Theorem onmcl 43759

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 7374	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ·o 𝑁) = (∅ ·o 𝑁))
2		simp3 1139	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → 𝑁 ∈ ω)
3		nnon 7823	. . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
4		om0r 8474	. . . . . 6 ⊢ (𝑁 ∈ On → (∅ ·o 𝑁) = ∅)
5	2, 3, 4	3syl 18	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → (∅ ·o 𝑁) = ∅)
6	1, 5	sylan9eqr 2793	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → (𝐴 ·o 𝑁) = ∅)
7		simpl2 1194	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → 𝐵 ∈ On)
8		omelon 9567	. . . . . 6 ⊢ ω ∈ On
9	7, 8	jctil 519	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → (ω ∈ On ∧ 𝐵 ∈ On))
10		peano1 7840	. . . . 5 ⊢ ∅ ∈ ω
11		oen0 8522	. . . . 5 ⊢ (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 𝐵))
12	9, 10, 11	sylancl 587	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → ∅ ∈ (ω ↑o 𝐵))
13	6, 12	eqeltrd 2836	. . 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 1137	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → 𝐴 ∈ On)
17	16	anim1i 616	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
18		ondif1 8436	. . . 4 ⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
19	17, 18	sylibr 234	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → 𝐴 ∈ (On ∖ 1o))
20		simpl2 1194	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → 𝐵 ∈ On)
21		oveq2 7375	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
22	21	eleq1d 2821	. . . . . 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 7375	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
25	24	eleq1d 2821	. . . . . 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 7375	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
28	27	eleq1d 2821	. . . . . 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 7375	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑁))
31	30	eleq1d 2821	. . . . . 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 4071	. . . . . . . . 9 ⊢ (𝐴 ∈ (On ∖ 1o) → 𝐴 ∈ On)
34		om0 8452	. . . . . . . . 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 587	. . . . . . . 8 ⊢ (𝐵 ∈ On → ∅ ∈ (ω ↑o 𝐵))
39	38	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) → ∅ ∈ (ω ↑o 𝐵))
40	36, 39	eqeltrd 2836	. . . . . 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 729	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) → 𝐴 ∈ On)
44		simpll 767	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) ∧ (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)) → 𝑦 ∈ ω)
45		onmsuc 8464	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
46	43, 44, 45	syl2an2r 686	. . . . . . . 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 778	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) ∧ (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)) → 𝐴 ∈ (ω ↑o 𝐵))
49		eqid 2736	. . . . . . . . . . . . . 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 875	. . . . . . . . . . . 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 729	. . . . . . . . . 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 43758	. . . . . . . . 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 838	. . . . . . . 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 2836	. . . . . . 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 7847	. . . 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 837	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐴 ∈ (ω ↑o 𝐵) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))
63		on0eqel 6448	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
64	16, 63	syl 17	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
65	14, 62, 64	mpjaodan 961	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → (𝐴 ∈ (ω ↑o 𝐵) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∖ cdif 3886  ∅c0 4273  Oncon0 6323  suc csuc 6325  (class class class)co 7367  ωcom 7817  1_oc1o 8398   +_o coa 8402   ·_o comu 8403   ↑_o coe 8404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-omul 8410  df-oexp 8411

This theorem is referenced by: (None)