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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 6711 | . . . . 5 ⊢ 𝐹 Fn 𝑍 |
| 4 | uzmptshftfval.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
| 5 | 4 | zcnd 12723 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 6 | uzmptshftfval.z | . . . . . . . . 9 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 7 | 6 | fvexi 6920 | . . . . . . . 8 ⊢ 𝑍 ∈ V |
| 8 | 7 | mptex 7243 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐵) ∈ V |
| 9 | 2, 8 | eqeltri 2837 | . . . . . 6 ⊢ 𝐹 ∈ V |
| 10 | 9 | shftfn 15112 | . . . . 5 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑁 ∈ ℂ) → (𝐹 shift 𝑁) Fn {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ 𝑍}) |
| 11 | 3, 5, 10 | sylancr 587 | . . . 4 ⊢ (𝜑 → (𝐹 shift 𝑁) Fn {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ 𝑍}) |
| 12 | uzmptshftfval.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 13 | shftuz 15108 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)} = (ℤ≥‘(𝑀 + 𝑁))) | |
| 14 | 4, 12, 13 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)} = (ℤ≥‘(𝑀 + 𝑁))) |
| 15 | 6 | eleq2i 2833 | . . . . . . 7 ⊢ ((𝑦 − 𝑁) ∈ 𝑍 ↔ (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) |
| 16 | 15 | rabbii 3442 | . . . . . 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 2802 | . . . . 5 ⊢ (𝜑 → {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ 𝑍} = 𝑊) |
| 19 | 18 | fneq2d 6662 | . . . 4 ⊢ (𝜑 → ((𝐹 shift 𝑁) Fn {𝑦 ∈ ℂ ∣ (𝑦 − 𝑁) ∈ 𝑍} ↔ (𝐹 shift 𝑁) Fn 𝑊)) |
| 20 | 11, 19 | mpbid 232 | . . 3 ⊢ (𝜑 → (𝐹 shift 𝑁) Fn 𝑊) |
| 21 | dffn5 6967 | . . 3 ⊢ ((𝐹 shift 𝑁) Fn 𝑊 ↔ (𝐹 shift 𝑁) = (𝑦 ∈ 𝑊 ↦ ((𝐹 shift 𝑁)‘𝑦))) | |
| 22 | 20, 21 | sylib 218 | . 2 ⊢ (𝜑 → (𝐹 shift 𝑁) = (𝑦 ∈ 𝑊 ↦ ((𝐹 shift 𝑁)‘𝑦))) |
| 23 | uzssz 12899 | . . . . . . . 8 ⊢ (ℤ≥‘(𝑀 + 𝑁)) ⊆ ℤ | |
| 24 | 17, 23 | eqsstri 4030 | . . . . . . 7 ⊢ 𝑊 ⊆ ℤ |
| 25 | zsscn 12621 | . . . . . . 7 ⊢ ℤ ⊆ ℂ | |
| 26 | 24, 25 | sstri 3993 | . . . . . 6 ⊢ 𝑊 ⊆ ℂ |
| 27 | 26 | sseli 3979 | . . . . 5 ⊢ (𝑦 ∈ 𝑊 → 𝑦 ∈ ℂ) |
| 28 | 9 | shftval 15113 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝐹 shift 𝑁)‘𝑦) = (𝐹‘(𝑦 − 𝑁))) |
| 29 | 5, 27, 28 | syl2an 596 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑊) → ((𝐹 shift 𝑁)‘𝑦) = (𝐹‘(𝑦 − 𝑁))) |
| 30 | 17 | eleq2i 2833 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑊 ↔ 𝑦 ∈ (ℤ≥‘(𝑀 + 𝑁))) |
| 31 | 12, 4 | jca 511 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
| 32 | eluzsub 12908 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑦 ∈ (ℤ≥‘(𝑀 + 𝑁))) → (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) | |
| 33 | 32 | 3expa 1119 | . . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑦 ∈ (ℤ≥‘(𝑀 + 𝑁))) → (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) |
| 34 | 31, 33 | sylan 580 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℤ≥‘(𝑀 + 𝑁))) → (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) |
| 35 | 30, 34 | sylan2b 594 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑊) → (𝑦 − 𝑁) ∈ (ℤ≥‘𝑀)) |
| 36 | 35, 6 | eleqtrrdi 2852 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑊) → (𝑦 − 𝑁) ∈ 𝑍) |
| 37 | uzmptshftfval.c | . . . . . 6 ⊢ (𝑥 = (𝑦 − 𝑁) → 𝐵 = 𝐶) | |
| 38 | 37, 2, 1 | fvmpt3i 7021 | . . . . 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 2777 | . . 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 2777 | 1 ⊢ (𝜑 → (𝐹 shift 𝑁) = (𝑦 ∈ 𝑊 ↦ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ↦ cmpt 5225 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 + caddc 11158 − cmin 11492 ℤcz 12613 ℤ≥cuz 12878 shift cshi 15105 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-shft 15106 |
| This theorem is referenced by: dvradcnv2 44366 binomcxplemnotnn0 44375 |
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