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Mirrors > Home > MPE Home > Th. List > Mathboxes > digfval | Structured version Visualization version GIF version |
Description: Operation to obtain the π th digit of a nonnegative real number π in the positional system with base π΅. (Contributed by AV, 23-May-2020.) |
Ref | Expression |
---|---|
digfval | β’ (π΅ β β β (digitβπ΅) = (π β β€, π β (0[,)+β) β¦ ((ββ((π΅β-π) Β· π)) mod π΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dig 47282 | . 2 β’ digit = (π β β β¦ (π β β€, π β (0[,)+β) β¦ ((ββ((πβ-π) Β· π)) mod π))) | |
2 | oveq1 7416 | . . . . 5 β’ (π = π΅ β (πβ-π) = (π΅β-π)) | |
3 | 2 | fvoveq1d 7431 | . . . 4 β’ (π = π΅ β (ββ((πβ-π) Β· π)) = (ββ((π΅β-π) Β· π))) |
4 | id 22 | . . . 4 β’ (π = π΅ β π = π΅) | |
5 | 3, 4 | oveq12d 7427 | . . 3 β’ (π = π΅ β ((ββ((πβ-π) Β· π)) mod π) = ((ββ((π΅β-π) Β· π)) mod π΅)) |
6 | 5 | mpoeq3dv 7488 | . 2 β’ (π = π΅ β (π β β€, π β (0[,)+β) β¦ ((ββ((πβ-π) Β· π)) mod π)) = (π β β€, π β (0[,)+β) β¦ ((ββ((π΅β-π) Β· π)) mod π΅))) |
7 | id 22 | . 2 β’ (π΅ β β β π΅ β β) | |
8 | zex 12567 | . . . 4 β’ β€ β V | |
9 | ovex 7442 | . . . 4 β’ (0[,)+β) β V | |
10 | 8, 9 | pm3.2i 472 | . . 3 β’ (β€ β V β§ (0[,)+β) β V) |
11 | eqid 2733 | . . . 4 β’ (π β β€, π β (0[,)+β) β¦ ((ββ((π΅β-π) Β· π)) mod π΅)) = (π β β€, π β (0[,)+β) β¦ ((ββ((π΅β-π) Β· π)) mod π΅)) | |
12 | 11 | mpoexg 8063 | . . 3 β’ ((β€ β V β§ (0[,)+β) β V) β (π β β€, π β (0[,)+β) β¦ ((ββ((π΅β-π) Β· π)) mod π΅)) β V) |
13 | 10, 12 | mp1i 13 | . 2 β’ (π΅ β β β (π β β€, π β (0[,)+β) β¦ ((ββ((π΅β-π) Β· π)) mod π΅)) β V) |
14 | 1, 6, 7, 13 | fvmptd3 7022 | 1 β’ (π΅ β β β (digitβπ΅) = (π β β€, π β (0[,)+β) β¦ ((ββ((π΅β-π) Β· π)) mod π΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3475 βcfv 6544 (class class class)co 7409 β cmpo 7411 0cc0 11110 Β· cmul 11115 +βcpnf 11245 -cneg 11445 βcn 12212 β€cz 12558 [,)cico 13326 βcfl 13755 mod cmo 13834 βcexp 14027 digitcdig 47281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-neg 11447 df-z 12559 df-dig 47282 |
This theorem is referenced by: digval 47284 |
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