Theorem fv0p1e1 9225

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 6008	. . 3 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1))
2		0p1e1 9224	. . 3 ⊢ (0 + 1) = 1
3	1, 2	eqtrdi 2278	. 2 ⊢ (𝑁 = 0 → (𝑁 + 1) = 1)
4	3	fveq2d 5631	1 ⊢ (𝑁 = 0 → (𝐹‘(𝑁 + 1)) = (𝐹‘1))

Syntax hints:   → wi 4   = wceq 1395  ‘cfv 5318  (class class class)co 6001  0cc0 7999  1c1 8000   + caddc 8002

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 8092  ax-icn 8094  ax-addcl 8095  ax-mulcl 8097  ax-addcom 8099  ax-i2m1 8104  ax-0id 8107

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 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6004

This theorem is referenced by:  mertenslem2  12047  fprodfac  12126