 | . . 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 708 | . 2 ⊢ (([Q] ∘ ·pQ ) ↾ (Q × Q)):(Q × Q)⟶Q |
11 | df-mq 9775 | . . 3 ⊢ ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q)) | |
12 | 11 | feq1i 6074 | . 2 ⊢ ( ·Q :(Q × Q)⟶Q ↔ (([Q] ∘ ·pQ ) ↾ (Q × Q)):(Q × Q)⟶Q) |
13 | 10, 12 | mpbir 221 | 1 ⊢ ·Q :(Q × Q)⟶Q |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3607 × cxp 5141 ↾ cres 5145 ∘ ccom 5147 ⟶wf 5922 Ncnpi 9704 ·pQ cmpq 9709 Qcnq 9712 [Q]cerq 9714 ·Q cmq 9716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-omul 7610 df-er 7787 df-ni 9732 df-mi 9734 df-lti 9735 df-mpq 9769 df-enq 9771 df-nq 9772 df-erq 9773 df-mq 9775 df-1nq 9776 |
This theorem is referenced by: mulcomnq 9813 mulerpq 9817 mulassnq 9819 distrnq 9821 recmulnq 9824 recclnq 9826 dmrecnq 9828 ltmnq 9832 prlem936 9907 |
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