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Mirrors > Home > MPE Home > Th. List > uzneg | Structured version Visualization version 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 12337 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) | |
2 | eluzel2 12329 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
3 | eluzelz 12334 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
4 | zre 12066 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
5 | zre 12066 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | leneg 11221 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ -𝑁 ≤ -𝑀)) | |
7 | 4, 5, 6 | syl2an 599 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ -𝑁 ≤ -𝑀)) |
8 | 2, 3, 7 | syl2anc 587 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀 ≤ 𝑁 ↔ -𝑁 ≤ -𝑀)) |
9 | 1, 8 | mpbid 235 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → -𝑁 ≤ -𝑀) |
10 | znegcl 12098 | . . . 4 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) | |
11 | znegcl 12098 | . . . 4 ⊢ (𝑀 ∈ ℤ → -𝑀 ∈ ℤ) | |
12 | eluz 12338 | . . . 4 ⊢ ((-𝑁 ∈ ℤ ∧ -𝑀 ∈ ℤ) → (-𝑀 ∈ (ℤ≥‘-𝑁) ↔ -𝑁 ≤ -𝑀)) | |
13 | 10, 11, 12 | syl2an 599 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (-𝑀 ∈ (ℤ≥‘-𝑁) ↔ -𝑁 ≤ -𝑀)) |
14 | 3, 2, 13 | syl2anc 587 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (-𝑀 ∈ (ℤ≥‘-𝑁) ↔ -𝑁 ≤ -𝑀)) |
15 | 9, 14 | mpbird 260 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → -𝑀 ∈ (ℤ≥‘-𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2114 class class class wbr 5030 ‘cfv 6339 ℝcr 10614 ≤ cle 10754 -cneg 10949 ℤcz 12062 ℤ≥cuz 12324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-z 12063 df-uz 12325 |
This theorem is referenced by: fsum2dsub 32157 |
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