Theorem onesuc 8360

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) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))

Proof of Theorem onesuc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limom 7728	. 2 ⊢ Lim ω
2		frsuc 8268	. . 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 7737	. . . 4 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
4	3	fvresd 6794	. . 3 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o) ↾ ω)‘suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵))
5		fvres 6793	. . . 4 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
6	5	fveq2d 6778	. . 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)))
7	2, 4, 6	3eqtr3d 2786	. 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)))
8	1, 7	oesuclem 8355	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ↦ cmpt 5157   ↾ cres 5591  Oncon0 6266  suc csuc 6268  ‘cfv 6433  (class class class)co 7275  ωcom 7712  reccrdg 8240  1_oc1o 8290   ·_o comu 8295   ↑_o coe 8296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-omul 8302  df-oexp 8303

This theorem is referenced by:  oe1  8375  nnesuc  8439