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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 6972 | . 2 ⊢ Lim ω | |
2 | frsuc 7419 | . . 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 6978 | . . . 4 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω) | |
4 | fvres 6117 | . . . 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 6117 | . . . 4 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)) | |
7 | 6 | fveq2d 6107 | . . 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 2652 | . 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 7492 | 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 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ↦ cmpt 4643 ↾ cres 5040 Oncon0 5640 suc csuc 5642 ‘cfv 5804 (class class class)co 6549 ωcom 6957 reccrdg 7392 1𝑜c1o 7440 ·𝑜 comu 7445 ↑𝑜 coe 7446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-omul 7452 df-oexp 7453 |
This theorem is referenced by: oe1 7511 nnesuc 7575 |
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