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Mirrors > Home > ILE Home > Th. List > fzoss2 | 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 9087 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝐾 ∈ ℤ) | |
2 | peano2zm 8851 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (𝐾 − 1) ∈ ℤ) | |
3 | 1, 2 | syl 14 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝐾 − 1) ∈ ℤ) |
4 | 1zzd 8840 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 1 ∈ ℤ) | |
5 | id 19 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
6 | 1 | zcnd 8932 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝐾 ∈ ℂ) |
7 | ax-1cn 7501 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | npcan 7754 | . . . . . . 7 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 − 1) + 1) = 𝐾) | |
9 | 6, 7, 8 | sylancl 405 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → ((𝐾 − 1) + 1) = 𝐾) |
10 | 9 | fveq2d 5324 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (ℤ≥‘((𝐾 − 1) + 1)) = (ℤ≥‘𝐾)) |
11 | 5, 10 | eleqtrrd 2168 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1))) |
12 | eluzsub 9111 | . . . 4 ⊢ (((𝐾 − 1) ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1))) → (𝑁 − 1) ∈ (ℤ≥‘(𝐾 − 1))) | |
13 | 3, 4, 11, 12 | syl3anc 1175 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑁 − 1) ∈ (ℤ≥‘(𝐾 − 1))) |
14 | fzss2 9541 | . . 3 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(𝐾 − 1)) → (𝑀...(𝐾 − 1)) ⊆ (𝑀...(𝑁 − 1))) | |
15 | 13, 14 | syl 14 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀...(𝐾 − 1)) ⊆ (𝑀...(𝑁 − 1))) |
16 | fzoval 9622 | . . 3 ⊢ (𝐾 ∈ ℤ → (𝑀..^𝐾) = (𝑀...(𝐾 − 1))) | |
17 | 1, 16 | syl 14 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀..^𝐾) = (𝑀...(𝐾 − 1))) |
18 | eluzelz 9091 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ ℤ) | |
19 | fzoval 9622 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) | |
20 | 18, 19 | syl 14 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
21 | 15, 17, 20 | 3sstr4d 3072 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀..^𝐾) ⊆ (𝑀..^𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1290 ∈ wcel 1439 ⊆ wss 3002 ‘cfv 5030 (class class class)co 5668 ℂcc 7411 1c1 7414 + caddc 7416 − cmin 7716 ℤcz 8813 ℤ≥cuz 9082 ...cfz 9487 ..^cfzo 9616 |
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 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-addcom 7508 ax-addass 7510 ax-distr 7512 ax-i2m1 7513 ax-0lt1 7514 ax-0id 7516 ax-rnegex 7517 ax-cnre 7519 ax-pre-ltirr 7520 ax-pre-ltwlin 7521 ax-pre-lttrn 7522 ax-pre-ltadd 7524 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2624 df-sbc 2844 df-csb 2937 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-iun 3740 df-br 3854 df-opab 3908 df-mpt 3909 df-id 4131 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fn 5033 df-f 5034 df-fv 5038 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-1st 5927 df-2nd 5928 df-pnf 7587 df-mnf 7588 df-xr 7589 df-ltxr 7590 df-le 7591 df-sub 7718 df-neg 7719 df-inn 8486 df-n0 8737 df-z 8814 df-uz 9083 df-fz 9488 df-fzo 9617 |
This theorem is referenced by: fzossrbm1 9647 fzosplit 9651 fzossfzop1 9686 |
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