Theorem onesuc 8551

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 7886	. 2 ⊢ Lim ω
2		frsuc 8460	. . 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 7895	. . . 4 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
4	3	fvresd 6907	. . 3 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o) ↾ ω)‘suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵))
5		fvres 6906	. . . 4 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
6	5	fveq2d 6891	. . 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 2777	. 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 8546	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3464   ↦ cmpt 5207   ↾ cres 5669  Oncon0 6365  suc csuc 6367  ‘cfv 6542  (class class class)co 7414  ωcom 7870  reccrdg 8432  1_oc1o 8482   ·_o comu 8487   ↑_o coe 8488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414  ax-un 7738

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6303  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7871  df-2nd 7998  df-frecs 8289  df-wrecs 8320  df-recs 8394  df-rdg 8433  df-1o 8489  df-omul 8494  df-oexp 8495

This theorem is referenced by:  oe1  8565  nnesuc  8629