Theorem omlimcl2 43259

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 6393	. . . . . 6 ⊢ (𝐴 ∈ On → Ord 𝐴)
2	1	ad2antrr 726	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Ord 𝐴)
3		ne0i 4340	. . . . . 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 6388	. . . . . 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 1423	. . . 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 6447	. . . . . 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 6446	. . . . . . 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 8617	. . . . 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 7809	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
22	21, 10	anim12ci 614	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐵 ∈ On ∧ ∪ 𝐴 ∈ On))
23		omcl 8575	. . . . . . . 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 8599	. . . . 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 7448	. . . . . 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 8565	. . . . . . 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 2776	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → (𝐵 ·o 𝐴) = ((𝐵 ·o ∪ 𝐴) +o 𝐵))
36		limeq 6395	. . . . 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 7851	. . 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 1539   ∈ wcel 2107   ≠ wne 2939  ∅c0 4332  ∪ cuni 4906  Ord word 6382  Oncon0 6383  Lim wlim 6384  suc csuc 6385  (class class class)co 7432   +_o coa 8504   ·_o comu 8505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-omul 8512

This theorem is referenced by:  onexlimgt  43260  succlg  43346  dflim5  43347