Theorem omlimcl2 44228

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 6371	. . . . . 6 ⊢ (𝐴 ∈ On → Ord 𝐴)
2	1	ad2antrr 739	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Ord 𝐴)
3		ne0i 4287	. . . . . 6 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅)
4	3	adantl 487	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → 𝐴 ≠ ∅)
5		id 23	. . . . 5 ⊢ (𝐴 = ∪ 𝐴 → 𝐴 = ∪ 𝐴)
6		df-lim 6366	. . . . . 6 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
7	6	biimpri 231	. . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴)
8	2, 4, 5, 7	syl2an3an 1449	. . . 4 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴)
9	8	ex 418	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 = ∪ 𝐴 → Lim 𝐴))
10		limelon 6427	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
11	10	ad3antlr 744	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → 𝐵 ∈ On)
12		simpll 779	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → 𝐴 ∈ On)
13	12	anim1i 627	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → (𝐴 ∈ On ∧ Lim 𝐴))
14		0ellim 6426	. . . . . . 7 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
15	14	adantl 487	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∅ ∈ 𝐵)
16	15	ad3antlr 744	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → ∅ ∈ 𝐵)
17		omlimcl 8579	. . . . 5 ⊢ (((𝐵 ∈ On ∧ (𝐴 ∈ On ∧ Lim 𝐴)) ∧ ∅ ∈ 𝐵) → Lim (𝐵 ·o 𝐴))
18	11, 13, 16, 17	syl21anc 851	. . . 4 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → Lim (𝐵 ·o 𝐴))
19	18	ex 418	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (Lim 𝐴 → Lim (𝐵 ·o 𝐴)))
20	9, 19	syld 48	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 = ∪ 𝐴 → Lim (𝐵 ·o 𝐴)))
21		onuni 7800	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
22	21, 10	anim12ci 626	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐵 ∈ On ∧ ∪ 𝐴 ∈ On))
23		omcl 8537	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ ∪ 𝐴 ∈ On) → (𝐵 ·o ∪ 𝐴) ∈ On)
24	22, 23	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐵 ·o ∪ 𝐴) ∈ On)
25		simpr 490	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐵 ∈ 𝐶 ∧ Lim 𝐵))
26	24, 25	jca 521	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ((𝐵 ·o ∪ 𝐴) ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)))
27	26	ad2antrr 739	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → ((𝐵 ·o ∪ 𝐴) ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)))
28		oalimcl 8561	. . . . 5 ⊢ (((𝐵 ·o ∪ 𝐴) ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim ((𝐵 ·o ∪ 𝐴) +o 𝐵))
29	27, 28	syl 18	. . . 4 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → Lim ((𝐵 ·o ∪ 𝐴) +o 𝐵))
30		simpr 490	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → 𝐴 = suc ∪ 𝐴)
31	30	oveq2d 7434	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → (𝐵 ·o 𝐴) = (𝐵 ·o suc ∪ 𝐴))
32	22	ad2antrr 739	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → (𝐵 ∈ On ∧ ∪ 𝐴 ∈ On))
33		omsuc 8527	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ ∪ 𝐴 ∈ On) → (𝐵 ·o suc ∪ 𝐴) = ((𝐵 ·o ∪ 𝐴) +o 𝐵))
34	32, 33	syl 18	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → (𝐵 ·o suc ∪ 𝐴) = ((𝐵 ·o ∪ 𝐴) +o 𝐵))
35	31, 34	eqtrd 2796	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → (𝐵 ·o 𝐴) = ((𝐵 ·o ∪ 𝐴) +o 𝐵))
36		limeq 6373	. . . . 5 ⊢ ((𝐵 ·o 𝐴) = ((𝐵 ·o ∪ 𝐴) +o 𝐵) → (Lim (𝐵 ·o 𝐴) ↔ Lim ((𝐵 ·o ∪ 𝐴) +o 𝐵)))
37	35, 36	syl 18	. . . 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 260	. . 3 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → Lim (𝐵 ·o 𝐴))
39	38	ex 418	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 = suc ∪ 𝐴 → Lim (𝐵 ·o 𝐴)))
40		orduniorsuc 7839	. . 3 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
41	2, 40	syl 18	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
42	20, 39, 41	mpjaod 874	1 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐵 ·o 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∅c0 4279  ∪ cuni 4867  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363  (class class class)co 7418   +_o coa 8466   ·_o comu 8467

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-oadd 8473  df-omul 8474

This theorem is used by:  onexlimgt  44229  succlg  44314  dflim5  44315