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Mirrors > Home > MPE Home > Th. List > mulclnq | Structured version Visualization version GIF version |
Description: Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulclnq | ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) ∈ Q) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulpqnq 10098 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵))) | |
2 | elpqn 10082 | . . . 4 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
3 | elpqn 10082 | . . . 4 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
4 | mulpqf 10103 | . . . . 5 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N) | |
5 | 4 | fovcl 7042 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ∈ (N × N)) |
6 | 2, 3, 5 | syl2an 589 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·pQ 𝐵) ∈ (N × N)) |
7 | nqercl 10088 | . . 3 ⊢ ((𝐴 ·pQ 𝐵) ∈ (N × N) → ([Q]‘(𝐴 ·pQ 𝐵)) ∈ Q) | |
8 | 6, 7 | syl 17 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ([Q]‘(𝐴 ·pQ 𝐵)) ∈ Q) |
9 | 1, 8 | eqeltrd 2859 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) ∈ Q) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2107 × cxp 5353 ‘cfv 6135 (class class class)co 6922 Ncnpi 10001 ·pQ cmpq 10006 Qcnq 10009 [Q]cerq 10011 ·Q cmq 10013 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-omul 7848 df-er 8026 df-ni 10029 df-mi 10031 df-lti 10032 df-mpq 10066 df-enq 10068 df-nq 10069 df-erq 10070 df-mq 10072 df-1nq 10073 |
This theorem is referenced by: ltrnq 10136 mpv 10168 dmmp 10170 mulclprlem 10176 mulclpr 10177 mulasspr 10181 distrlem1pr 10182 distrlem4pr 10183 distrlem5pr 10184 1idpr 10186 prlem934 10190 prlem936 10204 reclem3pr 10206 reclem4pr 10207 |
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