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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 12249 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝐾 ∈ ℤ) | |
2 | peano2zm 12026 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (𝐾 − 1) ∈ ℤ) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝐾 − 1) ∈ ℤ) |
4 | 1zzd 12014 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 1 ∈ ℤ) | |
5 | id 22 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
6 | 1 | zcnd 12089 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝐾 ∈ ℂ) |
7 | ax-1cn 10595 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | npcan 10895 | . . . . . . 7 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 − 1) + 1) = 𝐾) | |
9 | 6, 7, 8 | sylancl 588 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → ((𝐾 − 1) + 1) = 𝐾) |
10 | 9 | fveq2d 6674 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (ℤ≥‘((𝐾 − 1) + 1)) = (ℤ≥‘𝐾)) |
11 | 5, 10 | eleqtrrd 2916 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1))) |
12 | eluzsub 12275 | . . . 4 ⊢ (((𝐾 − 1) ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1))) → (𝑁 − 1) ∈ (ℤ≥‘(𝐾 − 1))) | |
13 | 3, 4, 11, 12 | syl3anc 1367 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑁 − 1) ∈ (ℤ≥‘(𝐾 − 1))) |
14 | fzss2 12948 | . . 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 13040 | . . 3 ⊢ (𝐾 ∈ ℤ → (𝑀..^𝐾) = (𝑀...(𝐾 − 1))) | |
17 | 1, 16 | syl 17 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀..^𝐾) = (𝑀...(𝐾 − 1))) |
18 | eluzelz 12254 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ ℤ) | |
19 | fzoval 13040 | . . 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 4014 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀..^𝐾) ⊆ (𝑀..^𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 1c1 10538 + caddc 10540 − cmin 10870 ℤcz 11982 ℤ≥cuz 12244 ...cfz 12893 ..^cfzo 13034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 |
This theorem is referenced by: fzossrbm1 13067 fzosplit 13071 elfzoext 13095 fzossfzop1 13116 uzindi 13351 ccatass 13942 ccatrn 13943 ccatalpha 13947 swrdval2 14008 pfxres 14041 pfxf 14042 pfxccat1 14064 pfxccatin12lem2a 14089 splfv1 14117 revccat 14128 repswpfx 14147 psgnunilem5 18622 efgsp1 18863 efgsres 18864 wlkres 27452 trlreslem 27481 crctcshwlkn0lem4 27591 wwlksm1edg 27659 wwlksnred 27670 clwwlkccatlem 27767 clwlkclwwlklem2fv1 27773 clwlkclwwlklem2 27778 clwwisshclwwslem 27792 clwwlkinwwlk 27818 clwwlkf 27826 wwlksubclwwlk 27837 trlsegvdeg 28006 iundisjfi 30519 fz1nntr 30527 wrdres 30613 pfxf1 30618 swrdrn2 30628 swrdrn3 30629 swrdf1 30630 swrdrndisj 30631 cycpmco2rn 30767 cycpmco2lem6 30773 cycpmco2lem7 30774 cycpmconjslem2 30797 measiuns 31476 signstfvp 31841 signstfvc 31844 signstres 31845 signsvfn 31852 prodfzo03 31874 breprexplemc 31903 pfxwlk 32370 iccpartres 43598 iccpartigtl 43603 iccelpart 43613 |
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