Theorem onesuc 7764

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 7227	. 2 ⊢ Lim ω
2		frsuc 7685	. . 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 7233	. . . 4 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
4		fvres 6348	. . . 4 ⊢ (suc 𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜) ↾ ω)‘suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘suc 𝐵))
5	3, 4	syl 17	. . 3 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜) ↾ ω)‘suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘suc 𝐵))
6		fvres 6348	. . . 4 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
7	6	fveq2d 6336	. . 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)))
8	2, 5, 7	3eqtr3d 2813	. 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)))
9	1, 8	oesuclem 7759	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ↑𝑜 suc 𝐵) = ((𝐴 ↑𝑜 𝐵) ·𝑜 𝐴))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  Vcvv 3351   ↦ cmpt 4863   ↾ cres 5251  Oncon0 5866  suc csuc 5868  ‘cfv 6031  (class class class)co 6793  ωcom 7212  reccrdg 7658  1_𝑜c1o 7706   ·_𝑜 comu 7711   ↑_𝑜 coe 7712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-omul 7718  df-oexp 7719

This theorem is referenced by:  oe1  7778  nnesuc  7842