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Mirrors > Home > MPE Home > Th. List > revval | Structured version Visualization version GIF version |
Description: Value of the word reversing function. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
Ref | Expression |
---|---|
revval | β’ (π β π β (reverseβπ) = (π₯ β (0..^(β―βπ)) β¦ (πβ(((β―βπ) β 1) β π₯)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3487 | . 2 β’ (π β π β π β V) | |
2 | fveq2 6884 | . . . . 5 β’ (π€ = π β (β―βπ€) = (β―βπ)) | |
3 | 2 | oveq2d 7420 | . . . 4 β’ (π€ = π β (0..^(β―βπ€)) = (0..^(β―βπ))) |
4 | id 22 | . . . . 5 β’ (π€ = π β π€ = π) | |
5 | 2 | oveq1d 7419 | . . . . . 6 β’ (π€ = π β ((β―βπ€) β 1) = ((β―βπ) β 1)) |
6 | 5 | oveq1d 7419 | . . . . 5 β’ (π€ = π β (((β―βπ€) β 1) β π₯) = (((β―βπ) β 1) β π₯)) |
7 | 4, 6 | fveq12d 6891 | . . . 4 β’ (π€ = π β (π€β(((β―βπ€) β 1) β π₯)) = (πβ(((β―βπ) β 1) β π₯))) |
8 | 3, 7 | mpteq12dv 5232 | . . 3 β’ (π€ = π β (π₯ β (0..^(β―βπ€)) β¦ (π€β(((β―βπ€) β 1) β π₯))) = (π₯ β (0..^(β―βπ)) β¦ (πβ(((β―βπ) β 1) β π₯)))) |
9 | df-reverse 14713 | . . 3 β’ reverse = (π€ β V β¦ (π₯ β (0..^(β―βπ€)) β¦ (π€β(((β―βπ€) β 1) β π₯)))) | |
10 | ovex 7437 | . . . 4 β’ (0..^(β―βπ)) β V | |
11 | 10 | mptex 7219 | . . 3 β’ (π₯ β (0..^(β―βπ)) β¦ (πβ(((β―βπ) β 1) β π₯))) β V |
12 | 8, 9, 11 | fvmpt 6991 | . 2 β’ (π β V β (reverseβπ) = (π₯ β (0..^(β―βπ)) β¦ (πβ(((β―βπ) β 1) β π₯)))) |
13 | 1, 12 | syl 17 | 1 β’ (π β π β (reverseβπ) = (π₯ β (0..^(β―βπ)) β¦ (πβ(((β―βπ) β 1) β π₯)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3468 β¦ cmpt 5224 βcfv 6536 (class class class)co 7404 0cc0 11109 1c1 11110 β cmin 11445 ..^cfzo 13630 β―chash 14293 reversecreverse 14712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-reverse 14713 |
This theorem is referenced by: revcl 14715 revlen 14716 revfv 14717 repswrevw 14741 revco 14789 |
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