Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > revfv | Structured version Visualization version GIF version |
Description: Reverse of a word at a point. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
Ref | Expression |
---|---|
revfv | ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑋 ∈ (0..^(#‘𝑊))) → ((reverse‘𝑊)‘𝑋) = (𝑊‘(((#‘𝑊) − 1) − 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | revval 13360 | . . 3 ⊢ (𝑊 ∈ Word 𝐴 → (reverse‘𝑊) = (𝑥 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘(((#‘𝑊) − 1) − 𝑥)))) | |
2 | 1 | fveq1d 6105 | . 2 ⊢ (𝑊 ∈ Word 𝐴 → ((reverse‘𝑊)‘𝑋) = ((𝑥 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘(((#‘𝑊) − 1) − 𝑥)))‘𝑋)) |
3 | oveq2 6557 | . . . 4 ⊢ (𝑥 = 𝑋 → (((#‘𝑊) − 1) − 𝑥) = (((#‘𝑊) − 1) − 𝑋)) | |
4 | 3 | fveq2d 6107 | . . 3 ⊢ (𝑥 = 𝑋 → (𝑊‘(((#‘𝑊) − 1) − 𝑥)) = (𝑊‘(((#‘𝑊) − 1) − 𝑋))) |
5 | eqid 2610 | . . 3 ⊢ (𝑥 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘(((#‘𝑊) − 1) − 𝑥))) = (𝑥 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘(((#‘𝑊) − 1) − 𝑥))) | |
6 | fvex 6113 | . . 3 ⊢ (𝑊‘(((#‘𝑊) − 1) − 𝑋)) ∈ V | |
7 | 4, 5, 6 | fvmpt 6191 | . 2 ⊢ (𝑋 ∈ (0..^(#‘𝑊)) → ((𝑥 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘(((#‘𝑊) − 1) − 𝑥)))‘𝑋) = (𝑊‘(((#‘𝑊) − 1) − 𝑋))) |
8 | 2, 7 | sylan9eq 2664 | 1 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑋 ∈ (0..^(#‘𝑊))) → ((reverse‘𝑊)‘𝑋) = (𝑊‘(((#‘𝑊) − 1) − 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 − cmin 10145 ..^cfzo 12334 #chash 12979 Word cword 13146 reversecreverse 13152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-reverse 13160 |
This theorem is referenced by: revs1 13365 revccat 13366 revrev 13367 revco 13431 |
Copyright terms: Public domain | W3C validator |