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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 10357 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵))) | |
2 | elpqn 10341 | . . . 4 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
3 | elpqn 10341 | . . . 4 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
4 | mulpqf 10362 | . . . . 5 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N) | |
5 | 4 | fovcl 7273 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ∈ (N × N)) |
6 | 2, 3, 5 | syl2an 597 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·pQ 𝐵) ∈ (N × N)) |
7 | nqercl 10347 | . . 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 2913 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) ∈ Q) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 × cxp 5548 ‘cfv 6350 (class class class)co 7150 Ncnpi 10260 ·pQ cmpq 10265 Qcnq 10268 [Q]cerq 10270 ·Q cmq 10272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-ni 10288 df-mi 10290 df-lti 10291 df-mpq 10325 df-enq 10327 df-nq 10328 df-erq 10329 df-mq 10331 df-1nq 10332 |
This theorem is referenced by: ltrnq 10395 mpv 10427 dmmp 10429 mulclprlem 10435 mulclpr 10436 mulasspr 10440 distrlem1pr 10441 distrlem4pr 10442 distrlem5pr 10443 1idpr 10445 prlem934 10449 prlem936 10463 reclem3pr 10465 reclem4pr 10466 |
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