Theorem omlimcl2 41991

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 6375	. . . . . 6 ⊢ (𝐴 ∈ On → Ord 𝐴)
2	1	ad2antrr 725	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Ord 𝐴)
3		ne0i 4335	. . . . . 6 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅)
4	3	adantl 483	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → 𝐴 ≠ ∅)
5		id 22	. . . . 5 ⊢ (𝐴 = ∪ 𝐴 → 𝐴 = ∪ 𝐴)
6		df-lim 6370	. . . . . 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 1423	. . . 4 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴)
9	8	ex 414	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 = ∪ 𝐴 → Lim 𝐴))
10		limelon 6429	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
11	10	ad3antlr 730	. . . . 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 616	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → (𝐴 ∈ On ∧ Lim 𝐴))
14		0ellim 6428	. . . . . . 7 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
15	14	adantl 483	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∅ ∈ 𝐵)
16	15	ad3antlr 730	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → ∅ ∈ 𝐵)
17		omlimcl 8578	. . . . 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 414	. . 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 7776	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
22	21, 10	anim12ci 615	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐵 ∈ On ∧ ∪ 𝐴 ∈ On))
23		omcl 8536	. . . . . . . 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 486	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐵 ∈ 𝐶 ∧ Lim 𝐵))
26	24, 25	jca 513	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ((𝐵 ·o ∪ 𝐴) ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)))
27	26	ad2antrr 725	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → ((𝐵 ·o ∪ 𝐴) ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)))
28		oalimcl 8560	. . . . 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 486	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → 𝐴 = suc ∪ 𝐴)
31	30	oveq2d 7425	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → (𝐵 ·o 𝐴) = (𝐵 ·o suc ∪ 𝐴))
32	22	ad2antrr 725	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → (𝐵 ∈ On ∧ ∪ 𝐴 ∈ On))
33		omsuc 8526	. . . . . . 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 2773	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc ∪ 𝐴) → (𝐵 ·o 𝐴) = ((𝐵 ·o ∪ 𝐴) +o 𝐵))
36		limeq 6377	. . . . 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 414	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 = suc ∪ 𝐴 → Lim (𝐵 ·o 𝐴)))
40		orduniorsuc 7818	. . 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 859	1 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐵 ·o 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∅c0 4323  ∪ cuni 4909  Ord word 6364  Oncon0 6365  Lim wlim 6366  suc csuc 6367  (class class class)co 7409   +_o coa 8463   ·_o comu 8464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  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 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-omul 8471

This theorem is referenced by:  onexlimgt  41992  succlg  42078  dflim5  42079