Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 8731 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ (𝑁 − 1) ∈ ℕ)) | |
2 | oveq1 5774 | . . . . . 6 ⊢ (𝑁 = 1 → (𝑁 − 1) = (1 − 1)) | |
3 | 1m1e0 8782 | . . . . . 6 ⊢ (1 − 1) = 0 | |
4 | 2, 3 | syl6eq 2186 | . . . . 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 8972 | . 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 5767 0cc0 7613 1c1 7614 − cmin 7926 ℕcn 8713 ℕ0cn0 8970 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sub 7928 df-inn 8714 df-n0 8971 |
This theorem is referenced by: elnn0nn 9012 nnaddm1cl 9108 nn0n0n1ge2 9114 fseq1m1p1 9868 nn0ennn 10199 expm1t 10314 expgt1 10324 bcn1 10497 bcm1k 10499 bcn2m1 10508 resqrexlemnm 10783 resqrexlemcvg 10784 resqrexlemga 10788 binomlem 11245 arisum 11260 arisum2 11261 cvgratnnlemnexp 11286 cvgratnnlemfm 11291 mertenslem2 11298 iddvdsexp 11506 dvdsfac 11547 oexpneg 11563 phibnd 11882 phiprmpw 11887 dvexp 12833 |
Copyright terms: Public domain | W3C validator |