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Mirrors > Home > ILE Home > Th. List > shftvalg | GIF version |
Description: Value of a sequence shifted by 𝐴. (Contributed by Scott Fenton, 16-Dec-2017.) |
Ref | Expression |
---|---|
shftvalg | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5903 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓 shift 𝐴) = (𝐹 shift 𝐴)) | |
2 | 1 | fveq1d 5536 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓 shift 𝐴)‘𝐵) = ((𝐹 shift 𝐴)‘𝐵)) |
3 | fveq1 5533 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝐵 − 𝐴)) = (𝐹‘(𝐵 − 𝐴))) | |
4 | 2, 3 | eqeq12d 2204 | . . . 4 ⊢ (𝑓 = 𝐹 → (((𝑓 shift 𝐴)‘𝐵) = (𝑓‘(𝐵 − 𝐴)) ↔ ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴)))) |
5 | 4 | imbi2d 230 | . . 3 ⊢ (𝑓 = 𝐹 → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝑓 shift 𝐴)‘𝐵) = (𝑓‘(𝐵 − 𝐴))) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴))))) |
6 | vex 2755 | . . . 4 ⊢ 𝑓 ∈ V | |
7 | 6 | shftval 10866 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝑓 shift 𝐴)‘𝐵) = (𝑓‘(𝐵 − 𝐴))) |
8 | 5, 7 | vtoclg 2812 | . 2 ⊢ (𝐹 ∈ 𝑉 → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴)))) |
9 | 8 | 3impib 1203 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ‘cfv 5235 (class class class)co 5896 ℂcc 7839 − cmin 8158 shift cshi 10855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-resscn 7933 ax-1cn 7934 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-addass 7943 ax-distr 7945 ax-i2m1 7946 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-sub 8160 df-shft 10856 |
This theorem is referenced by: seq3shft 10879 climshftlemg 11342 |
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