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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 10097 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵))) | |
2 | elpqn 10084 | . . . 4 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
3 | elpqn 10084 | . . . 4 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
4 | addpqf 10103 | . . . . 5 ⊢ +pQ :((N × N) × (N × N))⟶(N × N) | |
5 | 4 | fovcl 7044 | . . . 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 10090 | . . 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 5355 ‘cfv 6137 (class class class)co 6924 Ncnpi 10003 +pQ cplpq 10007 Qcnq 10011 [Q]cerq 10013 +Q cplq 10014 |
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 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
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 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-omul 7850 df-er 8028 df-ni 10031 df-pli 10032 df-mi 10033 df-lti 10034 df-plpq 10067 df-enq 10070 df-nq 10071 df-erq 10072 df-plq 10073 df-1nq 10075 |
This theorem is referenced by: halfnq 10135 plpv 10169 dmplp 10171 addclprlem2 10176 addclpr 10177 addasspr 10181 distrlem1pr 10184 distrlem4pr 10185 distrlem5pr 10186 ltaddpr 10193 ltexprlem6 10200 ltexprlem7 10201 prlem936 10206 |
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