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Mirrors > Home > ILE Home > Th. List > eluzsub | GIF version |
Description: Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
eluzsub | ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 9127 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → 𝑁 ∈ ℤ) | |
2 | 1 | 3ad2ant3 969 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → 𝑁 ∈ ℤ) |
3 | simp2 947 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → 𝐾 ∈ ℤ) | |
4 | 2, 3 | zsubcld 8972 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑁 − 𝐾) ∈ ℤ) |
5 | simp3 948 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) | |
6 | simp1 946 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → 𝑀 ∈ ℤ) | |
7 | 6, 3 | zaddcld 8971 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑀 + 𝐾) ∈ ℤ) |
8 | eluz1 9122 | . . . . . 6 ⊢ ((𝑀 + 𝐾) ∈ ℤ → (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) ↔ (𝑁 ∈ ℤ ∧ (𝑀 + 𝐾) ≤ 𝑁))) | |
9 | 7, 8 | syl 14 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) ↔ (𝑁 ∈ ℤ ∧ (𝑀 + 𝐾) ≤ 𝑁))) |
10 | 5, 9 | mpbid 146 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑁 ∈ ℤ ∧ (𝑀 + 𝐾) ≤ 𝑁)) |
11 | 10 | simprd 113 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑀 + 𝐾) ≤ 𝑁) |
12 | 6 | zred 8967 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → 𝑀 ∈ ℝ) |
13 | 3 | zred 8967 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → 𝐾 ∈ ℝ) |
14 | 2 | zred 8967 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → 𝑁 ∈ ℝ) |
15 | leaddsub 8013 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 + 𝐾) ≤ 𝑁 ↔ 𝑀 ≤ (𝑁 − 𝐾))) | |
16 | 12, 13, 14, 15 | syl3anc 1181 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → ((𝑀 + 𝐾) ≤ 𝑁 ↔ 𝑀 ≤ (𝑁 − 𝐾))) |
17 | 11, 16 | mpbid 146 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → 𝑀 ≤ (𝑁 − 𝐾)) |
18 | eluz1 9122 | . . 3 ⊢ (𝑀 ∈ ℤ → ((𝑁 − 𝐾) ∈ (ℤ≥‘𝑀) ↔ ((𝑁 − 𝐾) ∈ ℤ ∧ 𝑀 ≤ (𝑁 − 𝐾)))) | |
19 | 6, 18 | syl 14 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → ((𝑁 − 𝐾) ∈ (ℤ≥‘𝑀) ↔ ((𝑁 − 𝐾) ∈ ℤ ∧ 𝑀 ≤ (𝑁 − 𝐾)))) |
20 | 4, 17, 19 | mpbir2and 893 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 927 ∈ wcel 1445 class class class wbr 3867 ‘cfv 5049 (class class class)co 5690 ℝcr 7446 + caddc 7450 ≤ cle 7620 − cmin 7750 ℤcz 8848 ℤ≥cuz 9118 |
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 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-addass 7544 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-0id 7550 ax-rnegex 7551 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-ltadd 7558 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-inn 8521 df-n0 8772 df-z 8849 df-uz 9119 |
This theorem is referenced by: fzoss2 9732 shftuz 10382 climshftlemg 10861 isumshft 11049 |
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