Theorem oelim 8521

Description: Ordinal exponentiation with a limit exponent and nonzero base. Definition 8.30 of [TakeutiZaring] p. 67. Definition 2.6 of [Schloeder] p. 4. (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 6423	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		simpr 490	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → Lim 𝐵)
3	1, 2	jca 521	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (𝐵 ∈ On ∧ Lim 𝐵))
4		rdglim2a 8422	. . . 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 740	. . 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 8502	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝐵))
7		onelon 6382	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
8		oevn0 8502	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥))
9	7, 8	sylanl2 694	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥))
10	9	exp42 441	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝑥 ∈ 𝐵 → (∅ ∈ 𝐴 → (𝐴 ↑o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥)))))
11	10	com34 92	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → (𝑥 ∈ 𝐵 → (𝐴 ↑o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥)))))
12	11	imp41 431	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐴 ↑o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥))
13	12	iuneq2dv 4976	. . . . 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 2776	. . . 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 734	. . 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 694	1 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ↑o 𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ∅c0 4279  ∪ ciun 4951   ↦ cmpt 5186  Oncon0 6357  Lim wlim 6358  ‘cfv 6533  (class class class)co 7413  reccrdg 8398  1_oc1o 8448   ·_o comu 8453   ↑_o coe 8454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-oexp 8461

This theorem is used by:  oecl  8524  oe1m  8532  oen0  8574  oeordi  8575  oewordri  8580  oeworde  8581  oelim2  8583  oeoalem  8584  oeoelem  8586  oeeulem  8589  oe0suclim  44118  nnoeomeqom  44153