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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 3492 | . 2 β’ (π β π β π β V) | |
2 | fveq2 6902 | . . . . 5 β’ (π€ = π β (β―βπ€) = (β―βπ)) | |
3 | 2 | oveq2d 7442 | . . . 4 β’ (π€ = π β (0..^(β―βπ€)) = (0..^(β―βπ))) |
4 | id 22 | . . . . 5 β’ (π€ = π β π€ = π) | |
5 | 2 | oveq1d 7441 | . . . . . 6 β’ (π€ = π β ((β―βπ€) β 1) = ((β―βπ) β 1)) |
6 | 5 | oveq1d 7441 | . . . . 5 β’ (π€ = π β (((β―βπ€) β 1) β π₯) = (((β―βπ) β 1) β π₯)) |
7 | 4, 6 | fveq12d 6909 | . . . 4 β’ (π€ = π β (π€β(((β―βπ€) β 1) β π₯)) = (πβ(((β―βπ) β 1) β π₯))) |
8 | 3, 7 | mpteq12dv 5243 | . . 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 14749 | . . 3 β’ reverse = (π€ β V β¦ (π₯ β (0..^(β―βπ€)) β¦ (π€β(((β―βπ€) β 1) β π₯)))) | |
10 | ovex 7459 | . . . 4 β’ (0..^(β―βπ)) β V | |
11 | 10 | mptex 7241 | . . 3 β’ (π₯ β (0..^(β―βπ)) β¦ (πβ(((β―βπ) β 1) β π₯))) β V |
12 | 8, 9, 11 | fvmpt 7010 | . 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 3473 β¦ cmpt 5235 βcfv 6553 (class class class)co 7426 0cc0 11146 1c1 11147 β cmin 11482 ..^cfzo 13667 β―chash 14329 reversecreverse 14748 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-reverse 14749 |
This theorem is referenced by: revcl 14751 revlen 14752 revfv 14753 repswrevw 14777 revco 14825 |
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