Theorem fv0p1e1 9248

Description: Function value at N + 1 with N replaced by 0. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)

Assertion

Ref	Expression
fv0p1e1	|- ( N = 0 -> ( F ` ( N + 1 ) ) = ( F ` 1 ) )

Proof of Theorem fv0p1e1

Step	Hyp	Ref	Expression
1		oveq1 6020	. . 3 |- ( N = 0 -> ( N + 1 ) = ( 0 + 1 ) )
2		0p1e1 9247	. . 3 |- ( 0 + 1 ) = 1
3	1, 2	eqtrdi 2278	. 2 |- ( N = 0 -> ( N + 1 ) = 1 )
4	3	fveq2d 5639	1 |- ( N = 0 -> ( F ` ( N + 1 ) ) = ( F ` 1 ) )

Syntax hints:   -> wi 4   = wceq 1395   ` cfv 5324  (class class class)co 6013   0cc0 8022   1c1 8023   + caddc 8025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-1cn 8115  ax-icn 8117  ax-addcl 8118  ax-mulcl 8120  ax-addcom 8122  ax-i2m1 8127  ax-0id 8130

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-iota 5284  df-fv 5332  df-ov 6016

This theorem is referenced by:  mertenslem2  12087  fprodfac  12166  2wlklem  16171  clwwlkn2  16216