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Mirrors > Home > MPE Home > Th. List > oelim | Structured version Visualization version GIF version |
Description: Ordinal exponentiation with a limit exponent and nonzero mantissa. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 1-Jan-2005.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
oelim | ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ↑𝑜 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limelon 5705 | . . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On) | |
2 | simpr 476 | . . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → Lim 𝐵) | |
3 | 1, 2 | jca 553 | . 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (𝐵 ∈ On ∧ Lim 𝐵)) |
4 | rdglim2a 7416 | . . . 4 ⊢ ((𝐵 ∈ On ∧ Lim 𝐵) → (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥)) | |
5 | 4 | ad2antlr 759 | . . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥)) |
6 | oevn0 7482 | . . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝐵)) | |
7 | onelon 5665 | . . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On) | |
8 | oevn0 7482 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥)) | |
9 | 7, 8 | sylanl2 681 | . . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥)) |
10 | 9 | exp42 637 | . . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝑥 ∈ 𝐵 → (∅ ∈ 𝐴 → (𝐴 ↑𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥))))) |
11 | 10 | com34 89 | . . . . . . 7 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → (𝑥 ∈ 𝐵 → (𝐴 ↑𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥))))) |
12 | 11 | imp41 617 | . . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐴 ↑𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥)) |
13 | 12 | iuneq2dv 4478 | . . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∪ 𝑥 ∈ 𝐵 (𝐴 ↑𝑜 𝑥) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥)) |
14 | 6, 13 | eqeq12d 2625 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ↑𝑜 𝑥) ↔ (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥))) |
15 | 14 | adantlrr 753 | . . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ((𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ↑𝑜 𝑥) ↔ (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥))) |
16 | 5, 15 | mpbird 246 | . 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ↑𝑜 𝑥)) |
17 | 3, 16 | sylanl2 681 | 1 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ↑𝑜 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ∪ ciun 4455 ↦ cmpt 4643 Oncon0 5640 Lim wlim 5641 ‘cfv 5804 (class class class)co 6549 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-rep 4699 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-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oexp 7453 |
This theorem is referenced by: oecl 7504 oe1m 7512 oen0 7553 oeordi 7554 oewordri 7559 oeworde 7560 oelim2 7562 oeoalem 7563 oeoelem 7565 oeeulem 7568 |
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