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Theorem fv0p1e1 8858
 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 5788 . . 3 (𝑁 = 0 → (𝑁 + 1) = (0 + 1))
2 0p1e1 8857 . . 3 (0 + 1) = 1
31, 2eqtrdi 2189 . 2 (𝑁 = 0 → (𝑁 + 1) = 1)
43fveq2d 5432 1 (𝑁 = 0 → (𝐹‘(𝑁 + 1)) = (𝐹‘1))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332  ‘cfv 5130  (class class class)co 5781  0cc0 7643  1c1 7644   + caddc 7646 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-1cn 7736  ax-icn 7738  ax-addcl 7739  ax-mulcl 7741  ax-addcom 7743  ax-i2m1 7748  ax-0id 7751 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-iota 5095  df-fv 5138  df-ov 5784 This theorem is referenced by:  mertenslem2  11336
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