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Mirrors > Home > ILE Home > Th. List > nnanq0 | GIF version |
Description: Addition of nonnegative fractions with a common denominator. You can add two fractions with the same denominator by adding their numerators and keeping the same denominator. (Contributed by Jim Kingdon, 1-Dec-2019.) |
Ref | Expression |
---|---|
nnanq0 | ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → [〈(𝑁 +𝑜 𝑀), 𝐴〉] ~Q0 = ([〈𝑁, 𝐴〉] ~Q0 +Q0 [〈𝑀, 𝐴〉] ~Q0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addnnnq0 7006 | . . 3 ⊢ (((𝑁 ∈ ω ∧ 𝐴 ∈ N) ∧ (𝑀 ∈ ω ∧ 𝐴 ∈ N)) → ([〈𝑁, 𝐴〉] ~Q0 +Q0 [〈𝑀, 𝐴〉] ~Q0 ) = [〈((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)), (𝐴 ·𝑜 𝐴)〉] ~Q0 ) | |
2 | 1 | 3impdir 1230 | . 2 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ([〈𝑁, 𝐴〉] ~Q0 +Q0 [〈𝑀, 𝐴〉] ~Q0 ) = [〈((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)), (𝐴 ·𝑜 𝐴)〉] ~Q0 ) |
3 | pinn 6866 | . . . . . . . 8 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
4 | nnmcom 6250 | . . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝐴 ∈ ω) → (𝑁 ·𝑜 𝐴) = (𝐴 ·𝑜 𝑁)) | |
5 | 3, 4 | sylan2 280 | . . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 ·𝑜 𝐴) = (𝐴 ·𝑜 𝑁)) |
6 | 5 | 3adant2 962 | . . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 ·𝑜 𝐴) = (𝐴 ·𝑜 𝑁)) |
7 | 6 | oveq1d 5667 | . . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀))) |
8 | nndi 6247 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀))) | |
9 | 8 | 3coml 1150 | . . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀))) |
10 | 3, 9 | syl3an3 1209 | . . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀))) |
11 | 7, 10 | eqtr4d 2123 | . . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)) = (𝐴 ·𝑜 (𝑁 +𝑜 𝑀))) |
12 | 11 | opeq1d 3628 | . . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → 〈((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)), (𝐴 ·𝑜 𝐴)〉 = 〈(𝐴 ·𝑜 (𝑁 +𝑜 𝑀)), (𝐴 ·𝑜 𝐴)〉) |
13 | 12 | eceq1d 6326 | . 2 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → [〈((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)), (𝐴 ·𝑜 𝐴)〉] ~Q0 = [〈(𝐴 ·𝑜 (𝑁 +𝑜 𝑀)), (𝐴 ·𝑜 𝐴)〉] ~Q0 ) |
14 | simp3 945 | . . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → 𝐴 ∈ N) | |
15 | nnacl 6241 | . . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝑁 +𝑜 𝑀) ∈ ω) | |
16 | 15 | 3adant3 963 | . . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 +𝑜 𝑀) ∈ ω) |
17 | mulcanenq0ec 7002 | . . 3 ⊢ ((𝐴 ∈ N ∧ (𝑁 +𝑜 𝑀) ∈ ω ∧ 𝐴 ∈ N) → [〈(𝐴 ·𝑜 (𝑁 +𝑜 𝑀)), (𝐴 ·𝑜 𝐴)〉] ~Q0 = [〈(𝑁 +𝑜 𝑀), 𝐴〉] ~Q0 ) | |
18 | 14, 16, 14, 17 | syl3anc 1174 | . 2 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → [〈(𝐴 ·𝑜 (𝑁 +𝑜 𝑀)), (𝐴 ·𝑜 𝐴)〉] ~Q0 = [〈(𝑁 +𝑜 𝑀), 𝐴〉] ~Q0 ) |
19 | 2, 13, 18 | 3eqtrrd 2125 | 1 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → [〈(𝑁 +𝑜 𝑀), 𝐴〉] ~Q0 = ([〈𝑁, 𝐴〉] ~Q0 +Q0 [〈𝑀, 𝐴〉] ~Q0 )) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 924 = wceq 1289 ∈ wcel 1438 〈cop 3449 ωcom 4405 (class class class)co 5652 +𝑜 coa 6178 ·𝑜 comu 6179 [cec 6288 Ncnpi 6829 ~Q0 ceq0 6843 +Q0 cplq0 6846 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-iinf 4403 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-id 4120 df-iord 4193 df-on 4195 df-suc 4198 df-iom 4406 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-recs 6070 df-irdg 6135 df-oadd 6185 df-omul 6186 df-er 6290 df-ec 6292 df-qs 6296 df-ni 6861 df-mi 6863 df-enq0 6981 df-nq0 6982 df-plq0 6984 |
This theorem is referenced by: nq02m 7022 prarloclemcalc 7059 |
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