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Mirrors > Home > MPE Home > Th. List > fzoss2 | Structured version Visualization version GIF version |
Description: Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
Ref | Expression |
---|---|
fzoss2 | ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀..^𝐾) ⊆ (𝑀..^𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 12236 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝐾 ∈ ℤ) | |
2 | peano2zm 12013 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (𝐾 − 1) ∈ ℤ) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝐾 − 1) ∈ ℤ) |
4 | 1zzd 12001 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 1 ∈ ℤ) | |
5 | id 22 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
6 | 1 | zcnd 12076 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝐾 ∈ ℂ) |
7 | ax-1cn 10583 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | npcan 10883 | . . . . . . 7 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 − 1) + 1) = 𝐾) | |
9 | 6, 7, 8 | sylancl 586 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → ((𝐾 − 1) + 1) = 𝐾) |
10 | 9 | fveq2d 6667 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (ℤ≥‘((𝐾 − 1) + 1)) = (ℤ≥‘𝐾)) |
11 | 5, 10 | eleqtrrd 2913 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1))) |
12 | eluzsub 12262 | . . . 4 ⊢ (((𝐾 − 1) ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1))) → (𝑁 − 1) ∈ (ℤ≥‘(𝐾 − 1))) | |
13 | 3, 4, 11, 12 | syl3anc 1363 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑁 − 1) ∈ (ℤ≥‘(𝐾 − 1))) |
14 | fzss2 12935 | . . 3 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(𝐾 − 1)) → (𝑀...(𝐾 − 1)) ⊆ (𝑀...(𝑁 − 1))) | |
15 | 13, 14 | syl 17 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀...(𝐾 − 1)) ⊆ (𝑀...(𝑁 − 1))) |
16 | fzoval 13027 | . . 3 ⊢ (𝐾 ∈ ℤ → (𝑀..^𝐾) = (𝑀...(𝐾 − 1))) | |
17 | 1, 16 | syl 17 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀..^𝐾) = (𝑀...(𝐾 − 1))) |
18 | eluzelz 12241 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ ℤ) | |
19 | fzoval 13027 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) | |
20 | 18, 19 | syl 17 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
21 | 15, 17, 20 | 3sstr4d 4011 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀..^𝐾) ⊆ (𝑀..^𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 1c1 10526 + caddc 10528 − cmin 10858 ℤcz 11969 ℤ≥cuz 12231 ...cfz 12880 ..^cfzo 13021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 |
This theorem is referenced by: fzossrbm1 13054 fzosplit 13058 elfzoext 13082 fzossfzop1 13103 uzindi 13338 ccatass 13930 ccatrn 13931 ccatalpha 13935 swrdval2 13996 pfxres 14029 pfxf 14030 pfxccat1 14052 pfxccatin12lem2a 14077 splfv1 14105 revccat 14116 repswpfx 14135 psgnunilem5 18551 efgsp1 18792 efgsres 18793 wlkres 27379 trlreslem 27408 crctcshwlkn0lem4 27518 wwlksm1edg 27586 wwlksnred 27597 clwwlkccatlem 27694 clwlkclwwlklem2fv1 27700 clwlkclwwlklem2 27705 clwwisshclwwslem 27719 clwwlkinwwlk 27745 clwwlkf 27753 wwlksubclwwlk 27764 trlsegvdeg 27933 iundisjfi 30445 fz1nntr 30453 wrdres 30540 pfxf1 30545 swrdrn2 30555 swrdrn3 30556 swrdf1 30557 swrdrndisj 30558 cycpmco2rn 30694 cycpmco2lem6 30700 cycpmco2lem7 30701 cycpmconjslem2 30724 measiuns 31375 signstfvp 31740 signstfvc 31743 signstres 31744 signsvfn 31751 prodfzo03 31773 breprexplemc 31802 pfxwlk 32267 iccpartres 43455 iccpartigtl 43460 iccelpart 43470 |
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