Theorem onmcl 43872

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 7399	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ·o 𝑁) = (∅ ·o 𝑁))
2		simp3 1150	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → 𝑁 ∈ ω)
3		nnon 7848	. . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
4		om0r 8503	. . . . . 6 ⊢ (𝑁 ∈ On → (∅ ·o 𝑁) = ∅)
5	2, 3, 4	3syl 18	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → (∅ ·o 𝑁) = ∅)
6	1, 5	sylan9eqr 2818	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → (𝐴 ·o 𝑁) = ∅)
7		simpl2 1205	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → 𝐵 ∈ On)
8		omelon 9598	. . . . . 6 ⊢ ω ∈ On
9	7, 8	jctil 527	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → (ω ∈ On ∧ 𝐵 ∈ On))
10		peano1 7865	. . . . 5 ⊢ ∅ ∈ ω
11		oen0 8551	. . . . 5 ⊢ (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 𝐵))
12	9, 10, 11	sylancl 595	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ 𝐴 = ∅) → ∅ ∈ (ω ↑o 𝐵))
13	6, 12	eqeltrd 2861	. . 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 484	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → 𝑁 ∈ ω)
16		simp1 1148	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → 𝐴 ∈ On)
17	16	anim1i 624	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
18		ondif1 8465	. . . 4 ⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
19	17, 18	sylibr 236	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → 𝐴 ∈ (On ∖ 1o))
20		simpl2 1205	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → 𝐵 ∈ On)
21		oveq2 7400	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
22	21	eleq1d 2846	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ∈ (ω ↑o 𝐵) ↔ (𝐴 ·o ∅) ∈ (ω ↑o 𝐵)))
23	22	imbi2d 342	. . . . 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 7400	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
25	24	eleq1d 2846	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ∈ (ω ↑o 𝐵) ↔ (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)))
26	25	imbi2d 342	. . . . 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 7400	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
28	27	eleq1d 2846	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ∈ (ω ↑o 𝐵) ↔ (𝐴 ·o suc 𝑦) ∈ (ω ↑o 𝐵)))
29	28	imbi2d 342	. . . . 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 7400	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑁))
31	30	eleq1d 2846	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐴 ·o 𝑥) ∈ (ω ↑o 𝐵) ↔ (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))
32	31	imbi2d 342	. . . . 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 4084	. . . . . . . . 9 ⊢ (𝐴 ∈ (On ∖ 1o) → 𝐴 ∈ On)
34		om0 8481	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
35	33, 34	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (On ∖ 1o) → (𝐴 ·o ∅) = ∅)
36	35	adantr 484	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) → (𝐴 ·o ∅) = ∅)
37	8	jctl 531	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (ω ∈ On ∧ 𝐵 ∈ On))
38	37, 10, 11	sylancl 595	. . . . . . . 8 ⊢ (𝐵 ∈ On → ∅ ∈ (ω ↑o 𝐵))
39	38	adantl 485	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) → ∅ ∈ (ω ↑o 𝐵))
40	36, 39	eqeltrd 2861	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) → (𝐴 ·o ∅) ∈ (ω ↑o 𝐵))
41	40	adantr 484	. . . . 5 ⊢ (((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → (𝐴 ·o ∅) ∈ (ω ↑o 𝐵))
42	33	adantr 484	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
43	42	ad2antrl 738	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) → 𝐴 ∈ On)
44		simpll 776	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) ∧ (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)) → 𝑦 ∈ ω)
45		onmsuc 8493	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
46	43, 44, 45	syl2an2r 695	. . . . . . . 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 488	. . . . . . . . 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 787	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵))) ∧ (𝐴 ·o 𝑦) ∈ (ω ↑o 𝐵)) → 𝐴 ∈ (ω ↑o 𝐵))
49		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (ω ↑o 𝐵) = (ω ↑o 𝐵)
50	49	jctl 531	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ On → ((ω ↑o 𝐵) = (ω ↑o 𝐵) ∧ 𝐵 ∈ On))
51	50	olcd 885	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ On → ((ω ↑o 𝐵) = ∅ ∨ ((ω ↑o 𝐵) = (ω ↑o 𝐵) ∧ 𝐵 ∈ On)))
52	51	adantl 485	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) → ((ω ↑o 𝐵) = ∅ ∨ ((ω ↑o 𝐵) = (ω ↑o 𝐵) ∧ 𝐵 ∈ On)))
53	52	ad2antrl 738	. . . . . . . . . 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 484	. . . . . . . . 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 43871	. . . . . . . . 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 848	. . . . . . . 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 2861	. . . . . . 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 423	. . . . . 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 7873	. . . 4 ⊢ (𝑁 ∈ ω → (((𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ (ω ↑o 𝐵)) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))
61	60	expdimp 456	. . 3 ⊢ ((𝑁 ∈ ω ∧ (𝐴 ∈ (On ∖ 1o) ∧ 𝐵 ∈ On)) → (𝐴 ∈ (ω ↑o 𝐵) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))
62	15, 19, 20, 61	syl12anc 847	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐴 ∈ (ω ↑o 𝐵) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))
63		on0eqel 6467	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
64	16, 63	syl 17	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
65	14, 62, 64	mpjaodan 971	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → (𝐴 ∈ (ω ↑o 𝐵) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 858   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ∖ cdif 3901  ∅c0 4285  Oncon0 6342  suc csuc 6344  (class class class)co 7392  ωcom 7842  1_oc1o 8425   +_o coa 8429   ·_o comu 8430   ↑_o coe 8431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714  ax-inf2 9593

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-oadd 8436  df-omul 8437  df-oexp 8438

This theorem is referenced by: (None)