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Mirrors > Home > ILE Home > Th. List > uzneg | GIF version |
Description: Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.) |
Ref | Expression |
---|---|
uzneg | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → -𝑀 ∈ (ℤ≥‘-𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzle 9233 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) | |
2 | eluzel2 9226 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
3 | eluzelz 9230 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
4 | zre 8955 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
5 | zre 8955 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | leneg 8139 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ -𝑁 ≤ -𝑀)) | |
7 | 4, 5, 6 | syl2an 285 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ -𝑁 ≤ -𝑀)) |
8 | 2, 3, 7 | syl2anc 406 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀 ≤ 𝑁 ↔ -𝑁 ≤ -𝑀)) |
9 | 1, 8 | mpbid 146 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → -𝑁 ≤ -𝑀) |
10 | znegcl 8982 | . . . 4 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) | |
11 | znegcl 8982 | . . . 4 ⊢ (𝑀 ∈ ℤ → -𝑀 ∈ ℤ) | |
12 | eluz 9234 | . . . 4 ⊢ ((-𝑁 ∈ ℤ ∧ -𝑀 ∈ ℤ) → (-𝑀 ∈ (ℤ≥‘-𝑁) ↔ -𝑁 ≤ -𝑀)) | |
13 | 10, 11, 12 | syl2an 285 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (-𝑀 ∈ (ℤ≥‘-𝑁) ↔ -𝑁 ≤ -𝑀)) |
14 | 3, 2, 13 | syl2anc 406 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (-𝑀 ∈ (ℤ≥‘-𝑁) ↔ -𝑁 ≤ -𝑀)) |
15 | 9, 14 | mpbird 166 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → -𝑀 ∈ (ℤ≥‘-𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∈ wcel 1461 class class class wbr 3893 ‘cfv 5079 ℝcr 7539 ≤ cle 7718 -cneg 7850 ℤcz 8951 ℤ≥cuz 9221 |
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 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7629 ax-resscn 7630 ax-1cn 7631 ax-1re 7632 ax-icn 7633 ax-addcl 7634 ax-addrcl 7635 ax-mulcl 7636 ax-addcom 7638 ax-addass 7640 ax-distr 7642 ax-i2m1 7643 ax-0lt1 7644 ax-0id 7646 ax-rnegex 7647 ax-cnre 7649 ax-pre-ltirr 7650 ax-pre-ltwlin 7651 ax-pre-lttrn 7652 ax-pre-ltadd 7654 |
This theorem depends on definitions: df-bi 116 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-pnf 7719 df-mnf 7720 df-xr 7721 df-ltxr 7722 df-le 7723 df-sub 7851 df-neg 7852 df-inn 8624 df-z 8952 df-uz 9222 |
This theorem is referenced by: (None) |
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