Theorem digfval 42391

Description: Operation to obtain the k th digit of a nonnegative real number r in the positional system with base B. (Contributed by AV, 23-May-2020.)

Assertion

Ref	Expression
digfval	|- ( B e. NN -> (digit ` B ) = ( k e. ZZ , r e. ( 0 [,) +oo ) |-> ( ( |_ ` ( ( B ^ -u k ) x. r ) ) mod B ) ) )

Distinct variable group:   k, r, B

Proof of Theorem digfval

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dig 42390	. . 3 |- digit = ( b e. NN |-> ( k e. ZZ , r e. ( 0 [,) +oo ) |-> ( ( |_ ` ( ( b ^ -u k ) x. r ) ) mod b ) ) )
2	1	a1i 11	. 2 |- ( B e. NN -> digit = ( b e. NN |-> ( k e. ZZ , r e. ( 0 [,) +oo ) |-> ( ( |_ ` ( ( b ^ -u k ) x. r ) ) mod b ) ) ) )
3		oveq1 6657	. . . . . . 7 |- ( b = B -> ( b ^ -u k ) = ( B ^ -u k ) )
4	3	oveq1d 6665	. . . . . 6 |- ( b = B -> ( ( b ^ -u k ) x. r ) = ( ( B ^ -u k ) x. r ) )
5	4	fveq2d 6195	. . . . 5 |- ( b = B -> ( |_ ` ( ( b ^ -u k ) x. r ) ) = ( |_ ` ( ( B ^ -u k ) x. r ) ) )
6		id 22	. . . . 5 |- ( b = B -> b = B )
7	5, 6	oveq12d 6668	. . . 4 |- ( b = B -> ( ( |_ ` ( ( b ^ -u k ) x. r ) ) mod b ) = ( ( |_ ` ( ( B ^ -u k ) x. r ) ) mod B ) )
8	7	mpt2eq3dv 6721	. . 3 |- ( b = B -> ( k e. ZZ , r e. ( 0 [,) +oo ) |-> ( ( |_ ` ( ( b ^ -u k ) x. r ) ) mod b ) ) = ( k e. ZZ , r e. ( 0 [,) +oo ) |-> ( ( |_ ` ( ( B ^ -u k ) x. r ) ) mod B ) ) )
9	8	adantl 482	. 2 |- ( ( B e. NN /\ b = B ) -> ( k e. ZZ , r e. ( 0 [,) +oo ) |-> ( ( |_ ` ( ( b ^ -u k ) x. r ) ) mod b ) ) = ( k e. ZZ , r e. ( 0 [,) +oo ) |-> ( ( |_ ` ( ( B ^ -u k ) x. r ) ) mod B ) ) )
10		id 22	. 2 |- ( B e. NN -> B e. NN )
11		zex 11386	. . . 4 |- ZZ e. _V
12		ovex 6678	. . . 4 |- ( 0 [,) +oo ) e. _V
13	11, 12	pm3.2i 471	. . 3 |- ( ZZ e. _V /\ ( 0 [,) +oo ) e. _V )
14		eqid 2622	. . . 4 |- ( k e. ZZ , r e. ( 0 [,) +oo ) |-> ( ( |_ ` ( ( B ^ -u k ) x. r ) ) mod B ) ) = ( k e. ZZ , r e. ( 0 [,) +oo ) |-> ( ( |_ ` ( ( B ^ -u k ) x. r ) ) mod B ) )
15	14	mpt2exg 7245	. . 3 |- ( ( ZZ e. _V /\ ( 0 [,) +oo ) e. _V ) -> ( k e. ZZ , r e. ( 0 [,) +oo ) |-> ( ( |_ ` ( ( B ^ -u k ) x. r ) ) mod B ) ) e. _V )
16	13, 15	mp1i 13	. 2 |- ( B e. NN -> ( k e. ZZ , r e. ( 0 [,) +oo ) |-> ( ( |_ ` ( ( B ^ -u k ) x. r ) ) mod B ) ) e. _V )
17	2, 9, 10, 16	fvmptd 6288	1 |- ( B e. NN -> (digit ` B ) = ( k e. ZZ , r e. ( 0 [,) +oo ) |-> ( ( |_ ` ( ( B ^ -u k ) x. r ) ) mod B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   x. cmul 9941   +oocpnf 10071   -ucneg 10267   NNcn 11020   ZZcz 11377   [,)cico 12177   |_cfl 12591   mod cmo 12668   ^cexp 12860  digitcdig 42389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-neg 10269  df-z 11378  df-dig 42390

This theorem is referenced by:  digval  42392