Theorem oelim 8193

Description: Ordinal exponentiation with a limit exponent and nonzero base. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 1-Jan-2005.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
oelim	⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ↑o 𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem oelim

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limelon 6236	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		simpr 488	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → Lim 𝐵)
3	1, 2	jca 515	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (𝐵 ∈ On ∧ Lim 𝐵))
4		rdglim2a 8101	. . . 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 727	. . 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 8174	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝐵))
7		onelon 6198	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
8		oevn0 8174	. . . . . . . . . 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 439	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝑥 ∈ 𝐵 → (∅ ∈ 𝐴 → (𝐴 ↑o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥)))))
11	10	com34 91	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → (𝑥 ∈ 𝐵 → (𝐴 ↑o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥)))))
12	11	imp41 429	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐴 ↑o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥))
13	12	iuneq2dv 4906	. . . . 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 2755	. . . 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 721	. . 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 260	. 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))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2114  Vcvv 3399  ∅c0 4212  ∪ ciun 4882   ↦ cmpt 5111  Oncon0 6173  Lim wlim 6174  ‘cfv 6340  (class class class)co 7173  reccrdg 8077  1_oc1o 8127   ·_o comu 8132   ↑_o coe 8133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7482

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-wrecs 7979  df-recs 8040  df-rdg 8078  df-1o 8134  df-oexp 8140

This theorem is referenced by:  oecl  8196  oe1m  8205  oen0  8246  oeordi  8247  oewordri  8252  oeworde  8253  oelim2  8255  oeoalem  8256  oeoelem  8258  oeeulem  8261