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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 6692 | . . . . 5 ⊢ 𝐹 Fn 𝑍 |
4 | uzmptshftfval.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
5 | 4 | zcnd 12683 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
6 | uzmptshftfval.z | . . . . . . . . 9 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
7 | 6 | fvexi 6905 | . . . . . . . 8 ⊢ 𝑍 ∈ V |
8 | 7 | mptex 7229 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐵) ∈ V |
9 | 2, 8 | eqeltri 2824 | . . . . . 6 ⊢ 𝐹 ∈ V |
10 | 9 | shftfn 15038 | . . . . 5 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑁 ∈ ℂ) → (𝐹 shift 𝑁) Fn {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ 𝑍}) |
11 | 3, 5, 10 | sylancr 586 | . . . 4 ⊢ (𝜑 → (𝐹 shift 𝑁) Fn {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ 𝑍}) |
12 | uzmptshftfval.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
13 | shftuz 15034 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)} = (ℤ≥‘(𝑀 + 𝑁))) | |
14 | 4, 12, 13 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)} = (ℤ≥‘(𝑀 + 𝑁))) |
15 | 6 | eleq2i 2820 | . . . . . . 7 ⊢ ((𝑦 − 𝑁) ∈ 𝑍 ↔ (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) |
16 | 15 | rabbii 3433 | . . . . . 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 2792 | . . . . 5 ⊢ (𝜑 → {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ 𝑍} = 𝑊) |
19 | 18 | fneq2d 6642 | . . . 4 ⊢ (𝜑 → ((𝐹 shift 𝑁) Fn {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ 𝑍} ↔ (𝐹 shift 𝑁) Fn 𝑊)) |
20 | 11, 19 | mpbid 231 | . . 3 ⊢ (𝜑 → (𝐹 shift 𝑁) Fn 𝑊) |
21 | dffn5 6951 | . . 3 ⊢ ((𝐹 shift 𝑁) Fn 𝑊 ↔ (𝐹 shift 𝑁) = (𝑦 ∈ 𝑊 ↦ ((𝐹 shift 𝑁)‘𝑦))) | |
22 | 20, 21 | sylib 217 | . 2 ⊢ (𝜑 → (𝐹 shift 𝑁) = (𝑦 ∈ 𝑊 ↦ ((𝐹 shift 𝑁)‘𝑦))) |
23 | uzssz 12859 | . . . . . . . 8 ⊢ (ℤ≥‘(𝑀 + 𝑁)) ⊆ ℤ | |
24 | 17, 23 | eqsstri 4012 | . . . . . . 7 ⊢ 𝑊 ⊆ ℤ |
25 | zsscn 12582 | . . . . . . 7 ⊢ ℤ ⊆ ℂ | |
26 | 24, 25 | sstri 3987 | . . . . . 6 ⊢ 𝑊 ⊆ ℂ |
27 | 26 | sseli 3974 | . . . . 5 ⊢ (𝑦 ∈ 𝑊 → 𝑦 ∈ ℂ) |
28 | 9 | shftval 15039 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝐹 shift 𝑁)‘𝑦) = (𝐹‘(𝑦 − 𝑁))) |
29 | 5, 27, 28 | syl2an 595 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑊) → ((𝐹 shift 𝑁)‘𝑦) = (𝐹‘(𝑦 − 𝑁))) |
30 | 17 | eleq2i 2820 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑊 ↔ 𝑦 ∈ (ℤ≥‘(𝑀 + 𝑁))) |
31 | 12, 4 | jca 511 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
32 | eluzsub 12868 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑦 ∈ (ℤ≥‘(𝑀 + 𝑁))) → (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) | |
33 | 32 | 3expa 1116 | . . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑦 ∈ (ℤ≥‘(𝑀 + 𝑁))) → (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) |
34 | 31, 33 | sylan 579 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℤ≥‘(𝑀 + 𝑁))) → (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) |
35 | 30, 34 | sylan2b 593 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑊) → (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) |
36 | 35, 6 | eleqtrrdi 2839 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑊) → (𝑦 − 𝑁) ∈ 𝑍) |
37 | uzmptshftfval.c | . . . . . 6 ⊢ (𝑥 = (𝑦 − 𝑁) → 𝐵 = 𝐶) | |
38 | 37, 2, 1 | fvmpt3i 7004 | . . . . 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 2767 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑊) → ((𝐹 shift 𝑁)‘𝑦) = 𝐶) |
41 | 40 | mpteq2dva 5242 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝑊 ↦ ((𝐹 shift 𝑁)‘𝑦)) = (𝑦 ∈ 𝑊 ↦ 𝐶)) |
42 | 22, 41 | eqtrd 2767 | 1 ⊢ (𝜑 → (𝐹 shift 𝑁) = (𝑦 ∈ 𝑊 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {crab 3427 Vcvv 3469 ↦ cmpt 5225 Fn wfn 6537 ‘cfv 6542 (class class class)co 7414 ℂcc 11122 + caddc 11127 − cmin 11460 ℤcz 12574 ℤ≥cuz 12838 shift cshi 15031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-n0 12489 df-z 12575 df-uz 12839 df-shft 15032 |
This theorem is referenced by: dvradcnv2 43697 binomcxplemnotnn0 43706 |
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