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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 10956 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵))) | |
2 | elpqn 10940 | . . . 4 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
3 | elpqn 10940 | . . . 4 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
4 | mulpqf 10961 | . . . . 5 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N) | |
5 | 4 | fovcl 7543 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ∈ (N × N)) |
6 | 2, 3, 5 | syl2an 595 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·pQ 𝐵) ∈ (N × N)) |
7 | nqercl 10946 | . . 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 2828 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) ∈ Q) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 × cxp 5670 ‘cfv 6542 (class class class)co 7414 Ncnpi 10859 ·pQ cmpq 10864 Qcnq 10867 [Q]cerq 10869 ·Q cmq 10871 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-omul 8485 df-er 8718 df-ni 10887 df-mi 10889 df-lti 10890 df-mpq 10924 df-enq 10926 df-nq 10927 df-erq 10928 df-mq 10930 df-1nq 10931 |
This theorem is referenced by: ltrnq 10994 mpv 11026 dmmp 11028 mulclprlem 11034 mulclpr 11035 mulasspr 11039 distrlem1pr 11040 distrlem4pr 11041 distrlem5pr 11042 1idpr 11044 prlem934 11048 prlem936 11062 reclem3pr 11064 reclem4pr 11065 |
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