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Mirrors > Home > MPE Home > Th. List > addclnq | Structured version Visualization version GIF version |
Description: Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
addclnq | ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) ∈ Q) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addpqnq 10935 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵))) | |
2 | elpqn 10922 | . . . 4 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
3 | elpqn 10922 | . . . 4 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
4 | addpqf 10941 | . . . . 5 ⊢ +pQ :((N × N) × (N × N))⟶(N × N) | |
5 | 4 | fovcl 7533 | . . . 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 10928 | . . 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 2827 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) ∈ Q) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 × cxp 5667 ‘cfv 6537 (class class class)co 7405 Ncnpi 10841 +pQ cplpq 10845 Qcnq 10849 [Q]cerq 10851 +Q cplq 10852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-omul 8472 df-er 8705 df-ni 10869 df-pli 10870 df-mi 10871 df-lti 10872 df-plpq 10905 df-enq 10908 df-nq 10909 df-erq 10910 df-plq 10911 df-1nq 10913 |
This theorem is referenced by: halfnq 10973 plpv 11007 dmplp 11009 addclprlem2 11014 addclpr 11015 addasspr 11019 distrlem1pr 11022 distrlem4pr 11023 distrlem5pr 11024 ltaddpr 11031 ltexprlem6 11038 ltexprlem7 11039 prlem936 11044 |
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