Theorem onmcl 44099

Description: If an ordinal is less than a power of omega, the product with a natural number is also less than that power of omega. (Contributed by RP, 19-Feb-2025.)

Assertion

Ref	Expression
onmcl	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → (𝐴 ∈ (ω ↑o 𝐵) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))

Proof of Theorem onmcl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7430	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ·o 𝑁) = (∅ ·o 𝑁))
2		simp3 1156	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → 𝑁 ∈ ω)
3		nnon 7877	. . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
4		om0r 8533	. . . . . 6 ⊢ (𝑁 ∈ On → (∅ ·o 𝑁) = ∅)
5	2, 3, 4	3syl 19	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → (∅ ·o 𝑁) = ∅)
6	1, 5	sylan9eqr 2823	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → (𝐴 ·o 𝑁) = ∅)
7		simpl2 1211	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → 𝐵 ∈ On)
8		omelon 9625	. . . . . 6 ⊢ ω ∈ On
9	7, 8	jctil 529	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → (ω ∈ On ∧ 𝐵 ∈ On))
10		peano1 7894	. . . . 5 ⊢ ∅ ∈ ω
11		oen0 8581	. . . . 5 ⊢ (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 𝐵))
12	9, 10, 11	sylancl 598	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → ∅ ∈ (ω ↑o 𝐵))
13	6, 12	eqeltrd 2866	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵))
14	13	a1d 26	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → (𝐴 ∈ (ω ↑o 𝐵) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))
15	2	adantr 486	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → 𝑁 ∈ ω)
16		simp1 1154	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → 𝐴 ∈ On)
17	16	anim1i 627	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
18		ondif1 8495	. . . 4 ⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
19	17, 18	sylibr 237	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → 𝐴 ∈ (On ∖ 1o))
20		simpl2 1211	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → 𝐵 ∈ On)
21		oveq2 7431	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
22	21	eleq1d 2851	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ∈ (ω ↑o 𝐵) ↔ (𝐴 ·o ∅) ∈ (ω ↑o 𝐵)))
23	22	imbi2d 343	. . . . 5 ⊢ (𝑥 = ∅ → ((((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → (𝐴 ·o 𝑥) ∈ (ω ↑o 𝐵)) ↔ (((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → (𝐴 ·o ∅) ∈ (ω ↑o 𝐵))))
24		oveq2 7431	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
25	24	eleq1d 2851	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ∈ (ω ↑o 𝐵) ↔ (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)))
26	25	imbi2d 343	. . . . 5 ⊢ (𝑥 = 𝑦 → ((((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → (𝐴 ·o 𝑥) ∈ (ω ↑o 𝐵)) ↔ (((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵))))
27		oveq2 7431	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
28	27	eleq1d 2851	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ∈ (ω ↑o 𝐵) ↔ (𝐴 ·o suc 𝑦) ∈ (ω ↑o 𝐵)))
29	28	imbi2d 343	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → (𝐴 ·o 𝑥) ∈ (ω ↑o 𝐵)) ↔ (((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → (𝐴 ·o suc 𝑦) ∈ (ω ↑o 𝐵))))
30		oveq2 7431	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑁))
31	30	eleq1d 2851	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐴 ·o 𝑥) ∈ (ω ↑o 𝐵) ↔ (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))
32	31	imbi2d 343	. . . . 5 ⊢ (𝑥 = 𝑁 → ((((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → (𝐴 ·o 𝑥) ∈ (ω ↑o 𝐵)) ↔ (((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵))))
33		eldifi 4088	. . . . . . . . 9 ⊢ (𝐴 ∈ (On ∖ 1o) → 𝐴 ∈ On)
34		om0 8511	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
35	33, 34	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ (On ∖ 1o) → (𝐴 ·o ∅) = ∅)
36	35	adantr 486	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) → (𝐴 ·o ∅) = ∅)
37	8	jctl 533	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (ω ∈ On ∧ 𝐵 ∈ On))
38	37, 10, 11	sylancl 598	. . . . . . . 8 ⊢ (𝐵 ∈ On → ∅ ∈ (ω ↑o 𝐵))
39	38	adantl 487	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) → ∅ ∈ (ω ↑o 𝐵))
40	36, 39	eqeltrd 2866	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) → (𝐴 ·o ∅) ∈ (ω ↑o 𝐵))
41	40	adantr 486	. . . . 5 ⊢ (((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → (𝐴 ·o ∅) ∈ (ω ↑o 𝐵))
42	33	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
43	42	ad2antrl 741	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) → 𝐴 ∈ On)
44		simpll 779	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) ∧ (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)) → 𝑦 ∈ ω)
45		onmsuc 8523	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
46	43, 44, 45	syl2an2r 698	. . . . . . . 8 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) ∧ (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
47		simpr 490	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) ∧ (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)) → (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵))
48		simplrr 790	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) ∧ (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)) → 𝐴 ∈ (ω ↑o 𝐵))
49		eqid 2766	. . . . . . . . . . . . . 14 ⊢ (ω ↑o 𝐵) = (ω ↑o 𝐵)
50	49	jctl 533	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ On → ((ω ↑o 𝐵) = (ω ↑o 𝐵) ∧ 𝐵 ∈ On))
51	50	olcd 888	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ On → ((ω ↑o 𝐵) = ∅ ∨ ((ω ↑o 𝐵) = (ω ↑o 𝐵) ∧ 𝐵 ∈ On)))
52	51	adantl 487	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) → ((ω ↑o 𝐵) = ∅ ∨ ((ω ↑o 𝐵) = (ω ↑o 𝐵) ∧ 𝐵 ∈ On)))
53	52	ad2antrl 741	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) → ((ω ↑o 𝐵) = ∅ ∨ ((ω ↑o 𝐵) = (ω ↑o 𝐵) ∧ 𝐵 ∈ On)))
54	53	adantr 486	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) ∧ (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)) → ((ω ↑o 𝐵) = ∅ ∨ ((ω ↑o 𝐵) = (ω ↑o 𝐵) ∧ 𝐵 ∈ On)))
55		oacl2g 44098	. . . . . . . . 9 ⊢ ((((𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵) ∧ 𝐴 ∈ (ω ↑o 𝐵)) ∧ ((ω ↑o 𝐵) = ∅ ∨ ((ω ↑o 𝐵) = (ω ↑o 𝐵) ∧ 𝐵 ∈ On))) → ((𝐴 ·o 𝑦) +o 𝐴) ∈ (ω ↑o 𝐵))
56	47, 48, 54, 55	syl21anc 851	. . . . . . . 8 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) ∧ (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)) → ((𝐴 ·o 𝑦) +o 𝐴) ∈ (ω ↑o 𝐵))
57	46, 56	eqeltrd 2866	. . . . . . 7 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) ∧ (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)) → (𝐴 ·o suc 𝑦) ∈ (ω ↑o 𝐵))
58	57	exp31 425	. . . . . 6 ⊢ (𝑦 ∈ ω → (((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → ((𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵) → (𝐴 ·o suc 𝑦) ∈ (ω ↑o 𝐵))))
59	58	a2d 30	. . . . 5 ⊢ (𝑦 ∈ ω → ((((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)) → (((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → (𝐴 ·o suc 𝑦) ∈ (ω ↑o 𝐵))))
60	23, 26, 29, 32, 41, 59	finds 7902	. . . 4 ⊢ (𝑁 ∈ ω → (((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))
61	60	expdimp 458	. . 3 ⊢ ((𝑁 ∈ ω ∧ (𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On)) → (𝐴 ∈ (ω ↑o 𝐵) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))
62	15, 19, 20, 61	syl12anc 850	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐴 ∈ (ω ↑o 𝐵) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))
63		on0eqel 6493	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
64	16, 63	syl 18	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
65	14, 62, 64	mpjaodan 973	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → (𝐴 ∈ (ω ↑o 𝐵) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ∖ cdif 3905  ∅c0 4289  Oncon0 6367  suc csuc 6369  (class class class)co 7423  ωcom 7871  1_oc1o 8455   +_o coa 8459   ·_o comu 8460   ↑_o coe 8461

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745  ax-inf2 9620

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-omul 8467  df-oexp 8468

This theorem is used by: (None)