Theorem fv0p1e1 8828

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 5774	. . 3 |- ( N = 0 -> ( N + 1 ) = ( 0 + 1 ) )
2		0p1e1 8827	. . 3 |- ( 0 + 1 ) = 1
3	1, 2	syl6eq 2186	. 2 |- ( N = 0 -> ( N + 1 ) = 1 )
4	3	fveq2d 5418	1 |- ( N = 0 -> ( F ` ( N + 1 ) ) = ( F ` 1 ) )

Syntax hints:   -> wi 4   = wceq 1331   ` cfv 5118  (class class class)co 5767   0cc0 7613   1c1 7614   + caddc 7616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-1cn 7706  ax-icn 7708  ax-addcl 7709  ax-mulcl 7711  ax-addcom 7713  ax-i2m1 7718  ax-0id 7721

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770

This theorem is referenced by:  mertenslem2  11298