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Mirrors > Home > ILE Home > Th. List > eluzsubi | GIF version |
Description: Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.) |
Ref | Expression |
---|---|
eluzaddi.1 | ⊢ 𝑀 ∈ ℤ |
eluzaddi.2 | ⊢ 𝐾 ∈ ℤ |
Ref | Expression |
---|---|
eluzsubi | ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 8779 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → 𝑁 ∈ ℤ) | |
2 | eluzaddi.2 | . . 3 ⊢ 𝐾 ∈ ℤ | |
3 | zsubcl 8543 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 − 𝐾) ∈ ℤ) | |
4 | 1, 2, 3 | sylancl 404 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → (𝑁 − 𝐾) ∈ ℤ) |
5 | eluzaddi.1 | . . . . 5 ⊢ 𝑀 ∈ ℤ | |
6 | zaddcl 8542 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 + 𝐾) ∈ ℤ) | |
7 | 5, 2, 6 | mp2an 417 | . . . 4 ⊢ (𝑀 + 𝐾) ∈ ℤ |
8 | 7 | eluz1i 8777 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) ↔ (𝑁 ∈ ℤ ∧ (𝑀 + 𝐾) ≤ 𝑁)) |
9 | zre 8506 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
10 | 5 | zrei 8508 | . . . . . 6 ⊢ 𝑀 ∈ ℝ |
11 | 2 | zrei 8508 | . . . . . 6 ⊢ 𝐾 ∈ ℝ |
12 | leaddsub 7679 | . . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 + 𝐾) ≤ 𝑁 ↔ 𝑀 ≤ (𝑁 − 𝐾))) | |
13 | 10, 11, 12 | mp3an12 1259 | . . . . 5 ⊢ (𝑁 ∈ ℝ → ((𝑀 + 𝐾) ≤ 𝑁 ↔ 𝑀 ≤ (𝑁 − 𝐾))) |
14 | 9, 13 | syl 14 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((𝑀 + 𝐾) ≤ 𝑁 ↔ 𝑀 ≤ (𝑁 − 𝐾))) |
15 | 14 | biimpa 290 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 + 𝐾) ≤ 𝑁) → 𝑀 ≤ (𝑁 − 𝐾)) |
16 | 8, 15 | sylbi 119 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → 𝑀 ≤ (𝑁 − 𝐾)) |
17 | 5 | eluz1i 8777 | . 2 ⊢ ((𝑁 − 𝐾) ∈ (ℤ≥‘𝑀) ↔ ((𝑁 − 𝐾) ∈ ℤ ∧ 𝑀 ≤ (𝑁 − 𝐾))) |
18 | 4, 16, 17 | sylanbrc 408 | 1 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∈ wcel 1434 class class class wbr 3805 ‘cfv 4952 (class class class)co 5564 ℝcr 7112 + caddc 7116 ≤ cle 7286 − cmin 7416 ℤcz 8502 ℤ≥cuz 8770 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3916 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-cnex 7199 ax-resscn 7200 ax-1cn 7201 ax-1re 7202 ax-icn 7203 ax-addcl 7204 ax-addrcl 7205 ax-mulcl 7206 ax-addcom 7208 ax-addass 7210 ax-distr 7212 ax-i2m1 7213 ax-0lt1 7214 ax-0id 7216 ax-rnegex 7217 ax-cnre 7219 ax-pre-ltirr 7220 ax-pre-ltwlin 7221 ax-pre-lttrn 7222 ax-pre-ltadd 7224 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2612 df-sbc 2825 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-int 3657 df-br 3806 df-opab 3860 df-mpt 3861 df-id 4076 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-rn 4402 df-res 4403 df-ima 4404 df-iota 4917 df-fun 4954 df-fn 4955 df-f 4956 df-fv 4960 df-riota 5520 df-ov 5567 df-oprab 5568 df-mpt2 5569 df-pnf 7287 df-mnf 7288 df-xr 7289 df-ltxr 7290 df-le 7291 df-sub 7418 df-neg 7419 df-inn 8177 df-n0 8426 df-z 8503 df-uz 8771 |
This theorem is referenced by: (None) |
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