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Mirrors > Home > MPE Home > Th. List > Mathboxes > eluzge0nn0 | Structured version Visualization version GIF version |
Description: If an integer is greater than or equal to a nonnegative integer, then it is a nonnegative integer. (Contributed by Alexander van der Vekens, 27-Aug-2018.) |
Ref | Expression |
---|---|
eluzge0nn0 | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (0 ≤ 𝑀 → 𝑁 ∈ ℕ0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2 12243 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
2 | simpl2 1188 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) ∧ 0 ≤ 𝑀) → 𝑁 ∈ ℤ) | |
3 | zre 11979 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
4 | zre 11979 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
5 | 0red 10638 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 0 ∈ ℝ) | |
6 | simpl 485 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ∈ ℝ) | |
7 | simpr 487 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ∈ ℝ) | |
8 | 5, 6, 7 | 3jca 1124 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ)) |
9 | 3, 4, 8 | syl2an 597 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ)) |
10 | letr 10728 | . . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((0 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁) → 0 ≤ 𝑁)) | |
11 | 9, 10 | syl 17 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((0 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁) → 0 ≤ 𝑁)) |
12 | 11 | expcomd 419 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 → (0 ≤ 𝑀 → 0 ≤ 𝑁))) |
13 | 12 | ex 415 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ ℤ → (𝑀 ≤ 𝑁 → (0 ≤ 𝑀 → 0 ≤ 𝑁)))) |
14 | 13 | 3imp1 1343 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) ∧ 0 ≤ 𝑀) → 0 ≤ 𝑁) |
15 | elnn0z 11988 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) | |
16 | 2, 14, 15 | sylanbrc 585 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) ∧ 0 ≤ 𝑀) → 𝑁 ∈ ℕ0) |
17 | 16 | ex 415 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (0 ≤ 𝑀 → 𝑁 ∈ ℕ0)) |
18 | 1, 17 | sylbi 219 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (0 ≤ 𝑀 → 𝑁 ∈ ℕ0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 class class class wbr 5059 ‘cfv 6350 ℝcr 10530 0cc0 10531 ≤ cle 10670 ℕ0cn0 11891 ℤcz 11975 ℤ≥cuz 12237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 |
This theorem is referenced by: (None) |
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