Theorem fmtno 41441

Description: The N th Fermat number. (Contributed by AV, 13-Jun-2021.)

Assertion

Ref	Expression
fmtno	|- ( N e. NN0 -> (FermatNo ` N ) = ( ( 2 ^ ( 2 ^ N ) ) + 1 ) )

Proof of Theorem fmtno

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fmtno 41440	. . 3 |- FermatNo = ( n e. NN0 |-> ( ( 2 ^ ( 2 ^ n ) ) + 1 ) )
2	1	a1i 11	. 2 |- ( N e. NN0 -> FermatNo = ( n e. NN0 |-> ( ( 2 ^ ( 2 ^ n ) ) + 1 ) ) )
3		oveq2 6658	. . . . 5 |- ( n = N -> ( 2 ^ n ) = ( 2 ^ N ) )
4	3	oveq2d 6666	. . . 4 |- ( n = N -> ( 2 ^ ( 2 ^ n ) ) = ( 2 ^ ( 2 ^ N ) ) )
5	4	oveq1d 6665	. . 3 |- ( n = N -> ( ( 2 ^ ( 2 ^ n ) ) + 1 ) = ( ( 2 ^ ( 2 ^ N ) ) + 1 ) )
6	5	adantl 482	. 2 |- ( ( N e. NN0 /\ n = N ) -> ( ( 2 ^ ( 2 ^ n ) ) + 1 ) = ( ( 2 ^ ( 2 ^ N ) ) + 1 ) )
7		id 22	. 2 |- ( N e. NN0 -> N e. NN0 )
8		ovexd 6680	. 2 |- ( N e. NN0 -> ( ( 2 ^ ( 2 ^ N ) ) + 1 ) e. _V )
9	2, 6, 7, 8	fvmptd 6288	1 |- ( N e. NN0 -> (FermatNo ` N ) = ( ( 2 ^ ( 2 ^ N ) ) + 1 ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   2c2 11070   NN0cn0 11292   ^cexp 12860  FermatNocfmtno 41439

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

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-fmtno 41440

This theorem is referenced by:  fmtnoge3  41442  fmtnom1nn  41444  fmtnoodd  41445  fmtnof1  41447  fmtnorec1  41449  fmtnosqrt  41451  fmtno0  41452  fmtno1  41453  fmtnorec2lem  41454  fmtnorec3  41460  fmtnorec4  41461  fmtno2  41462  fmtno3  41463  fmtno4  41464  fmtnoprmfac1lem  41476  fmtno4prm  41487  2pwp1prmfmtno  41504