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Theorem onesuc 8144
Description: Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.)
Assertion
Ref Expression
onesuc ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴o suc 𝐵) = ((𝐴o 𝐵) ·o 𝐴))

Proof of Theorem onesuc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 limom 7584 . 2 Lim ω
2 frsuc 8061 . . 3 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o) ↾ ω)‘𝐵)))
3 peano2 7591 . . . 4 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
43fvresd 6683 . . 3 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o) ↾ ω)‘suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵))
5 fvres 6682 . . . 4 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
65fveq2d 6667 . . 3 (𝐵 ∈ ω → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o) ↾ ω)‘𝐵)) = ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
72, 4, 63eqtr3d 2861 . 2 (𝐵 ∈ ω → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
81, 7oesuclem 8139 1 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴o suc 𝐵) = ((𝐴o 𝐵) ·o 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cmpt 5137  cres 5550  Oncon0 6184  suc csuc 6186  cfv 6348  (class class class)co 7145  ωcom 7569  reccrdg 8034  1oc1o 8084   ·o comu 8089  o coe 8090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-omul 8096  df-oexp 8097
This theorem is referenced by:  oe1  8159  nnesuc  8223
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