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Mirrors > Home > ILE Home > Th. List > nnm1nn0 | GIF version |
Description: A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nnm1nn0 | ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn1m1nn 8738 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ (𝑁 − 1) ∈ ℕ)) | |
2 | oveq1 5781 | . . . . . 6 ⊢ (𝑁 = 1 → (𝑁 − 1) = (1 − 1)) | |
3 | 1m1e0 8789 | . . . . . 6 ⊢ (1 − 1) = 0 | |
4 | 2, 3 | syl6eq 2188 | . . . . 5 ⊢ (𝑁 = 1 → (𝑁 − 1) = 0) |
5 | 4 | orim1i 749 | . . . 4 ⊢ ((𝑁 = 1 ∨ (𝑁 − 1) ∈ ℕ) → ((𝑁 − 1) = 0 ∨ (𝑁 − 1) ∈ ℕ)) |
6 | 1, 5 | syl 14 | . . 3 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) = 0 ∨ (𝑁 − 1) ∈ ℕ)) |
7 | 6 | orcomd 718 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) ∈ ℕ ∨ (𝑁 − 1) = 0)) |
8 | elnn0 8979 | . 2 ⊢ ((𝑁 − 1) ∈ ℕ0 ↔ ((𝑁 − 1) ∈ ℕ ∨ (𝑁 − 1) = 0)) | |
9 | 7, 8 | sylibr 133 | 1 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 697 = wceq 1331 ∈ wcel 1480 (class class class)co 5774 0cc0 7620 1c1 7621 − cmin 7933 ℕcn 8720 ℕ0cn0 8977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 df-inn 8721 df-n0 8978 |
This theorem is referenced by: elnn0nn 9019 nnaddm1cl 9115 nn0n0n1ge2 9121 fseq1m1p1 9875 nn0ennn 10206 expm1t 10321 expgt1 10331 bcn1 10504 bcm1k 10506 bcn2m1 10515 resqrexlemnm 10790 resqrexlemcvg 10791 resqrexlemga 10795 binomlem 11252 arisum 11267 arisum2 11268 cvgratnnlemnexp 11293 cvgratnnlemfm 11298 mertenslem2 11305 iddvdsexp 11517 dvdsfac 11558 oexpneg 11574 phibnd 11893 phiprmpw 11898 dvexp 12844 |
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