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Mirrors > Home > MPE Home > Th. List > onesuc | Structured version Visualization version GIF version |
Description: Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.) |
Ref | Expression |
---|---|
onesuc | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ↑𝑜 suc 𝐵) = ((𝐴 ↑𝑜 𝐵) ·𝑜 𝐴)) |
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)) |
Colors of variables: wff setvar class |
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 1𝑜c1o 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 |
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