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Mirrors > Home > MPE Home > Th. List > eluzsub | Structured version Visualization version GIF version |
Description: Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by SN, 7-Feb-2025.) |
Ref | Expression |
---|---|
eluzsub | ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1135 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → 𝑀 ∈ ℤ) | |
2 | eluzelz 12885 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → 𝑁 ∈ ℤ) | |
3 | 2 | 3ad2ant3 1134 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → 𝑁 ∈ ℤ) |
4 | simp2 1136 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → 𝐾 ∈ ℤ) | |
5 | 3, 4 | zsubcld 12724 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑁 − 𝐾) ∈ ℤ) |
6 | eluzle 12888 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → (𝑀 + 𝐾) ≤ 𝑁) | |
7 | 6 | 3ad2ant3 1134 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑀 + 𝐾) ≤ 𝑁) |
8 | zre 12614 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
9 | zre 12614 | . . . 4 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
10 | eluzelre 12886 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → 𝑁 ∈ ℝ) | |
11 | leaddsub 11736 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 + 𝐾) ≤ 𝑁 ↔ 𝑀 ≤ (𝑁 − 𝐾))) | |
12 | 8, 9, 10, 11 | syl3an 1159 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → ((𝑀 + 𝐾) ≤ 𝑁 ↔ 𝑀 ≤ (𝑁 − 𝐾))) |
13 | 7, 12 | mpbid 232 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → 𝑀 ≤ (𝑁 − 𝐾)) |
14 | eluz2 12881 | . 2 ⊢ ((𝑁 − 𝐾) ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑁 − 𝐾) ∈ ℤ ∧ 𝑀 ≤ (𝑁 − 𝐾))) | |
15 | 1, 5, 13, 14 | syl3anbrc 1342 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 ∈ wcel 2105 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 + caddc 11155 ≤ cle 11293 − cmin 11489 ℤcz 12610 ℤ≥cuz 12875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 |
This theorem is referenced by: eluzsubi 12908 fzoss2 13723 expmulnbnd 14270 shftuz 15104 climshftlem 15606 isumshft 15871 efgredleme 19775 uniioombllem3 25633 ulmshftlem 26446 ulmshft 26447 revpfxsfxrev 35099 caushft 37747 uzmptshftfval 44341 stoweidlem14 45969 nnsum4primeseven 47724 nnsum4primesevenALTV 47725 |
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