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Mirrors > Home > ILE Home > Th. List > nnnegz | GIF version |
Description: The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.) |
Ref | Expression |
---|---|
nnnegz | ⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnre 8951 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
2 | 1 | renegcld 8362 | . 2 ⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℝ) |
3 | nncn 8952 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
4 | negneg 8232 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → --𝑁 = 𝑁) | |
5 | 4 | eleq1d 2258 | . . . . 5 ⊢ (𝑁 ∈ ℂ → (--𝑁 ∈ ℕ ↔ 𝑁 ∈ ℕ)) |
6 | 5 | biimprd 158 | . . . 4 ⊢ (𝑁 ∈ ℂ → (𝑁 ∈ ℕ → --𝑁 ∈ ℕ)) |
7 | 3, 6 | mpcom 36 | . . 3 ⊢ (𝑁 ∈ ℕ → --𝑁 ∈ ℕ) |
8 | 7 | 3mix3d 1176 | . 2 ⊢ (𝑁 ∈ ℕ → (-𝑁 = 0 ∨ -𝑁 ∈ ℕ ∨ --𝑁 ∈ ℕ)) |
9 | elz 9280 | . 2 ⊢ (-𝑁 ∈ ℤ ↔ (-𝑁 ∈ ℝ ∧ (-𝑁 = 0 ∨ -𝑁 ∈ ℕ ∨ --𝑁 ∈ ℕ))) | |
10 | 2, 8, 9 | sylanbrc 417 | 1 ⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 979 = wceq 1364 ∈ wcel 2160 ℂcc 7834 ℝcr 7835 0cc0 7836 -cneg 8154 ℕcn 8944 ℤcz 9278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-setind 4551 ax-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-addcom 7936 ax-addass 7938 ax-distr 7940 ax-i2m1 7941 ax-0id 7944 ax-rnegex 7945 ax-cnre 7947 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5234 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-sub 8155 df-neg 8156 df-inn 8945 df-z 9279 |
This theorem is referenced by: znegcl 9309 neg1z 9310 zeo 9383 btwnz 9397 expaddzaplem 10589 mulgnegnn 13065 mulgneg2 13089 |
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