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Mirrors > Home > MPE Home > Th. List > Mathboxes > uzmptshftfval | Structured version Visualization version GIF version |
Description: When 𝐹 is a maps-to function on some set of upper integers 𝑍 that returns a set 𝐵, (𝐹 shift 𝑁) is another maps-to function on the shifted set of upper integers 𝑊. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
Ref | Expression |
---|---|
uzmptshftfval.f | ⊢ 𝐹 = (𝑥 ∈ 𝑍 ↦ 𝐵) |
uzmptshftfval.b | ⊢ 𝐵 ∈ V |
uzmptshftfval.c | ⊢ (𝑥 = (𝑦 − 𝑁) → 𝐵 = 𝐶) |
uzmptshftfval.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
uzmptshftfval.w | ⊢ 𝑊 = (ℤ≥‘(𝑀 + 𝑁)) |
uzmptshftfval.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
uzmptshftfval.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
Ref | Expression |
---|---|
uzmptshftfval | ⊢ (𝜑 → (𝐹 shift 𝑁) = (𝑦 ∈ 𝑊 ↦ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzmptshftfval.b | . . . . . 6 ⊢ 𝐵 ∈ V | |
2 | uzmptshftfval.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑍 ↦ 𝐵) | |
3 | 1, 2 | fnmpti 6256 | . . . . 5 ⊢ 𝐹 Fn 𝑍 |
4 | uzmptshftfval.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
5 | 4 | zcnd 11812 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
6 | uzmptshftfval.z | . . . . . . . . 9 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
7 | 6 | fvexi 6448 | . . . . . . . 8 ⊢ 𝑍 ∈ V |
8 | 7 | mptex 6743 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐵) ∈ V |
9 | 2, 8 | eqeltri 2903 | . . . . . 6 ⊢ 𝐹 ∈ V |
10 | 9 | shftfn 14191 | . . . . 5 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑁 ∈ ℂ) → (𝐹 shift 𝑁) Fn {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ 𝑍}) |
11 | 3, 5, 10 | sylancr 583 | . . . 4 ⊢ (𝜑 → (𝐹 shift 𝑁) Fn {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ 𝑍}) |
12 | uzmptshftfval.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
13 | shftuz 14187 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)} = (ℤ≥‘(𝑀 + 𝑁))) | |
14 | 4, 12, 13 | syl2anc 581 | . . . . . 6 ⊢ (𝜑 → {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)} = (ℤ≥‘(𝑀 + 𝑁))) |
15 | 6 | eleq2i 2899 | . . . . . . 7 ⊢ ((𝑦 − 𝑁) ∈ 𝑍 ↔ (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) |
16 | 15 | rabbii 3399 | . . . . . 6 ⊢ {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ 𝑍} = {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)} |
17 | uzmptshftfval.w | . . . . . 6 ⊢ 𝑊 = (ℤ≥‘(𝑀 + 𝑁)) | |
18 | 14, 16, 17 | 3eqtr4g 2887 | . . . . 5 ⊢ (𝜑 → {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ 𝑍} = 𝑊) |
19 | 18 | fneq2d 6216 | . . . 4 ⊢ (𝜑 → ((𝐹 shift 𝑁) Fn {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ 𝑍} ↔ (𝐹 shift 𝑁) Fn 𝑊)) |
20 | 11, 19 | mpbid 224 | . . 3 ⊢ (𝜑 → (𝐹 shift 𝑁) Fn 𝑊) |
21 | dffn5 6489 | . . 3 ⊢ ((𝐹 shift 𝑁) Fn 𝑊 ↔ (𝐹 shift 𝑁) = (𝑦 ∈ 𝑊 ↦ ((𝐹 shift 𝑁)‘𝑦))) | |
22 | 20, 21 | sylib 210 | . 2 ⊢ (𝜑 → (𝐹 shift 𝑁) = (𝑦 ∈ 𝑊 ↦ ((𝐹 shift 𝑁)‘𝑦))) |
23 | uzssz 11989 | . . . . . . . 8 ⊢ (ℤ≥‘(𝑀 + 𝑁)) ⊆ ℤ | |
24 | 17, 23 | eqsstri 3861 | . . . . . . 7 ⊢ 𝑊 ⊆ ℤ |
25 | zsscn 11713 | . . . . . . 7 ⊢ ℤ ⊆ ℂ | |
26 | 24, 25 | sstri 3837 | . . . . . 6 ⊢ 𝑊 ⊆ ℂ |
27 | 26 | sseli 3824 | . . . . 5 ⊢ (𝑦 ∈ 𝑊 → 𝑦 ∈ ℂ) |
28 | 9 | shftval 14192 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝐹 shift 𝑁)‘𝑦) = (𝐹‘(𝑦 − 𝑁))) |
29 | 5, 27, 28 | syl2an 591 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑊) → ((𝐹 shift 𝑁)‘𝑦) = (𝐹‘(𝑦 − 𝑁))) |
30 | 17 | eleq2i 2899 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑊 ↔ 𝑦 ∈ (ℤ≥‘(𝑀 + 𝑁))) |
31 | 12, 4 | jca 509 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
32 | eluzsub 11999 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑦 ∈ (ℤ≥‘(𝑀 + 𝑁))) → (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) | |
33 | 32 | 3expa 1153 | . . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑦 ∈ (ℤ≥‘(𝑀 + 𝑁))) → (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) |
34 | 31, 33 | sylan 577 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℤ≥‘(𝑀 + 𝑁))) → (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) |
35 | 30, 34 | sylan2b 589 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑊) → (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) |
36 | 35, 6 | syl6eleqr 2918 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑊) → (𝑦 − 𝑁) ∈ 𝑍) |
37 | uzmptshftfval.c | . . . . . 6 ⊢ (𝑥 = (𝑦 − 𝑁) → 𝐵 = 𝐶) | |
38 | 37, 2, 1 | fvmpt3i 6535 | . . . . 5 ⊢ ((𝑦 − 𝑁) ∈ 𝑍 → (𝐹‘(𝑦 − 𝑁)) = 𝐶) |
39 | 36, 38 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑊) → (𝐹‘(𝑦 − 𝑁)) = 𝐶) |
40 | 29, 39 | eqtrd 2862 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑊) → ((𝐹 shift 𝑁)‘𝑦) = 𝐶) |
41 | 40 | mpteq2dva 4968 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝑊 ↦ ((𝐹 shift 𝑁)‘𝑦)) = (𝑦 ∈ 𝑊 ↦ 𝐶)) |
42 | 22, 41 | eqtrd 2862 | 1 ⊢ (𝜑 → (𝐹 shift 𝑁) = (𝑦 ∈ 𝑊 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 {crab 3122 Vcvv 3415 ↦ cmpt 4953 Fn wfn 6119 ‘cfv 6124 (class class class)co 6906 ℂcc 10251 + caddc 10256 − cmin 10586 ℤcz 11705 ℤ≥cuz 11969 shift cshi 14184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-n0 11620 df-z 11706 df-uz 11970 df-shft 14185 |
This theorem is referenced by: dvradcnv2 39387 binomcxplemnotnn0 39396 |
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