Theorem blenval 42365

Description: The binary length of an integer. (Contributed by AV, 20-May-2020.)

Assertion

Ref	Expression
blenval	|- ( N e. V -> (#b ` N ) = if ( N = 0 , 1 , ( ( |_ ` ( 2 logb ( abs ` N ) ) ) + 1 ) ) )

Proof of Theorem blenval

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-blen 42364	. . 3 |- #b = ( n e. _V |-> if ( n = 0 , 1 , ( ( |_ ` ( 2 logb ( abs ` n ) ) ) + 1 ) ) )
2	1	a1i 11	. 2 |- ( N e. V -> #b = ( n e. _V |-> if ( n = 0 , 1 , ( ( |_ ` ( 2 logb ( abs ` n ) ) ) + 1 ) ) ) )
3		eqeq1 2626	. . . 4 |- ( n = N -> ( n = 0 <-> N = 0 ) )
4		fveq2 6191	. . . . . . 7 |- ( n = N -> ( abs ` n ) = ( abs ` N ) )
5	4	oveq2d 6666	. . . . . 6 |- ( n = N -> ( 2 logb ( abs ` n ) ) = ( 2 logb ( abs ` N ) ) )
6	5	fveq2d 6195	. . . . 5 |- ( n = N -> ( |_ ` ( 2 logb ( abs ` n ) ) ) = ( |_ ` ( 2 logb ( abs ` N ) ) ) )
7	6	oveq1d 6665	. . . 4 |- ( n = N -> ( ( |_ ` ( 2 logb ( abs ` n ) ) ) + 1 ) = ( ( |_ ` ( 2 logb ( abs ` N ) ) ) + 1 ) )
8	3, 7	ifbieq2d 4111	. . 3 |- ( n = N -> if ( n = 0 , 1 , ( ( |_ ` ( 2 logb ( abs ` n ) ) ) + 1 ) ) = if ( N = 0 , 1 , ( ( |_ ` ( 2 logb ( abs ` N ) ) ) + 1 ) ) )
9	8	adantl 482	. 2 |- ( ( N e. V /\ n = N ) -> if ( n = 0 , 1 , ( ( |_ ` ( 2 logb ( abs ` n ) ) ) + 1 ) ) = if ( N = 0 , 1 , ( ( |_ ` ( 2 logb ( abs ` N ) ) ) + 1 ) ) )
10		elex 3212	. 2 |- ( N e. V -> N e. _V )
11		1ex 10035	. . . 4 |- 1 e. _V
12		ovex 6678	. . . 4 |- ( ( |_ ` ( 2 logb ( abs ` N ) ) ) + 1 ) e. _V
13	11, 12	ifex 4156	. . 3 |- if ( N = 0 , 1 , ( ( |_ ` ( 2 logb ( abs ` N ) ) ) + 1 ) ) e. _V
14	13	a1i 11	. 2 |- ( N e. V -> if ( N = 0 , 1 , ( ( |_ ` ( 2 logb ( abs ` N ) ) ) + 1 ) ) e. _V )
15	2, 9, 10, 14	fvmptd 6288	1 |- ( N e. V -> (#b ` N ) = if ( N = 0 , 1 , ( ( |_ ` ( 2 logb ( abs ` N ) ) ) + 1 ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   2c2 11070   |_cfl 12591   abscabs 13974   log_b clogb 24502  #bcblen 42363

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-1cn 9994

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-blen 42364

This theorem is referenced by:  blen0  42366  blenn0  42367