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Theorem fv0p1e1 8803
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
StepHypRef Expression
1 oveq1 5749 . . 3 (𝑁 = 0 → (𝑁 + 1) = (0 + 1))
2 0p1e1 8802 . . 3 (0 + 1) = 1
31, 2syl6eq 2166 . 2 (𝑁 = 0 → (𝑁 + 1) = 1)
43fveq2d 5393 1 (𝑁 = 0 → (𝐹‘(𝑁 + 1)) = (𝐹‘1))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  cfv 5093  (class class class)co 5742  0cc0 7588  1c1 7589   + caddc 7591
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-1cn 7681  ax-icn 7683  ax-addcl 7684  ax-mulcl 7686  ax-addcom 7688  ax-i2m1 7693  ax-0id 7696
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745
This theorem is referenced by:  mertenslem2  11273
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