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Mirrors > Home > MPE Home > Th. List > fzoshftral | Structured version Visualization version GIF version |
Description: Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 12681. (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 12725 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) | |
2 | 1 | 3ad2ant2 1165 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
3 | 2 | raleqdv 3328 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀..^𝑁)𝜑 ↔ ∀𝑗 ∈ (𝑀...(𝑁 − 1))𝜑)) |
4 | peano2zm 11709 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
5 | fzshftral 12681 | . . 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 1204 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀...(𝑁 − 1))𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)...((𝑁 − 1) + 𝐾))[(𝑘 − 𝐾) / 𝑗]𝜑)) |
7 | zaddcl 11706 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ ℤ) | |
8 | 7 | 3adant1 1161 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ ℤ) |
9 | fzoval 12725 | . . . . 5 ⊢ ((𝑁 + 𝐾) ∈ ℤ → ((𝑀 + 𝐾)..^(𝑁 + 𝐾)) = ((𝑀 + 𝐾)...((𝑁 + 𝐾) − 1))) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑀 + 𝐾)..^(𝑁 + 𝐾)) = ((𝑀 + 𝐾)...((𝑁 + 𝐾) − 1))) |
11 | zcn 11670 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
12 | 11 | adantr 473 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝑁 ∈ ℂ) |
13 | zcn 11670 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ) | |
14 | 13 | adantl 474 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝐾 ∈ ℂ) |
15 | 1cnd 10324 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 1 ∈ ℂ) | |
16 | 12, 14, 15 | addsubd 10706 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑁 + 𝐾) − 1) = ((𝑁 − 1) + 𝐾)) |
17 | 16 | 3adant1 1161 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑁 + 𝐾) − 1) = ((𝑁 − 1) + 𝐾)) |
18 | 17 | oveq2d 6895 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑀 + 𝐾)...((𝑁 + 𝐾) − 1)) = ((𝑀 + 𝐾)...((𝑁 − 1) + 𝐾))) |
19 | 10, 18 | eqtr2d 2835 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑀 + 𝐾)...((𝑁 − 1) + 𝐾)) = ((𝑀 + 𝐾)..^(𝑁 + 𝐾))) |
20 | 19 | raleqdv 3328 | . 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 297 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀..^𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)..^(𝑁 + 𝐾))[(𝑘 − 𝐾) / 𝑗]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ∀wral 3090 [wsbc 3634 (class class class)co 6879 ℂcc 10223 1c1 10226 + caddc 10228 − cmin 10557 ℤcz 11665 ...cfz 12579 ..^cfzo 12719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-om 7301 df-1st 7402 df-2nd 7403 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-er 7983 df-en 8197 df-dom 8198 df-sdom 8199 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-nn 11314 df-n0 11580 df-z 11666 df-uz 11930 df-fz 12580 df-fzo 12720 |
This theorem is referenced by: swrdspsleq 13702 |
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