Theorem onesuc 8322

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 7703	. 2 ⊢ Lim ω
2		frsuc 8238	. . 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 7711	. . . 4 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
4	3	fvresd 6776	. . 3 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o) ↾ ω)‘suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵))
5		fvres 6775	. . . 4 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
6	5	fveq2d 6760	. . 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 8317	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ↦ cmpt 5153   ↾ cres 5582  Oncon0 6251  suc csuc 6253  ‘cfv 6418  (class class class)co 7255  ωcom 7687  reccrdg 8211  1_oc1o 8260   ·_o comu 8265   ↑_o coe 8266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-omul 8272  df-oexp 8273

This theorem is referenced by:  oe1  8337  nnesuc  8401