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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 6414 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +𝑜 𝐵)) | |
2 | pinn 6407 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
3 | pinn 6407 | . . . . 5 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
4 | nnacl 6059 | . . . . 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 6412 | . . . . 5 ⊢ (𝐵 ∈ N ↔ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵)) | |
7 | nnaordi 6081 | . . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝐵 → (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵))) | |
8 | ne0i 3230 | . . . . . . . 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 6406 | . . . 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 2114 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) ∈ N) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1393 ≠ wne 2204 ∅c0 3224 ωcom 4313 (class class class)co 5512 +𝑜 coa 5998 Ncnpi 6370 +N cpli 6371 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-ni 6402 df-pli 6403 |
This theorem is referenced by: addasspig 6428 distrpig 6431 ltapig 6436 1lt2pi 6438 indpi 6440 addcmpblnq 6465 addpipqqslem 6467 addclnq 6473 addassnqg 6480 distrnqg 6485 ltanqg 6498 1lt2nq 6504 ltexnqq 6506 archnqq 6515 prarloclemarch2 6517 nqnq0a 6552 nntopi 6968 |
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