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Mirrors > Home > ILE Home > Th. List > addpipqqslem | GIF version |
Description: Lemma for addpipqqs 6354. (Contributed by Jim Kingdon, 11-Sep-2019.) |
Ref | Expression |
---|---|
addpipqqslem | ⊢ (((A ∈ N ∧ B ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → 〈((A ·N 𝐷) +N (B ·N 𝐶)), (B ·N 𝐷)〉 ∈ (N × N)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulclpi 6312 | . . . 4 ⊢ ((A ∈ N ∧ 𝐷 ∈ N) → (A ·N 𝐷) ∈ N) | |
2 | mulclpi 6312 | . . . 4 ⊢ ((B ∈ N ∧ 𝐶 ∈ N) → (B ·N 𝐶) ∈ N) | |
3 | addclpi 6311 | . . . 4 ⊢ (((A ·N 𝐷) ∈ N ∧ (B ·N 𝐶) ∈ N) → ((A ·N 𝐷) +N (B ·N 𝐶)) ∈ N) | |
4 | 1, 2, 3 | syl2an 273 | . . 3 ⊢ (((A ∈ N ∧ 𝐷 ∈ N) ∧ (B ∈ N ∧ 𝐶 ∈ N)) → ((A ·N 𝐷) +N (B ·N 𝐶)) ∈ N) |
5 | 4 | an42s 523 | . 2 ⊢ (((A ∈ N ∧ B ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((A ·N 𝐷) +N (B ·N 𝐶)) ∈ N) |
6 | mulclpi 6312 | . . 3 ⊢ ((B ∈ N ∧ 𝐷 ∈ N) → (B ·N 𝐷) ∈ N) | |
7 | 6 | ad2ant2l 477 | . 2 ⊢ (((A ∈ N ∧ B ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (B ·N 𝐷) ∈ N) |
8 | opelxpi 4319 | . 2 ⊢ ((((A ·N 𝐷) +N (B ·N 𝐶)) ∈ N ∧ (B ·N 𝐷) ∈ N) → 〈((A ·N 𝐷) +N (B ·N 𝐶)), (B ·N 𝐷)〉 ∈ (N × N)) | |
9 | 5, 7, 8 | syl2anc 391 | 1 ⊢ (((A ∈ N ∧ B ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → 〈((A ·N 𝐷) +N (B ·N 𝐶)), (B ·N 𝐷)〉 ∈ (N × N)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 〈cop 3370 × cxp 4286 (class class class)co 5455 Ncnpi 6256 +N cpli 6257 ·N cmi 6258 |
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-omul 5945 df-ni 6288 df-pli 6289 df-mi 6290 |
This theorem is referenced by: addpipqqs 6354 |
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