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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 8596 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ (𝑁 − 1) ∈ ℕ)) | |
2 | oveq1 5713 | . . . . . 6 ⊢ (𝑁 = 1 → (𝑁 − 1) = (1 − 1)) | |
3 | 1m1e0 8647 | . . . . . 6 ⊢ (1 − 1) = 0 | |
4 | 2, 3 | syl6eq 2148 | . . . . 5 ⊢ (𝑁 = 1 → (𝑁 − 1) = 0) |
5 | 4 | orim1i 718 | . . . 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 689 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) ∈ ℕ ∨ (𝑁 − 1) = 0)) |
8 | elnn0 8831 | . 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 670 = wceq 1299 ∈ wcel 1448 (class class class)co 5706 0cc0 7500 1c1 7501 − cmin 7804 ℕcn 8578 ℕ0cn0 8829 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-addcom 7595 ax-addass 7597 ax-distr 7599 ax-i2m1 7600 ax-0id 7603 ax-rnegex 7604 ax-cnre 7606 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-br 3876 df-opab 3930 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-iota 5024 df-fun 5061 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-sub 7806 df-inn 8579 df-n0 8830 |
This theorem is referenced by: elnn0nn 8871 nnaddm1cl 8967 nn0n0n1ge2 8973 fseq1m1p1 9716 nn0ennn 10047 expm1t 10162 expgt1 10172 bcn1 10345 bcm1k 10347 bcn2m1 10356 resqrexlemnm 10630 resqrexlemcvg 10631 resqrexlemga 10635 binomlem 11091 arisum 11106 arisum2 11107 cvgratnnlemnexp 11132 cvgratnnlemfm 11137 mertenslem2 11144 iddvdsexp 11312 dvdsfac 11353 oexpneg 11369 phibnd 11685 phiprmpw 11690 |
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