Theorem oelim 8364

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 6329	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		simpr 485	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → Lim 𝐵)
3	1, 2	jca 512	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (𝐵 ∈ On ∧ Lim 𝐵))
4		rdglim2a 8264	. . . 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 724	. . 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 8345	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝐵))
7		onelon 6291	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
8		oevn0 8345	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥))
9	7, 8	sylanl2 678	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥))
10	9	exp42 436	. . . . . . . 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 426	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐴 ↑o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥))
13	12	iuneq2dv 4948	. . . . 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 2754	. . . 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 718	. . 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 256	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ↑o 𝑥))
17	3, 16	sylanl2 678	1 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ↑o 𝑥))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ∅c0 4256  ∪ ciun 4924   ↦ cmpt 5157  Oncon0 6266  Lim wlim 6267  ‘cfv 6433  (class class class)co 7275  reccrdg 8240  1_oc1o 8290   ·_o comu 8295   ↑_o coe 8296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oexp 8303

This theorem is referenced by:  oecl  8367  oe1m  8376  oen0  8417  oeordi  8418  oewordri  8423  oeworde  8424  oelim2  8426  oeoalem  8427  oeoelem  8429  oeeulem  8432