Theorem omlimcl2 43231

Description: The product of a limit ordinal with any nonzero ordinal is a limit ordinal. (Contributed by RP, 8-Jan-2025.)

Assertion

Ref	Expression
omlimcl2	⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐵 ·o 𝐴))

Proof of Theorem omlimcl2

Step	Hyp	Ref	Expression
1		eloni 6342	. . . . . 6 ⊢ (𝐴 ∈ On → Ord 𝐴)
2	1	ad2antrr 726	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Ord 𝐴)
3		ne0i 4304	. . . . . 6 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅)
4	3	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → 𝐴 ≠ ∅)
5		id 22	. . . . 5 ⊢ (𝐴 = ∪ 𝐴 → 𝐴 = ∪ 𝐴)
6		df-lim 6337	. . . . . 6 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
7	6	biimpri 228	. . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴)
8	2, 4, 5, 7	syl2an3an 1424	. . . 4 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴)
9	8	ex 412	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 = ∪ 𝐴 → Lim 𝐴))
10		limelon 6397	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
11	10	ad3antlr 731	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → 𝐵 ∈ On)
12		simpll 766	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → 𝐴 ∈ On)
13	12	anim1i 615	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → (𝐴 ∈ On ∧ Lim 𝐴))
14		0ellim 6396	. . . . . . 7 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
15	14	adantl 481	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∅ ∈ 𝐵)
16	15	ad3antlr 731	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → ∅ ∈ 𝐵)
17		omlimcl 8542	. . . . 5 ⊢ (((𝐵 ∈ On ∧ (𝐴 ∈ On ∧ Lim 𝐴)) ∧ ∅ ∈ 𝐵) → Lim (𝐵 ·o 𝐴))
18	11, 13, 16, 17	syl21anc 837	. . . 4 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → Lim (𝐵 ·o 𝐴))
19	18	ex 412	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (Lim 𝐴 → Lim (𝐵 ·o 𝐴)))
20	9, 19	syld 47	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 = ∪ 𝐴 → Lim (𝐵 ·o 𝐴)))
21		onuni 7764	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
22	21, 10	anim12ci 614	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐵 ∈ On ∧ ∪ 𝐴 ∈ On))
23		omcl 8500	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ ∪ 𝐴 ∈ On) → (𝐵 ·o ∪ 𝐴) ∈ On)
24	22, 23	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐵 ·o ∪ 𝐴) ∈ On)
25		simpr 484	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐵 ∈ 𝐶 ∧ Lim 𝐵))
26	24, 25	jca 511	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ((𝐵 ·o ∪ 𝐴) ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)))
27	26	ad2antrr 726	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → ((𝐵 ·o ∪ 𝐴) ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)))
28		oalimcl 8524	. . . . 5 ⊢ (((𝐵 ·o ∪ 𝐴) ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim ((𝐵 ·o ∪ 𝐴) +o 𝐵))
29	27, 28	syl 17	. . . 4 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → Lim ((𝐵 ·o ∪ 𝐴) +o 𝐵))
30		simpr 484	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → 𝐴 = suc ∪ 𝐴)
31	30	oveq2d 7403	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → (𝐵 ·o 𝐴) = (𝐵 ·o suc ∪ 𝐴))
32	22	ad2antrr 726	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → (𝐵 ∈ On ∧ ∪ 𝐴 ∈ On))
33		omsuc 8490	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ ∪ 𝐴 ∈ On) → (𝐵 ·o suc ∪ 𝐴) = ((𝐵 ·o ∪ 𝐴) +o 𝐵))
34	32, 33	syl 17	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → (𝐵 ·o suc ∪ 𝐴) = ((𝐵 ·o ∪ 𝐴) +o 𝐵))
35	31, 34	eqtrd 2764	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → (𝐵 ·o 𝐴) = ((𝐵 ·o ∪ 𝐴) +o 𝐵))
36		limeq 6344	. . . . 5 ⊢ ((𝐵 ·o 𝐴) = ((𝐵 ·o ∪ 𝐴) +o 𝐵) → (Lim (𝐵 ·o 𝐴) ↔ Lim ((𝐵 ·o ∪ 𝐴) +o 𝐵)))
37	35, 36	syl 17	. . . 4 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → (Lim (𝐵 ·o 𝐴) ↔ Lim ((𝐵 ·o ∪ 𝐴) +o 𝐵)))
38	29, 37	mpbird 257	. . 3 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → Lim (𝐵 ·o 𝐴))
39	38	ex 412	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 = suc ∪ 𝐴 → Lim (𝐵 ·o 𝐴)))
40		orduniorsuc 7805	. . 3 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
41	2, 40	syl 17	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
42	20, 39, 41	mpjaod 860	1 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐵 ·o 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∅c0 4296  ∪ cuni 4871  Ord word 6331  Oncon0 6332  Lim wlim 6333  suc csuc 6334  (class class class)co 7387   +_o coa 8431   ·_o comu 8432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-oadd 8438  df-omul 8439

This theorem is referenced by:  onexlimgt  43232  succlg  43317  dflim5  43318