Theorem omwordri 7539

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 6557	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
2		oveq2 6557	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 ∅))
3	1, 2	sseq12d 3597	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 ∅) ⊆ (𝐵 ·𝑜 ∅)))
4		oveq2 6557	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
5		oveq2 6557	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝑦))
6	4, 5	sseq12d 3597	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦)))
7		oveq2 6557	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
8		oveq2 6557	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 suc 𝑦))
9	7, 8	sseq12d 3597	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦)))
10		oveq2 6557	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐶))
11		oveq2 6557	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝐶))
12	10, 11	sseq12d 3597	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
13		om0 7484	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
14		0ss 3924	. . . . . . 7 ⊢ ∅ ⊆ (𝐵 ·𝑜 ∅)
15	13, 14	syl6eqss 3618	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 ∅) ⊆ (𝐵 ·𝑜 ∅))
16	15	ad2antrr 758	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ·𝑜 ∅) ⊆ (𝐵 ·𝑜 ∅))
17		omcl 7503	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 𝑦) ∈ On)
18	17	3adant2 1073	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 𝑦) ∈ On)
19		omcl 7503	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 𝑦) ∈ On)
20	19	3adant1 1072	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 𝑦) ∈ On)
21		simp1 1054	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
22		oawordri 7517	. . . . . . . . . . . . 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 1318	. . . . . . . . . . . 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 444	. . . . . . . . . . 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 748	. . . . . . . . . 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 7516	. . . . . . . . . . . . 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 1363	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)))
28	27	biimpa 500	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐵 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
29	28	adantrr 749	. . . . . . . . . 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 3578	. . . . . . . . 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 7493	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
32	31	3adant2 1073	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
33	32	adantr 480	. . . . . . . . 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 7493	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
35	34	3adant1 1072	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
36	35	adantr 480	. . . . . . . . 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 3611	. . . . . . . 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 1280	. . . . . . 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 85	. . . . . 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 615	. . . . 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 3176	. . . . . . . 8 ⊢ 𝑥 ∈ V
42		ss2iun 4472	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 ·𝑜 𝑦))
43		omlim 7500	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦))
44	43	ad2ant2rl 781	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → (𝐴 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦))
45		omlim 7500	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ·𝑜 𝑦))
46	45	adantl 481	. . . . . . . . . . 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 3597	. . . . . . . . . 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 235	. . . . . . . . 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 870	. . . . . . . 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 715	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥)))
51	50	expcom 450	. . . . . 6 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥))))
52	51	adantrd 483	. . . . 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 6956	. . . 4 ⊢ (𝐶 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
54	53	expd 451	. . 3 ⊢ (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶))))
55	54	3impib 1254	. 2 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
56	55	3coml 1264	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549   +_𝑜 coa 7444   ·_𝑜 comu 7445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-omul 7452

This theorem is referenced by:  omword2  7541  oewordri  7559  oeordsuc  7561