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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 7479 | . . 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 1305 | . 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 7339 | . . . . . . . 8 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
4 | nnmcom 6515 | . . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝐴 ∈ ω) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁)) | |
5 | 3, 4 | sylan2 286 | . . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁)) |
6 | 5 | 3adant2 1018 | . . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁)) |
7 | 6 | oveq1d 5912 | . . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀))) |
8 | nndi 6512 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀))) | |
9 | 8 | 3coml 1212 | . . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀))) |
10 | 3, 9 | syl3an3 1284 | . . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀))) |
11 | 7, 10 | eqtr4d 2225 | . . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)) = (𝐴 ·o (𝑁 +o 𝑀))) |
12 | 11 | opeq1d 3799 | . . 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 6596 | . 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 1001 | . . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → 𝐴 ∈ N) | |
15 | nnacl 6506 | . . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝑁 +o 𝑀) ∈ ω) | |
16 | 15 | 3adant3 1019 | . . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 +o 𝑀) ∈ ω) |
17 | mulcanenq0ec 7475 | . . 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 1249 | . 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 2227 | 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 980 = wceq 1364 ∈ wcel 2160 〈cop 3610 ωcom 4607 (class class class)co 5897 +o coa 6439 ·o comu 6440 [cec 6558 Ncnpi 7302 ~Q0 ceq0 7316 +Q0 cplq0 7319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-irdg 6396 df-oadd 6446 df-omul 6447 df-er 6560 df-ec 6562 df-qs 6566 df-ni 7334 df-mi 7336 df-enq0 7454 df-nq0 7455 df-plq0 7457 |
This theorem is referenced by: nq02m 7495 prarloclemcalc 7532 |
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