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Mirrors > Home > ILE Home > Th. List > addassnq0lemcl | GIF version |
Description: A natural number closure law. Lemma for addassnq0 6560. (Contributed by Jim Kingdon, 3-Dec-2019.) |
Ref | Expression |
---|---|
addassnq0lemcl | ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ N) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ N)) → (((𝐼 ·𝑜 𝐿) +𝑜 (𝐽 ·𝑜 𝐾)) ∈ ω ∧ (𝐽 ·𝑜 𝐿) ∈ N)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 6407 | . . . . 5 ⊢ (𝐿 ∈ N → 𝐿 ∈ ω) | |
2 | nnmcl 6060 | . . . . 5 ⊢ ((𝐼 ∈ ω ∧ 𝐿 ∈ ω) → (𝐼 ·𝑜 𝐿) ∈ ω) | |
3 | 1, 2 | sylan2 270 | . . . 4 ⊢ ((𝐼 ∈ ω ∧ 𝐿 ∈ N) → (𝐼 ·𝑜 𝐿) ∈ ω) |
4 | 3 | ad2ant2rl 480 | . . 3 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ N) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ N)) → (𝐼 ·𝑜 𝐿) ∈ ω) |
5 | pinn 6407 | . . . . 5 ⊢ (𝐽 ∈ N → 𝐽 ∈ ω) | |
6 | nnmcl 6060 | . . . . 5 ⊢ ((𝐽 ∈ ω ∧ 𝐾 ∈ ω) → (𝐽 ·𝑜 𝐾) ∈ ω) | |
7 | 5, 6 | sylan 267 | . . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ ω) → (𝐽 ·𝑜 𝐾) ∈ ω) |
8 | 7 | ad2ant2lr 479 | . . 3 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ N) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ N)) → (𝐽 ·𝑜 𝐾) ∈ ω) |
9 | nnacl 6059 | . . 3 ⊢ (((𝐼 ·𝑜 𝐿) ∈ ω ∧ (𝐽 ·𝑜 𝐾) ∈ ω) → ((𝐼 ·𝑜 𝐿) +𝑜 (𝐽 ·𝑜 𝐾)) ∈ ω) | |
10 | 4, 8, 9 | syl2anc 391 | . 2 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ N) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ N)) → ((𝐼 ·𝑜 𝐿) +𝑜 (𝐽 ·𝑜 𝐾)) ∈ ω) |
11 | mulpiord 6415 | . . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐿 ∈ N) → (𝐽 ·N 𝐿) = (𝐽 ·𝑜 𝐿)) | |
12 | mulclpi 6426 | . . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐿 ∈ N) → (𝐽 ·N 𝐿) ∈ N) | |
13 | 11, 12 | eqeltrrd 2115 | . . 3 ⊢ ((𝐽 ∈ N ∧ 𝐿 ∈ N) → (𝐽 ·𝑜 𝐿) ∈ N) |
14 | 13 | ad2ant2l 477 | . 2 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ N) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ N)) → (𝐽 ·𝑜 𝐿) ∈ N) |
15 | 10, 14 | jca 290 | 1 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ N) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ N)) → (((𝐼 ·𝑜 𝐿) +𝑜 (𝐽 ·𝑜 𝐾)) ∈ ω ∧ (𝐽 ·𝑜 𝐿) ∈ N)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1393 ωcom 4313 (class class class)co 5512 +𝑜 coa 5998 ·𝑜 comu 5999 Ncnpi 6370 ·N cmi 6372 |
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-omul 6006 df-ni 6402 df-mi 6404 |
This theorem is referenced by: addassnq0 6560 |
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