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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. Theorem 2.20 of [Schloeder] p. 6. (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 7434 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵)) | |
2 | 1 | eleq1d 2814 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 +o 𝑥) ∈ ω ↔ (𝐴 +o 𝐵) ∈ ω)) |
3 | 2 | imbi2d 339 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 +o 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 +o 𝐵) ∈ ω))) |
4 | oveq2 7434 | . . . . 5 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅)) | |
5 | 4 | eleq1d 2814 | . . . 4 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝑥) ∈ ω ↔ (𝐴 +o ∅) ∈ ω)) |
6 | oveq2 7434 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦)) | |
7 | 6 | eleq1d 2814 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ∈ ω ↔ (𝐴 +o 𝑦) ∈ ω)) |
8 | oveq2 7434 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦)) | |
9 | 8 | eleq1d 2814 | . . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) ∈ ω ↔ (𝐴 +o suc 𝑦) ∈ ω)) |
10 | nna0 8631 | . . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴) | |
11 | 10 | eleq1d 2814 | . . . . 5 ⊢ (𝐴 ∈ ω → ((𝐴 +o ∅) ∈ ω ↔ 𝐴 ∈ ω)) |
12 | 11 | ibir 267 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 +o ∅) ∈ ω) |
13 | peano2 7902 | . . . . . 6 ⊢ ((𝐴 +o 𝑦) ∈ ω → suc (𝐴 +o 𝑦) ∈ ω) | |
14 | nnasuc 8633 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦)) | |
15 | 14 | eleq1d 2814 | . . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +o suc 𝑦) ∈ ω ↔ suc (𝐴 +o 𝑦) ∈ ω)) |
16 | 13, 15 | imbitrrid 245 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +o 𝑦) ∈ ω → (𝐴 +o suc 𝑦) ∈ ω)) |
17 | 16 | expcom 412 | . . . 4 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 +o 𝑦) ∈ ω → (𝐴 +o suc 𝑦) ∈ ω))) |
18 | 5, 7, 9, 12, 17 | finds2 7912 | . . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 +o 𝑥) ∈ ω)) |
19 | 3, 18 | vtoclga 3565 | . 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 +o 𝐵) ∈ ω)) |
20 | 19 | impcom 406 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∅c0 4326 suc csuc 6376 (class class class)co 7426 ωcom 7876 +o coa 8490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-oadd 8497 |
This theorem is referenced by: nnmcl 8639 nnacli 8641 nnarcl 8643 nnaord 8646 nnawordi 8648 nnaass 8649 nndi 8650 nnaword 8654 nnawordex 8664 oaabslem 8674 eldifsucnn 8691 unfilem1 9341 unfiOLD 9344 ttrcltr 9747 nnadju 10228 nnadjuALT 10229 ficardun 10231 ficardunOLD 10232 ficardun2 10233 ficardun2OLD 10234 pwsdompw 10235 addclpi 10923 hashgadd 14376 hashdom 14378 precsexlem6 28130 precsexlem7 28131 om2noseqlt 28192 finxpreclem4 36906 nnamecl 42747 naddcnff 42822 naddwordnexlem3 42860 finona1cl 42914 |
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