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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 6693 | . . . . 5 ⊢ 𝐹 Fn 𝑍 |
4 | uzmptshftfval.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
5 | 4 | zcnd 12692 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
6 | uzmptshftfval.z | . . . . . . . . 9 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
7 | 6 | fvexi 6904 | . . . . . . . 8 ⊢ 𝑍 ∈ V |
8 | 7 | mptex 7229 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐵) ∈ V |
9 | 2, 8 | eqeltri 2821 | . . . . . 6 ⊢ 𝐹 ∈ V |
10 | 9 | shftfn 15047 | . . . . 5 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑁 ∈ ℂ) → (𝐹 shift 𝑁) Fn {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ 𝑍}) |
11 | 3, 5, 10 | sylancr 585 | . . . 4 ⊢ (𝜑 → (𝐹 shift 𝑁) Fn {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ 𝑍}) |
12 | uzmptshftfval.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
13 | shftuz 15043 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)} = (ℤ≥‘(𝑀 + 𝑁))) | |
14 | 4, 12, 13 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)} = (ℤ≥‘(𝑀 + 𝑁))) |
15 | 6 | eleq2i 2817 | . . . . . . 7 ⊢ ((𝑦 − 𝑁) ∈ 𝑍 ↔ (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) |
16 | 15 | rabbii 3425 | . . . . . 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 2790 | . . . . 5 ⊢ (𝜑 → {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ 𝑍} = 𝑊) |
19 | 18 | fneq2d 6643 | . . . 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 6950 | . . 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 12868 | . . . . . . . 8 ⊢ (ℤ≥‘(𝑀 + 𝑁)) ⊆ ℤ | |
24 | 17, 23 | eqsstri 4008 | . . . . . . 7 ⊢ 𝑊 ⊆ ℤ |
25 | zsscn 12591 | . . . . . . 7 ⊢ ℤ ⊆ ℂ | |
26 | 24, 25 | sstri 3983 | . . . . . 6 ⊢ 𝑊 ⊆ ℂ |
27 | 26 | sseli 3969 | . . . . 5 ⊢ (𝑦 ∈ 𝑊 → 𝑦 ∈ ℂ) |
28 | 9 | shftval 15048 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝐹 shift 𝑁)‘𝑦) = (𝐹‘(𝑦 − 𝑁))) |
29 | 5, 27, 28 | syl2an 594 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑊) → ((𝐹 shift 𝑁)‘𝑦) = (𝐹‘(𝑦 − 𝑁))) |
30 | 17 | eleq2i 2817 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑊 ↔ 𝑦 ∈ (ℤ≥‘(𝑀 + 𝑁))) |
31 | 12, 4 | jca 510 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
32 | eluzsub 12877 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑦 ∈ (ℤ≥‘(𝑀 + 𝑁))) → (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) | |
33 | 32 | 3expa 1115 | . . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑦 ∈ (ℤ≥‘(𝑀 + 𝑁))) → (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) |
34 | 31, 33 | sylan 578 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℤ≥‘(𝑀 + 𝑁))) → (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) |
35 | 30, 34 | sylan2b 592 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑊) → (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) |
36 | 35, 6 | eleqtrrdi 2836 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑊) → (𝑦 − 𝑁) ∈ 𝑍) |
37 | uzmptshftfval.c | . . . . . 6 ⊢ (𝑥 = (𝑦 − 𝑁) → 𝐵 = 𝐶) | |
38 | 37, 2, 1 | fvmpt3i 7003 | . . . . 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 2765 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑊) → ((𝐹 shift 𝑁)‘𝑦) = 𝐶) |
41 | 40 | mpteq2dva 5244 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝑊 ↦ ((𝐹 shift 𝑁)‘𝑦)) = (𝑦 ∈ 𝑊 ↦ 𝐶)) |
42 | 22, 41 | eqtrd 2765 | 1 ⊢ (𝜑 → (𝐹 shift 𝑁) = (𝑦 ∈ 𝑊 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {crab 3419 Vcvv 3463 ↦ cmpt 5227 Fn wfn 6538 ‘cfv 6543 (class class class)co 7413 ℂcc 11131 + caddc 11136 − cmin 11469 ℤcz 12583 ℤ≥cuz 12847 shift cshi 15040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-n0 12498 df-z 12584 df-uz 12848 df-shft 15041 |
This theorem is referenced by: dvradcnv2 43845 binomcxplemnotnn0 43854 |
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