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Mirrors > Home > ILE Home > Th. List > shftval3 | GIF version |
Description: Value of a sequence shifted by 𝐴 − 𝐵. (Contributed by NM, 20-Jul-2005.) |
Ref | Expression |
---|---|
shftfval.1 | ⊢ 𝐹 ∈ V |
Ref | Expression |
---|---|
shftval3 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift (𝐴 − 𝐵))‘𝐴) = (𝐹‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 7782 | . . 3 ⊢ 0 ∈ ℂ | |
2 | shftfval.1 | . . . 4 ⊢ 𝐹 ∈ V | |
3 | 2 | shftval2 10630 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 0 ∈ ℂ) → ((𝐹 shift (𝐴 − 𝐵))‘(𝐴 + 0)) = (𝐹‘(𝐵 + 0))) |
4 | 1, 3 | mp3an3 1305 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift (𝐴 − 𝐵))‘(𝐴 + 0)) = (𝐹‘(𝐵 + 0))) |
5 | addid1 7924 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) | |
6 | 5 | adantr 274 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 0) = 𝐴) |
7 | 6 | fveq2d 5433 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift (𝐴 − 𝐵))‘(𝐴 + 0)) = ((𝐹 shift (𝐴 − 𝐵))‘𝐴)) |
8 | addid1 7924 | . . . 4 ⊢ (𝐵 ∈ ℂ → (𝐵 + 0) = 𝐵) | |
9 | 8 | adantl 275 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 + 0) = 𝐵) |
10 | 9 | fveq2d 5433 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐹‘(𝐵 + 0)) = (𝐹‘𝐵)) |
11 | 4, 7, 10 | 3eqtr3d 2181 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift (𝐴 − 𝐵))‘𝐴) = (𝐹‘𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ‘cfv 5131 (class class class)co 5782 ℂcc 7642 0cc0 7644 + caddc 7647 − cmin 7957 shift cshi 10618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-resscn 7736 ax-1cn 7737 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-sub 7959 df-shft 10619 |
This theorem is referenced by: (None) |
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