![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nnacl | GIF version |
Description: Closure of addition of natural numbers. Proposition 8.9 of [TakeutiZaring] p. 59. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
nnacl | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) ∈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5571 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐵)) | |
2 | 1 | eleq1d 2151 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 +𝑜 𝑥) ∈ ω ↔ (𝐴 +𝑜 𝐵) ∈ ω)) |
3 | 2 | imbi2d 228 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 +𝑜 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 +𝑜 𝐵) ∈ ω))) |
4 | oveq2 5571 | . . . . 5 ⊢ (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅)) | |
5 | 4 | eleq1d 2151 | . . . 4 ⊢ (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) ∈ ω ↔ (𝐴 +𝑜 ∅) ∈ ω)) |
6 | oveq2 5571 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦)) | |
7 | 6 | eleq1d 2151 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ ω ↔ (𝐴 +𝑜 𝑦) ∈ ω)) |
8 | oveq2 5571 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 suc 𝑦)) | |
9 | 8 | eleq1d 2151 | . . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝑥) ∈ ω ↔ (𝐴 +𝑜 suc 𝑦) ∈ ω)) |
10 | nna0 6138 | . . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴) | |
11 | 10 | eleq1d 2151 | . . . . 5 ⊢ (𝐴 ∈ ω → ((𝐴 +𝑜 ∅) ∈ ω ↔ 𝐴 ∈ ω)) |
12 | 11 | ibir 175 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 +𝑜 ∅) ∈ ω) |
13 | peano2 4364 | . . . . . 6 ⊢ ((𝐴 +𝑜 𝑦) ∈ ω → suc (𝐴 +𝑜 𝑦) ∈ ω) | |
14 | nnasuc 6140 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦)) | |
15 | 14 | eleq1d 2151 | . . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +𝑜 suc 𝑦) ∈ ω ↔ suc (𝐴 +𝑜 𝑦) ∈ ω)) |
16 | 13, 15 | syl5ibr 154 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +𝑜 𝑦) ∈ ω → (𝐴 +𝑜 suc 𝑦) ∈ ω)) |
17 | 16 | expcom 114 | . . . 4 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 +𝑜 𝑦) ∈ ω → (𝐴 +𝑜 suc 𝑦) ∈ ω))) |
18 | 5, 7, 9, 12, 17 | finds2 4370 | . . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 +𝑜 𝑥) ∈ ω)) |
19 | 3, 18 | vtoclga 2673 | . 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 +𝑜 𝐵) ∈ ω)) |
20 | 19 | impcom 123 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) ∈ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 ∅c0 3267 suc csuc 4148 ωcom 4359 (class class class)co 5563 +𝑜 coa 6082 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3913 ax-sep 3916 ax-nul 3924 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-iinf 4357 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2612 df-sbc 2825 df-csb 2918 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-nul 3268 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-int 3657 df-iun 3700 df-br 3806 df-opab 3860 df-mpt 3861 df-tr 3896 df-id 4076 df-iord 4149 df-on 4151 df-suc 4154 df-iom 4360 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-rn 4402 df-res 4403 df-ima 4404 df-iota 4917 df-fun 4954 df-fn 4955 df-f 4956 df-f1 4957 df-fo 4958 df-f1o 4959 df-fv 4960 df-ov 5566 df-oprab 5567 df-mpt2 5568 df-1st 5818 df-2nd 5819 df-recs 5974 df-irdg 6039 df-oadd 6089 |
This theorem is referenced by: nnmcl 6145 nnacli 6146 nnaass 6149 nndi 6150 nndir 6154 nnaordi 6168 nnaord 6169 nnaword 6171 addclpi 6631 nnppipi 6647 archnqq 6721 addcmpblnq0 6747 addclnq0 6755 nnanq0 6762 distrnq0 6763 addassnq0lemcl 6765 prarloclemlt 6797 prarloclemlo 6798 prarloclem3 6801 omgadd 9878 hashunlem 9880 hashun 9881 |
Copyright terms: Public domain | W3C validator |