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Mirrors > Home > MPE Home > Th. List > nnnegz | Structured version Visualization version 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 10904 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
2 | 1 | renegcld 10336 | . 2 ⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℝ) |
3 | nncn 10905 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
4 | negneg 10210 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → --𝑁 = 𝑁) | |
5 | 4 | eleq1d 2672 | . . . . 5 ⊢ (𝑁 ∈ ℂ → (--𝑁 ∈ ℕ ↔ 𝑁 ∈ ℕ)) |
6 | 5 | biimprd 237 | . . . 4 ⊢ (𝑁 ∈ ℂ → (𝑁 ∈ ℕ → --𝑁 ∈ ℕ)) |
7 | 3, 6 | mpcom 37 | . . 3 ⊢ (𝑁 ∈ ℕ → --𝑁 ∈ ℕ) |
8 | 7 | 3mix3d 1231 | . 2 ⊢ (𝑁 ∈ ℕ → (-𝑁 = 0 ∨ -𝑁 ∈ ℕ ∨ --𝑁 ∈ ℕ)) |
9 | elz 11256 | . 2 ⊢ (-𝑁 ∈ ℤ ↔ (-𝑁 ∈ ℝ ∧ (-𝑁 = 0 ∨ -𝑁 ∈ ℕ ∨ --𝑁 ∈ ℕ))) | |
10 | 2, 8, 9 | sylanbrc 695 | 1 ⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1030 = wceq 1475 ∈ wcel 1977 ℂcc 9813 ℝcr 9814 0cc0 9815 -cneg 10146 ℕcn 10897 ℤcz 11254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-neg 10148 df-nn 10898 df-z 11255 |
This theorem is referenced by: znegcl 11289 neg1z 11290 zeo 11339 btwnz 11355 uzwo3 11659 expneg 12730 expaddzlem 12765 mulgnegnn 17374 mulgneg2 17398 knoppndvlem18 31690 jm2.19lem1 36574 hoicvr 39438 proththd 40069 |
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