Theorem omlimcl2 42942

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 6376	. . . . . 6 ⊢ (𝐴 ∈ On → Ord 𝐴)
2	1	ad2antrr 724	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Ord 𝐴)
3		ne0i 4335	. . . . . 6 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅)
4	3	adantl 480	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → 𝐴 ≠ ∅)
5		id 22	. . . . 5 ⊢ (𝐴 = ∪ 𝐴 → 𝐴 = ∪ 𝐴)
6		df-lim 6371	. . . . . 6 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
7	6	biimpri 227	. . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴)
8	2, 4, 5, 7	syl2an3an 1419	. . . 4 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴)
9	8	ex 411	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 = ∪ 𝐴 → Lim 𝐴))
10		limelon 6430	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
11	10	ad3antlr 729	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → 𝐵 ∈ On)
12		simpll 765	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → 𝐴 ∈ On)
13	12	anim1i 613	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → (𝐴 ∈ On ∧ Lim 𝐴))
14		0ellim 6429	. . . . . . 7 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
15	14	adantl 480	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∅ ∈ 𝐵)
16	15	ad3antlr 729	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → ∅ ∈ 𝐵)
17		omlimcl 8598	. . . . 5 ⊢ (((𝐵 ∈ On ∧ (𝐴 ∈ On ∧ Lim 𝐴)) ∧ ∅ ∈ 𝐵) → Lim (𝐵 ·o 𝐴))
18	11, 13, 16, 17	syl21anc 836	. . . 4 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → Lim (𝐵 ·o 𝐴))
19	18	ex 411	. . 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 7787	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
22	21, 10	anim12ci 612	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐵 ∈ On ∧ ∪ 𝐴 ∈ On))
23		omcl 8556	. . . . . . . 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 483	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐵 ∈ 𝐶 ∧ Lim 𝐵))
26	24, 25	jca 510	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ((𝐵 ·o ∪ 𝐴) ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)))
27	26	ad2antrr 724	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → ((𝐵 ·o ∪ 𝐴) ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)))
28		oalimcl 8580	. . . . 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 483	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → 𝐴 = suc ∪ 𝐴)
31	30	oveq2d 7430	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → (𝐵 ·o 𝐴) = (𝐵 ·o suc ∪ 𝐴))
32	22	ad2antrr 724	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → (𝐵 ∈ On ∧ ∪ 𝐴 ∈ On))
33		omsuc 8546	. . . . . . 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 2766	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → (𝐵 ·o 𝐴) = ((𝐵 ·o ∪ 𝐴) +o 𝐵))
36		limeq 6378	. . . . 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 256	. . 3 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → Lim (𝐵 ·o 𝐴))
39	38	ex 411	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 = suc ∪ 𝐴 → Lim (𝐵 ·o 𝐴)))
40		orduniorsuc 7829	. . 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 858	1 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐵 ·o 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∅c0 4323  ∪ cuni 4906  Ord word 6365  Oncon0 6366  Lim wlim 6367  suc csuc 6368  (class class class)co 7414   +_o coa 8483   ·_o comu 8484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7736

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-oadd 8490  df-omul 8491

This theorem is referenced by:  onexlimgt  42943  succlg  43029  dflim5  43030