Theorem omlim 7500

Description: Ordinal multiplication with a limit ordinal. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 3-Aug-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
omlim	⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ·𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ·𝑜 𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem omlim

Dummy variable 𝑦 is distinct from all other variables.

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)), ∅)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))
5	4	adantl 481	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))
6		omv 7479	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝐵))
7		onelon 5665	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
8		omv 7479	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))
9	7, 8	sylan2 490	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ 𝐵)) → (𝐴 ·𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))
10	9	anassrs 678	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ 𝐵) → (𝐴 ·𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))
11	10	iuneq2dv 4478	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ 𝑥 ∈ 𝐵 (𝐴 ·𝑜 𝑥) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))
12	6, 11	eqeq12d 2625	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ·𝑜 𝑥) ↔ (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥)))
13	12	adantrr 749	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → ((𝐴 ·𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ·𝑜 𝑥) ↔ (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥)))
14	5, 13	mpbird 246	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (𝐴 ·𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ·𝑜 𝑥))
15	3, 14	sylan2 490	1 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ·𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ·𝑜 𝑥))

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   +_𝑜 coa 7444   ·_𝑜 comu 7445

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-omul 7452

This theorem is referenced by:  omcl  7503  om0r  7506  om1r  7510  omordi  7533  omwordri  7539  omordlim  7544  omlimcl  7545  odi  7546  omass  7547  omeulem1  7549  oeoalem  7563  oeoelem  7565  omabslem  7613  omabs  7614