Theorem omwordri 7652

Description: Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Dec-2004.)

Assertion

Ref	Expression
omwordri	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))

Proof of Theorem omwordri

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

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
2		oveq2 6658	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 ∅))
3	1, 2	sseq12d 3634	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 ∅) ⊆ (𝐵 ·𝑜 ∅)))
4		oveq2 6658	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
5		oveq2 6658	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝑦))
6	4, 5	sseq12d 3634	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦)))
7		oveq2 6658	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
8		oveq2 6658	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 suc 𝑦))
9	7, 8	sseq12d 3634	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦)))
10		oveq2 6658	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐶))
11		oveq2 6658	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝐶))
12	10, 11	sseq12d 3634	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
13		om0 7597	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
14		0ss 3972	. . . . . . 7 ⊢ ∅ ⊆ (𝐵 ·𝑜 ∅)
15	13, 14	syl6eqss 3655	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 ∅) ⊆ (𝐵 ·𝑜 ∅))
16	15	ad2antrr 762	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ·𝑜 ∅) ⊆ (𝐵 ·𝑜 ∅))
17		omcl 7616	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 𝑦) ∈ On)
18	17	3adant2 1080	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 𝑦) ∈ On)
19		omcl 7616	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 𝑦) ∈ On)
20	19	3adant1 1079	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 𝑦) ∈ On)
21		simp1 1061	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
22		oawordri 7630	. . . . . . . . . . . . 13 ⊢ (((𝐴 ·𝑜 𝑦) ∈ On ∧ (𝐵 ·𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴)))
23	18, 20, 21, 22	syl3anc 1326	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴)))
24	23	imp 445	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦)) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴))
25	24	adantrl 752	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦))) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴))
26		oaword 7629	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐵 ·𝑜 𝑦) ∈ On) → (𝐴 ⊆ 𝐵 ↔ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)))
27	20, 26	syld3an3 1371	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)))
28	27	biimpa 501	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐵 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
29	28	adantrr 753	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦))) → ((𝐵 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
30	25, 29	sstrd 3613	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦))) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
31		omsuc 7606	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
32	31	3adant2 1080	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
33	32	adantr 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦))) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
34		omsuc 7606	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
35	34	3adant1 1079	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
36	35	adantr 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦))) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
37	30, 33, 36	3sstr4d 3648	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦))) → (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦))
38	37	exp520 1288	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (𝐴 ⊆ 𝐵 → ((𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦))))))
39	38	com3r 87	. . . . . 6 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → (𝐴 ⊆ 𝐵 → ((𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦))))))
40	39	imp4c 617	. . . . 5 ⊢ (𝑦 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦))))
41		vex 3203	. . . . . . . 8 ⊢ 𝑥 ∈ V
42		ss2iun 4536	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 ·𝑜 𝑦))
43		omlim 7613	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦))
44	43	ad2ant2rl 785	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → (𝐴 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦))
45		omlim 7613	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ·𝑜 𝑦))
46	45	adantl 482	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → (𝐵 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ·𝑜 𝑦))
47	44, 46	sseq12d 3634	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 ·𝑜 𝑦)))
48	42, 47	syl5ibr 236	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥)))
49	48	anandirs 874	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥)))
50	41, 49	mpanr1 719	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥)))
51	50	expcom 451	. . . . . 6 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥))))
52	51	adantrd 484	. . . . 5 ⊢ (Lim 𝑥 → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥))))
53	3, 6, 9, 12, 16, 40, 52	tfinds3 7064	. . . 4 ⊢ (𝐶 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
54	53	expd 452	. . 3 ⊢ (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶))))
55	54	3impib 1262	. 2 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
56	55	3coml 1272	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ∪ ciun 4520  Oncon0 5723  Lim wlim 5724  suc csuc 5725  (class class class)co 6650   +_𝑜 coa 7557   ·_𝑜 comu 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-omul 7565

This theorem is referenced by:  omword2  7654  oewordri  7672  oeordsuc  7674