Theorem fv0p1e1 9097

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 5925	. . 3 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1))
2		0p1e1 9096	. . 3 ⊢ (0 + 1) = 1
3	1, 2	eqtrdi 2242	. 2 ⊢ (𝑁 = 0 → (𝑁 + 1) = 1)
4	3	fveq2d 5558	1 ⊢ (𝑁 = 0 → (𝐹‘(𝑁 + 1)) = (𝐹‘1))

Syntax hints:   → wi 4   = wceq 1364  ‘cfv 5254  (class class class)co 5918  0cc0 7872  1c1 7873   + caddc 7875

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-1cn 7965  ax-icn 7967  ax-addcl 7968  ax-mulcl 7970  ax-addcom 7972  ax-i2m1 7977  ax-0id 7980

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921

This theorem is referenced by:  mertenslem2  11679  fprodfac  11758