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Mirrors > Home > ILE Home > Th. List > nnmcl | GIF version |
Description: Closure of multiplication of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
nnmcl | ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (A ·𝑜 B) ∈ 𝜔) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5463 | . . . . 5 ⊢ (x = B → (A ·𝑜 x) = (A ·𝑜 B)) | |
2 | 1 | eleq1d 2103 | . . . 4 ⊢ (x = B → ((A ·𝑜 x) ∈ 𝜔 ↔ (A ·𝑜 B) ∈ 𝜔)) |
3 | 2 | imbi2d 219 | . . 3 ⊢ (x = B → ((A ∈ 𝜔 → (A ·𝑜 x) ∈ 𝜔) ↔ (A ∈ 𝜔 → (A ·𝑜 B) ∈ 𝜔))) |
4 | oveq2 5463 | . . . . 5 ⊢ (x = ∅ → (A ·𝑜 x) = (A ·𝑜 ∅)) | |
5 | 4 | eleq1d 2103 | . . . 4 ⊢ (x = ∅ → ((A ·𝑜 x) ∈ 𝜔 ↔ (A ·𝑜 ∅) ∈ 𝜔)) |
6 | oveq2 5463 | . . . . 5 ⊢ (x = y → (A ·𝑜 x) = (A ·𝑜 y)) | |
7 | 6 | eleq1d 2103 | . . . 4 ⊢ (x = y → ((A ·𝑜 x) ∈ 𝜔 ↔ (A ·𝑜 y) ∈ 𝜔)) |
8 | oveq2 5463 | . . . . 5 ⊢ (x = suc y → (A ·𝑜 x) = (A ·𝑜 suc y)) | |
9 | 8 | eleq1d 2103 | . . . 4 ⊢ (x = suc y → ((A ·𝑜 x) ∈ 𝜔 ↔ (A ·𝑜 suc y) ∈ 𝜔)) |
10 | nnm0 5993 | . . . . 5 ⊢ (A ∈ 𝜔 → (A ·𝑜 ∅) = ∅) | |
11 | peano1 4260 | . . . . 5 ⊢ ∅ ∈ 𝜔 | |
12 | 10, 11 | syl6eqel 2125 | . . . 4 ⊢ (A ∈ 𝜔 → (A ·𝑜 ∅) ∈ 𝜔) |
13 | nnacl 5998 | . . . . . . . 8 ⊢ (((A ·𝑜 y) ∈ 𝜔 ∧ A ∈ 𝜔) → ((A ·𝑜 y) +𝑜 A) ∈ 𝜔) | |
14 | 13 | expcom 109 | . . . . . . 7 ⊢ (A ∈ 𝜔 → ((A ·𝑜 y) ∈ 𝜔 → ((A ·𝑜 y) +𝑜 A) ∈ 𝜔)) |
15 | 14 | adantr 261 | . . . . . 6 ⊢ ((A ∈ 𝜔 ∧ y ∈ 𝜔) → ((A ·𝑜 y) ∈ 𝜔 → ((A ·𝑜 y) +𝑜 A) ∈ 𝜔)) |
16 | nnmsuc 5995 | . . . . . . 7 ⊢ ((A ∈ 𝜔 ∧ y ∈ 𝜔) → (A ·𝑜 suc y) = ((A ·𝑜 y) +𝑜 A)) | |
17 | 16 | eleq1d 2103 | . . . . . 6 ⊢ ((A ∈ 𝜔 ∧ y ∈ 𝜔) → ((A ·𝑜 suc y) ∈ 𝜔 ↔ ((A ·𝑜 y) +𝑜 A) ∈ 𝜔)) |
18 | 15, 17 | sylibrd 158 | . . . . 5 ⊢ ((A ∈ 𝜔 ∧ y ∈ 𝜔) → ((A ·𝑜 y) ∈ 𝜔 → (A ·𝑜 suc y) ∈ 𝜔)) |
19 | 18 | expcom 109 | . . . 4 ⊢ (y ∈ 𝜔 → (A ∈ 𝜔 → ((A ·𝑜 y) ∈ 𝜔 → (A ·𝑜 suc y) ∈ 𝜔))) |
20 | 5, 7, 9, 12, 19 | finds2 4267 | . . 3 ⊢ (x ∈ 𝜔 → (A ∈ 𝜔 → (A ·𝑜 x) ∈ 𝜔)) |
21 | 3, 20 | vtoclga 2613 | . 2 ⊢ (B ∈ 𝜔 → (A ∈ 𝜔 → (A ·𝑜 B) ∈ 𝜔)) |
22 | 21 | impcom 116 | 1 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (A ·𝑜 B) ∈ 𝜔) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 ∅c0 3218 suc csuc 4068 𝜔com 4256 (class class class)co 5455 +𝑜 coa 5937 ·𝑜 comu 5938 |
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-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 |
This theorem is referenced by: nnmcli 6001 nndi 6004 nnmass 6005 nnmsucr 6006 nnmordi 6025 nnmord 6026 nnmword 6027 mulclpi 6312 enq0tr 6417 addcmpblnq0 6426 mulcmpblnq0 6427 mulcanenq0ec 6428 addclnq0 6434 mulclnq0 6435 nqpnq0nq 6436 distrnq0 6442 addassnq0lemcl 6444 addassnq0 6445 |
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