Theorem onmcl 42760

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 7427	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ·o 𝑁) = (∅ ·o 𝑁))
2		simp3 1136	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → 𝑁 ∈ ω)
3		nnon 7876	. . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
4		om0r 8560	. . . . . 6 ⊢ (𝑁 ∈ On → (∅ ·o 𝑁) = ∅)
5	2, 3, 4	3syl 18	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → (∅ ·o 𝑁) = ∅)
6	1, 5	sylan9eqr 2790	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → (𝐴 ·o 𝑁) = ∅)
7		simpl2 1190	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → 𝐵 ∈ On)
8		omelon 9670	. . . . . 6 ⊢ ω ∈ On
9	7, 8	jctil 519	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → (ω ∈ On ∧ 𝐵 ∈ On))
10		peano1 7894	. . . . 5 ⊢ ∅ ∈ ω
11		oen0 8607	. . . . 5 ⊢ (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 𝐵))
12	9, 10, 11	sylancl 585	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → ∅ ∈ (ω ↑o 𝐵))
13	6, 12	eqeltrd 2829	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵))
14	13	a1d 25	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → (𝐴 ∈ (ω ↑o 𝐵) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))
15	2	adantr 480	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → 𝑁 ∈ ω)
16		simp1 1134	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → 𝐴 ∈ On)
17	16	anim1i 614	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
18		ondif1 8522	. . . 4 ⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
19	17, 18	sylibr 233	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → 𝐴 ∈ (On ∖ 1o))
20		simpl2 1190	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → 𝐵 ∈ On)
21		oveq2 7428	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
22	21	eleq1d 2814	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ∈ (ω ↑o 𝐵) ↔ (𝐴 ·o ∅) ∈ (ω ↑o 𝐵)))
23	22	imbi2d 340	. . . . 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 7428	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
25	24	eleq1d 2814	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ∈ (ω ↑o 𝐵) ↔ (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)))
26	25	imbi2d 340	. . . . 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 7428	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
28	27	eleq1d 2814	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ∈ (ω ↑o 𝐵) ↔ (𝐴 ·o suc 𝑦) ∈ (ω ↑o 𝐵)))
29	28	imbi2d 340	. . . . 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 7428	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑁))
31	30	eleq1d 2814	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐴 ·o 𝑥) ∈ (ω ↑o 𝐵) ↔ (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))
32	31	imbi2d 340	. . . . 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 4125	. . . . . . . . 9 ⊢ (𝐴 ∈ (On ∖ 1o) → 𝐴 ∈ On)
34		om0 8538	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
35	33, 34	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (On ∖ 1o) → (𝐴 ·o ∅) = ∅)
36	35	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) → (𝐴 ·o ∅) = ∅)
37	8	jctl 523	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (ω ∈ On ∧ 𝐵 ∈ On))
38	37, 10, 11	sylancl 585	. . . . . . . 8 ⊢ (𝐵 ∈ On → ∅ ∈ (ω ↑o 𝐵))
39	38	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) → ∅ ∈ (ω ↑o 𝐵))
40	36, 39	eqeltrd 2829	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) → (𝐴 ·o ∅) ∈ (ω ↑o 𝐵))
41	40	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → (𝐴 ·o ∅) ∈ (ω ↑o 𝐵))
42	33	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
43	42	ad2antrl 727	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) → 𝐴 ∈ On)
44		simpll 766	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) ∧ (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)) → 𝑦 ∈ ω)
45		onmsuc 8550	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
46	43, 44, 45	syl2an2r 684	. . . . . . . 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 484	. . . . . . . . 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 777	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) ∧ (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)) → 𝐴 ∈ (ω ↑o 𝐵))
49		eqid 2728	. . . . . . . . . . . . . 14 ⊢ (ω ↑o 𝐵) = (ω ↑o 𝐵)
50	49	jctl 523	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ On → ((ω ↑o 𝐵) = (ω ↑o 𝐵) ∧ 𝐵 ∈ On))
51	50	olcd 873	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ On → ((ω ↑o 𝐵) = ∅ ∨ ((ω ↑o 𝐵) = (ω ↑o 𝐵) ∧ 𝐵 ∈ On)))
52	51	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) → ((ω ↑o 𝐵) = ∅ ∨ ((ω ↑o 𝐵) = (ω ↑o 𝐵) ∧ 𝐵 ∈ On)))
53	52	ad2antrl 727	. . . . . . . . . 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 480	. . . . . . . . 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 42759	. . . . . . . . 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 837	. . . . . . . 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 2829	. . . . . . 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 419	. . . . . 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 29	. . . . 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 7904	. . . 4 ⊢ (𝑁 ∈ ω → (((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))
61	60	expdimp 452	. . 3 ⊢ ((𝑁 ∈ ω ∧ (𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On)) → (𝐴 ∈ (ω ↑o 𝐵) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))
62	15, 19, 20, 61	syl12anc 836	. 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 17	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
65	14, 62, 64	mpjaodan 957	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → (𝐴 ∈ (ω ↑o 𝐵) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ∖ cdif 3944  ∅c0 4323  Oncon0 6369  suc csuc 6371  (class class class)co 7420  ωcom 7870  1_oc1o 8480   +_o coa 8484   ·_o comu 8485   ↑_o coe 8486

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 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740  ax-inf2 9665

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-oadd 8491  df-omul 8492  df-oexp 8493

This theorem is referenced by: (None)