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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 7257 | . . 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 1272 | . 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 7117 | . . . . . . . 8 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
4 | nnmcom 6385 | . . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝐴 ∈ ω) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁)) | |
5 | 3, 4 | sylan2 284 | . . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁)) |
6 | 5 | 3adant2 1000 | . . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁)) |
7 | 6 | oveq1d 5789 | . . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀))) |
8 | nndi 6382 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀))) | |
9 | 8 | 3coml 1188 | . . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀))) |
10 | 3, 9 | syl3an3 1251 | . . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀))) |
11 | 7, 10 | eqtr4d 2175 | . . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)) = (𝐴 ·o (𝑁 +o 𝑀))) |
12 | 11 | opeq1d 3711 | . . 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 6465 | . 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 983 | . . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → 𝐴 ∈ N) | |
15 | nnacl 6376 | . . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝑁 +o 𝑀) ∈ ω) | |
16 | 15 | 3adant3 1001 | . . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 +o 𝑀) ∈ ω) |
17 | mulcanenq0ec 7253 | . . 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 1216 | . 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 2177 | 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 962 = wceq 1331 ∈ wcel 1480 〈cop 3530 ωcom 4504 (class class class)co 5774 +o coa 6310 ·o comu 6311 [cec 6427 Ncnpi 7080 ~Q0 ceq0 7094 +Q0 cplq0 7097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-mi 7114 df-enq0 7232 df-nq0 7233 df-plq0 7235 |
This theorem is referenced by: nq02m 7273 prarloclemcalc 7310 |
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