Theorem fv0p1e1 9153

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 5953	. . 3 |- ( N = 0 -> ( N + 1 ) = ( 0 + 1 ) )
2		0p1e1 9152	. . 3 |- ( 0 + 1 ) = 1
3	1, 2	eqtrdi 2254	. 2 |- ( N = 0 -> ( N + 1 ) = 1 )
4	3	fveq2d 5582	1 |- ( N = 0 -> ( F ` ( N + 1 ) ) = ( F ` 1 ) )

Syntax hints:   -> wi 4   = wceq 1373   ` cfv 5272  (class class class)co 5946   0cc0 7927   1c1 7928   + caddc 7930

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-1cn 8020  ax-icn 8022  ax-addcl 8023  ax-mulcl 8025  ax-addcom 8027  ax-i2m1 8032  ax-0id 8035

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-iota 5233  df-fv 5280  df-ov 5949

This theorem is referenced by:  mertenslem2  11880  fprodfac  11959