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Mirrors > Home > ILE Home > Th. List > fzoshftral | GIF version |
Description: Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 9775. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
Ref | Expression |
---|---|
fzoshftral | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀..^𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)..^(𝑁 + 𝐾))[(𝑘 − 𝐾) / 𝑗]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzoval 9812 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) | |
2 | 1 | 3ad2ant2 984 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
3 | 2 | raleqdv 2604 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀..^𝑁)𝜑 ↔ ∀𝑗 ∈ (𝑀...(𝑁 − 1))𝜑)) |
4 | peano2zm 8990 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
5 | fzshftral 9775 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀...(𝑁 − 1))𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)...((𝑁 − 1) + 𝐾))[(𝑘 − 𝐾) / 𝑗]𝜑)) | |
6 | 4, 5 | syl3an2 1231 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀...(𝑁 − 1))𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)...((𝑁 − 1) + 𝐾))[(𝑘 − 𝐾) / 𝑗]𝜑)) |
7 | zaddcl 8992 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ ℤ) | |
8 | 7 | 3adant1 980 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ ℤ) |
9 | fzoval 9812 | . . . . 5 ⊢ ((𝑁 + 𝐾) ∈ ℤ → ((𝑀 + 𝐾)..^(𝑁 + 𝐾)) = ((𝑀 + 𝐾)...((𝑁 + 𝐾) − 1))) | |
10 | 8, 9 | syl 14 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑀 + 𝐾)..^(𝑁 + 𝐾)) = ((𝑀 + 𝐾)...((𝑁 + 𝐾) − 1))) |
11 | zcn 8957 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
12 | 11 | adantr 272 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝑁 ∈ ℂ) |
13 | zcn 8957 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ) | |
14 | 13 | adantl 273 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝐾 ∈ ℂ) |
15 | 1cnd 7700 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 1 ∈ ℂ) | |
16 | 12, 14, 15 | addsubd 8011 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑁 + 𝐾) − 1) = ((𝑁 − 1) + 𝐾)) |
17 | 16 | 3adant1 980 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑁 + 𝐾) − 1) = ((𝑁 − 1) + 𝐾)) |
18 | 17 | oveq2d 5742 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑀 + 𝐾)...((𝑁 + 𝐾) − 1)) = ((𝑀 + 𝐾)...((𝑁 − 1) + 𝐾))) |
19 | 10, 18 | eqtr2d 2146 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑀 + 𝐾)...((𝑁 − 1) + 𝐾)) = ((𝑀 + 𝐾)..^(𝑁 + 𝐾))) |
20 | 19 | raleqdv 2604 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑘 ∈ ((𝑀 + 𝐾)...((𝑁 − 1) + 𝐾))[(𝑘 − 𝐾) / 𝑗]𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)..^(𝑁 + 𝐾))[(𝑘 − 𝐾) / 𝑗]𝜑)) |
21 | 3, 6, 20 | 3bitrd 213 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀..^𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)..^(𝑁 + 𝐾))[(𝑘 − 𝐾) / 𝑗]𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 943 = wceq 1312 ∈ wcel 1461 ∀wral 2388 [wsbc 2876 (class class class)co 5726 ℂcc 7539 1c1 7542 + caddc 7544 − cmin 7850 ℤcz 8952 ...cfz 9677 ..^cfzo 9806 |
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 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-addcom 7639 ax-addass 7641 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-0id 7647 ax-rnegex 7648 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-ltwlin 7652 ax-pre-lttrn 7653 ax-pre-ltadd 7655 |
This theorem depends on definitions: df-bi 116 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724 df-sub 7852 df-neg 7853 df-inn 8625 df-n0 8876 df-z 8953 df-uz 9223 df-fz 9678 df-fzo 9807 |
This theorem is referenced by: (None) |
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