Theorem oelim 7734

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.)

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 5901	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		simpr 479	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → Lim 𝐵)
3	1, 2	jca 555	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (𝐵 ∈ On ∧ Lim 𝐵))
4		rdglim2a 7649	. . . 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 765	. . 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 7715	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝐵))
7		onelon 5861	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
8		oevn0 7715	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥))
9	7, 8	sylanl2 686	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥))
10	9	exp42 640	. . . . . . . 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 620	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐴 ↑𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥))
13	12	iuneq2dv 4650	. . . . 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 2739	. . . 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 759	. . 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 247	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ↑𝑜 𝑥))
17	3, 16	sylanl2 686	1 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ↑𝑜 𝑥))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1596   ∈ wcel 2103  Vcvv 3304  ∅c0 4023  ∪ ciun 4628   ↦ cmpt 4837  Oncon0 5836  Lim wlim 5837  ‘cfv 6001  (class class class)co 6765  reccrdg 7625  1_𝑜c1o 7673   ·_𝑜 comu 7678   ↑_𝑜 coe 7679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oexp 7686

This theorem is referenced by:  oecl  7737  oe1m  7745  oen0  7786  oeordi  7787  oewordri  7792  oeworde  7793  oelim2  7795  oeoalem  7796  oeoelem  7798  oeeulem  7801