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Mirrors > Home > ILE Home > Th. List > addclpi | GIF version |
Description: Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.) |
Ref | Expression |
---|---|
addclpi | ⊢ ((A ∈ N ∧ B ∈ N) → (A +N B) ∈ N) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addpiord 6300 | . 2 ⊢ ((A ∈ N ∧ B ∈ N) → (A +N B) = (A +𝑜 B)) | |
2 | pinn 6293 | . . 3 ⊢ (A ∈ N → A ∈ 𝜔) | |
3 | pinn 6293 | . . . . 5 ⊢ (B ∈ N → B ∈ 𝜔) | |
4 | nnacl 5998 | . . . . 5 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (A +𝑜 B) ∈ 𝜔) | |
5 | 3, 4 | sylan2 270 | . . . 4 ⊢ ((A ∈ 𝜔 ∧ B ∈ N) → (A +𝑜 B) ∈ 𝜔) |
6 | elni2 6298 | . . . . 5 ⊢ (B ∈ N ↔ (B ∈ 𝜔 ∧ ∅ ∈ B)) | |
7 | nnaordi 6017 | . . . . . . . 8 ⊢ ((B ∈ 𝜔 ∧ A ∈ 𝜔) → (∅ ∈ B → (A +𝑜 ∅) ∈ (A +𝑜 B))) | |
8 | ne0i 3224 | . . . . . . . 8 ⊢ ((A +𝑜 ∅) ∈ (A +𝑜 B) → (A +𝑜 B) ≠ ∅) | |
9 | 7, 8 | syl6 29 | . . . . . . 7 ⊢ ((B ∈ 𝜔 ∧ A ∈ 𝜔) → (∅ ∈ B → (A +𝑜 B) ≠ ∅)) |
10 | 9 | expcom 109 | . . . . . 6 ⊢ (A ∈ 𝜔 → (B ∈ 𝜔 → (∅ ∈ B → (A +𝑜 B) ≠ ∅))) |
11 | 10 | imp32 244 | . . . . 5 ⊢ ((A ∈ 𝜔 ∧ (B ∈ 𝜔 ∧ ∅ ∈ B)) → (A +𝑜 B) ≠ ∅) |
12 | 6, 11 | sylan2b 271 | . . . 4 ⊢ ((A ∈ 𝜔 ∧ B ∈ N) → (A +𝑜 B) ≠ ∅) |
13 | elni 6292 | . . . 4 ⊢ ((A +𝑜 B) ∈ N ↔ ((A +𝑜 B) ∈ 𝜔 ∧ (A +𝑜 B) ≠ ∅)) | |
14 | 5, 12, 13 | sylanbrc 394 | . . 3 ⊢ ((A ∈ 𝜔 ∧ B ∈ N) → (A +𝑜 B) ∈ N) |
15 | 2, 14 | sylan 267 | . 2 ⊢ ((A ∈ N ∧ B ∈ N) → (A +𝑜 B) ∈ N) |
16 | 1, 15 | eqeltrd 2111 | 1 ⊢ ((A ∈ N ∧ B ∈ N) → (A +N B) ∈ N) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 ≠ wne 2201 ∅c0 3218 𝜔com 4256 (class class class)co 5455 +𝑜 coa 5937 Ncnpi 6256 +N cpli 6257 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-id 4021 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-oadd 5944 df-ni 6288 df-pli 6289 |
This theorem is referenced by: addasspig 6314 distrpig 6317 ltapig 6322 1lt2pi 6324 indpi 6326 addcmpblnq 6351 addpipqqslem 6353 addclnq 6359 addassnqg 6366 distrnqg 6371 ltanqg 6384 1lt2nq 6389 ltexnqq 6391 archnqq 6400 prarloclemarch2 6402 nqnq0a 6437 |
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