Theorem fv0p1e1 8963

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 5843	. . 3 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1))
2		0p1e1 8962	. . 3 ⊢ (0 + 1) = 1
3	1, 2	eqtrdi 2213	. 2 ⊢ (𝑁 = 0 → (𝑁 + 1) = 1)
4	3	fveq2d 5484	1 ⊢ (𝑁 = 0 → (𝐹‘(𝑁 + 1)) = (𝐹‘1))

Syntax hints:   → wi 4   = wceq 1342  ‘cfv 5182  (class class class)co 5836  0cc0 7744  1c1 7745   + caddc 7747

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146  ax-1cn 7837  ax-icn 7839  ax-addcl 7840  ax-mulcl 7842  ax-addcom 7844  ax-i2m1 7849  ax-0id 7852

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-v 2723  df-un 3115  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-iota 5147  df-fv 5190  df-ov 5839

This theorem is referenced by:  mertenslem2  11463  fprodfac  11542