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Mirrors > Home > MPE Home > Th. List > shftval | Structured version Visualization version GIF version |
Description: Value of a sequence shifted by 𝐴. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.) |
Ref | Expression |
---|---|
shftfval.1 | ⊢ 𝐹 ∈ V |
Ref | Expression |
---|---|
shftval | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shftfval.1 | . . . . 5 ⊢ 𝐹 ∈ V | |
2 | 1 | shftfib 14423 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴) “ {𝐵}) = (𝐹 “ {(𝐵 − 𝐴)})) |
3 | 2 | eleq2d 2875 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑥 ∈ ((𝐹 shift 𝐴) “ {𝐵}) ↔ 𝑥 ∈ (𝐹 “ {(𝐵 − 𝐴)}))) |
4 | 3 | iotabidv 6308 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (℩𝑥𝑥 ∈ ((𝐹 shift 𝐴) “ {𝐵})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐵 − 𝐴)}))) |
5 | dffv3 6641 | . 2 ⊢ ((𝐹 shift 𝐴)‘𝐵) = (℩𝑥𝑥 ∈ ((𝐹 shift 𝐴) “ {𝐵})) | |
6 | dffv3 6641 | . 2 ⊢ (𝐹‘(𝐵 − 𝐴)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐵 − 𝐴)})) | |
7 | 4, 5, 6 | 3eqtr4g 2858 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 “ cima 5522 ℩cio 6281 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 − cmin 10859 shift cshi 14417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 df-shft 14418 |
This theorem is referenced by: shftval2 14426 shftval4 14428 shftval5 14429 shftf 14430 seqshft 14436 climshftlem 14923 isumshft 15186 cvgrat 15231 abelthlem6 25031 shftvalg 33076 uzmptshftfval 41050 |
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