Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nnacl | Structured version Visualization version 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 6658 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐵)) | |
2 | 1 | eleq1d 2686 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 +𝑜 𝑥) ∈ ω ↔ (𝐴 +𝑜 𝐵) ∈ ω)) |
3 | 2 | imbi2d 330 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 +𝑜 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 +𝑜 𝐵) ∈ ω))) |
4 | oveq2 6658 | . . . . 5 ⊢ (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅)) | |
5 | 4 | eleq1d 2686 | . . . 4 ⊢ (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) ∈ ω ↔ (𝐴 +𝑜 ∅) ∈ ω)) |
6 | oveq2 6658 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦)) | |
7 | 6 | eleq1d 2686 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ ω ↔ (𝐴 +𝑜 𝑦) ∈ ω)) |
8 | oveq2 6658 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 suc 𝑦)) | |
9 | 8 | eleq1d 2686 | . . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝑥) ∈ ω ↔ (𝐴 +𝑜 suc 𝑦) ∈ ω)) |
10 | nna0 7684 | . . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴) | |
11 | 10 | eleq1d 2686 | . . . . 5 ⊢ (𝐴 ∈ ω → ((𝐴 +𝑜 ∅) ∈ ω ↔ 𝐴 ∈ ω)) |
12 | 11 | ibir 257 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 +𝑜 ∅) ∈ ω) |
13 | peano2 7086 | . . . . . 6 ⊢ ((𝐴 +𝑜 𝑦) ∈ ω → suc (𝐴 +𝑜 𝑦) ∈ ω) | |
14 | nnasuc 7686 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦)) | |
15 | 14 | eleq1d 2686 | . . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +𝑜 suc 𝑦) ∈ ω ↔ suc (𝐴 +𝑜 𝑦) ∈ ω)) |
16 | 13, 15 | syl5ibr 236 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +𝑜 𝑦) ∈ ω → (𝐴 +𝑜 suc 𝑦) ∈ ω)) |
17 | 16 | expcom 451 | . . . 4 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 +𝑜 𝑦) ∈ ω → (𝐴 +𝑜 suc 𝑦) ∈ ω))) |
18 | 5, 7, 9, 12, 17 | finds2 7094 | . . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 +𝑜 𝑥) ∈ ω)) |
19 | 3, 18 | vtoclga 3272 | . 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 +𝑜 𝐵) ∈ ω)) |
20 | 19 | impcom 446 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∅c0 3915 suc csuc 5725 (class class class)co 6650 ωcom 7065 +𝑜 coa 7557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 |
This theorem is referenced by: nnmcl 7692 nnacli 7694 nnarcl 7696 nnaord 7699 nnawordi 7701 nnaass 7702 nndi 7703 nnaword 7707 nnawordex 7717 oaabslem 7723 unfilem1 8224 unfi 8227 nnacda 9023 ficardun 9024 ficardun2 9025 pwsdompw 9026 addclpi 9714 hashgadd 13166 hashdom 13168 finxpreclem4 33231 |
Copyright terms: Public domain | W3C validator |