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Mirrors > Home > ILE Home > Th. List > shftval2 | GIF version |
Description: Value of a sequence shifted by 𝐴 − 𝐵. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.) |
Ref | Expression |
---|---|
shftfval.1 | ⊢ 𝐹 ∈ V |
Ref | Expression |
---|---|
shftval2 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐹 shift (𝐴 − 𝐵))‘(𝐴 + 𝐶)) = (𝐹‘(𝐵 + 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subcl 7954 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) | |
2 | 1 | 3adant3 1001 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) |
3 | addcl 7738 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + 𝐶) ∈ ℂ) | |
4 | 3 | 3adant2 1000 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + 𝐶) ∈ ℂ) |
5 | shftfval.1 | . . . 4 ⊢ 𝐹 ∈ V | |
6 | 5 | shftval 10590 | . . 3 ⊢ (((𝐴 − 𝐵) ∈ ℂ ∧ (𝐴 + 𝐶) ∈ ℂ) → ((𝐹 shift (𝐴 − 𝐵))‘(𝐴 + 𝐶)) = (𝐹‘((𝐴 + 𝐶) − (𝐴 − 𝐵)))) |
7 | 2, 4, 6 | syl2anc 408 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐹 shift (𝐴 − 𝐵))‘(𝐴 + 𝐶)) = (𝐹‘((𝐴 + 𝐶) − (𝐴 − 𝐵)))) |
8 | pnncan 7996 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐶) − (𝐴 − 𝐵)) = (𝐶 + 𝐵)) | |
9 | 8 | 3com23 1187 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐶) − (𝐴 − 𝐵)) = (𝐶 + 𝐵)) |
10 | addcom 7892 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 + 𝐶) = (𝐶 + 𝐵)) | |
11 | 10 | 3adant1 999 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 + 𝐶) = (𝐶 + 𝐵)) |
12 | 9, 11 | eqtr4d 2173 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐶) − (𝐴 − 𝐵)) = (𝐵 + 𝐶)) |
13 | 12 | fveq2d 5418 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐹‘((𝐴 + 𝐶) − (𝐴 − 𝐵))) = (𝐹‘(𝐵 + 𝐶))) |
14 | 7, 13 | eqtrd 2170 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐹 shift (𝐴 − 𝐵))‘(𝐴 + 𝐶)) = (𝐹‘(𝐵 + 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 Vcvv 2681 ‘cfv 5118 (class class class)co 5767 ℂcc 7611 + caddc 7616 − cmin 7926 shift cshi 10579 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-resscn 7705 ax-1cn 7706 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sub 7928 df-shft 10580 |
This theorem is referenced by: shftval3 10592 |
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