Theorem omordi 3164

Description: Ordering property of ordinal multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63.

Assertion

Ref	Expression
omordi	⊢ (((B ∈ On ∧ C ∈ On) ∧ ∅ ∈ C) → (A ∈ B → (C ·o A) ∈ (C ·o B)))

Proof of Theorem omordi

Step	Hyp	Ref	Expression
1		onelon 2223	. . . . . 6 ⊢ ((B ∈ On ∧ A ∈ B) → A ∈ On)
2	1	exp 291	. . . . 5 ⊢ (B ∈ On → (A ∈ B → A ∈ On))
3		eleq2 1150	. . . . . . . . . 10 ⊢ (x = ∅ → (A ∈ x ↔ A ∈ ∅))
4		opreq2 3007	. . . . . . . . . . 11 ⊢ (x = ∅ → (C ·o x) = (C ·o ∅))
5	4	eleq2d 1156	. . . . . . . . . 10 ⊢ (x = ∅ → ((C ·o A) ∈ (C ·o x) ↔ (C ·o A) ∈ (C ·o ∅)))
6	3, 5	imbi12d 474	. . . . . . . . 9 ⊢ (x = ∅ → ((A ∈ x → (C ·o A) ∈ (C ·o x)) ↔ (A ∈ ∅ → (C ·o A) ∈ (C ·o ∅))))
7		eleq2 1150	. . . . . . . . . 10 ⊢ (x = y → (A ∈ x ↔ A ∈ y))
8		opreq2 3007	. . . . . . . . . . 11 ⊢ (x = y → (C ·o x) = (C ·o y))
9	8	eleq2d 1156	. . . . . . . . . 10 ⊢ (x = y → ((C ·o A) ∈ (C ·o x) ↔ (C ·o A) ∈ (C ·o y)))
10	7, 9	imbi12d 474	. . . . . . . . 9 ⊢ (x = y → ((A ∈ x → (C ·o A) ∈ (C ·o x)) ↔ (A ∈ y → (C ·o A) ∈ (C ·o y))))
11		eleq2 1150	. . . . . . . . . 10 ⊢ (x = suc y → (A ∈ x ↔ A ∈ suc y))
12		opreq2 3007	. . . . . . . . . . 11 ⊢ (x = suc y → (C ·o x) = (C ·o suc y))
13	12	eleq2d 1156	. . . . . . . . . 10 ⊢ (x = suc y → ((C ·o A) ∈ (C ·o x) ↔ (C ·o A) ∈ (C ·o suc y)))
14	11, 13	imbi12d 474	. . . . . . . . 9 ⊢ (x = suc y → ((A ∈ x → (C ·o A) ∈ (C ·o x)) ↔ (A ∈ suc y → (C ·o A) ∈ (C ·o suc y))))
15		eleq2 1150	. . . . . . . . . 10 ⊢ (x = B → (A ∈ x ↔ A ∈ B))
16		opreq2 3007	. . . . . . . . . . 11 ⊢ (x = B → (C ·o x) = (C ·o B))
17	16	eleq2d 1156	. . . . . . . . . 10 ⊢ (x = B → ((C ·o A) ∈ (C ·o x) ↔ (C ·o A) ∈ (C ·o B)))
18	15, 17	imbi12d 474	. . . . . . . . 9 ⊢ (x = B → ((A ∈ x → (C ·o A) ∈ (C ·o x)) ↔ (A ∈ B → (C ·o A) ∈ (C ·o B))))
19		noel 1711	. . . . . . . . . . 11 ⊢ ¬ A ∈ ∅
20	19	pm2.21i 73	. . . . . . . . . 10 ⊢ (A ∈ ∅ → (C ·o A) ∈ (C ·o ∅))
21	20	a1i 7	. . . . . . . . 9 ⊢ (((A ∈ On ∧ C ∈ On) ∧ ∅ ∈ C) → (A ∈ ∅ → (C ·o A) ∈ (C ·o ∅)))
22		oaword1 3154	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((C ·o y) ∈ On ∧ C ∈ On) → (C ·o y) ⊆ ((C ·o y) +o C))
23	22	sseld 1506	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((C ·o y) ∈ On ∧ C ∈ On) → ((C ·o A) ∈ (C ·o y) → (C ·o A) ∈ ((C ·o y) +o C)))
24	23	syl3d 26	. . . . . . . . . . . . . . . . . . 19 ⊢ (((C ·o y) ∈ On ∧ C ∈ On) → ((A ∈ y → (C ·o A) ∈ (C ·o y)) → (A ∈ y → (C ·o A) ∈ ((C ·o y) +o C))))
25	24	imp 277	. . . . . . . . . . . . . . . . . 18 ⊢ ((((C ·o y) ∈ On ∧ C ∈ On) ∧ (A ∈ y → (C ·o A) ∈ (C ·o y))) → (A ∈ y → (C ·o A) ∈ ((C ·o y) +o C)))
26	25	adantrl 311	. . . . . . . . . . . . . . . . 17 ⊢ ((((C ·o y) ∈ On ∧ C ∈ On) ∧ (∅ ∈ C ∧ (A ∈ y → (C ·o A) ∈ (C ·o y)))) → (A ∈ y → (C ·o A) ∈ ((C ·o y) +o C)))
27		opreq2 3007	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (A = y → (C ·o A) = (C ·o y))
28	27	eleq1d 1155	. . . . . . . . . . . . . . . . . . . 20 ⊢ (A = y → ((C ·o A) ∈ ((C ·o y) +o C) ↔ (C ·o y) ∈ ((C ·o y) +o C)))
29		oaord1 3153	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((C ·o y) ∈ On ∧ C ∈ On) → (∅ ∈ C ↔ (C ·o y) ∈ ((C ·o y) +o C)))
30	29	biimpa 324	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((C ·o y) ∈ On ∧ C ∈ On) ∧ ∅ ∈ C) → (C ·o y) ∈ ((C ·o y) +o C))
31	28, 30	syl5bir 184	. . . . . . . . . . . . . . . . . . 19 ⊢ (A = y → ((((C ·o y) ∈ On ∧ C ∈ On) ∧ ∅ ∈ C) → (C ·o A) ∈ ((C ·o y) +o C)))
32	31	com12 13	. . . . . . . . . . . . . . . . . 18 ⊢ ((((C ·o y) ∈ On ∧ C ∈ On) ∧ ∅ ∈ C) → (A = y → (C ·o A) ∈ ((C ·o y) +o C)))
33	32	adantrr 312	. . . . . . . . . . . . . . . . 17 ⊢ ((((C ·o y) ∈ On ∧ C ∈ On) ∧ (∅ ∈ C ∧ (A ∈ y → (C ·o A) ∈ (C ·o y)))) → (A = y → (C ·o A) ∈ ((C ·o y) +o C)))
34	26, 33	jaod 329	. . . . . . . . . . . . . . . 16 ⊢ ((((C ·o y) ∈ On ∧ C ∈ On) ∧ (∅ ∈ C ∧ (A ∈ y → (C ·o A) ∈ (C ·o y)))) → ((A ∈ y ∨ A = y) → (C ·o A) ∈ ((C ·o y) +o C)))
35		omcl 3139	. . . . . . . . . . . . . . . . 17 ⊢ ((C ∈ On ∧ y ∈ On) → (C ·o y) ∈ On)
36		pm3.26 256	. . . . . . . . . . . . . . . . 17 ⊢ ((C ∈ On ∧ y ∈ On) → C ∈ On)
37	35, 36	jca 236	. . . . . . . . . . . . . . . 16 ⊢ ((C ∈ On ∧ y ∈ On) → ((C ·o y) ∈ On ∧ C ∈ On))
38	34, 37	sylan 343	. . . . . . . . . . . . . . 15 ⊢ (((C ∈ On ∧ y ∈ On) ∧ (∅ ∈ C ∧ (A ∈ y → (C ·o A) ∈ (C ·o y)))) → ((A ∈ y ∨ A = y) → (C ·o A) ∈ ((C ·o y) +o C)))
39		elsuci 2289	. . . . . . . . . . . . . . 15 ⊢ (A ∈ suc y → (A ∈ y ∨ A = y))
40	38, 39	syl5 22	. . . . . . . . . . . . . 14 ⊢ (((C ∈ On ∧ y ∈ On) ∧ (∅ ∈ C ∧ (A ∈ y → (C ·o A) ∈ (C ·o y)))) → (A ∈ suc y → (C ·o A) ∈ ((C ·o y) +o C)))
41		omsuc 3133	. . . . . . . . . . . . . . . 16 ⊢ ((C ∈ On ∧ y ∈ On) → (C ·o suc y) = ((C ·o y) +o C))
42	41	eleq2d 1156	. . . . . . . . . . . . . . 15 ⊢ ((C ∈ On ∧ y ∈ On) → ((C ·o A) ∈ (C ·o suc y) ↔ (C ·o A) ∈ ((C ·o y) +o C)))
43	42	adantr 306	. . . . . . . . . . . . . 14 ⊢ (((C ∈ On ∧ y ∈ On) ∧ (∅ ∈ C ∧ (A ∈ y → (C ·o A) ∈ (C ·o y)))) → ((C ·o A) ∈ (C ·o suc y) ↔ (C ·o A) ∈ ((C ·o y) +o C)))
44	40, 43	sylibrd 179	. . . . . . . . . . . . 13 ⊢ (((C ∈ On ∧ y ∈ On) ∧ (∅ ∈ C ∧ (A ∈ y → (C ·o A) ∈ (C ·o y)))) → (A ∈ suc y → (C ·o A) ∈ (C ·o suc y)))
45	44	exp43 301	. . . . . . . . . . . 12 ⊢ (C ∈ On → (y ∈ On → (∅ ∈ C → ((A ∈ y → (C ·o A) ∈ (C ·o y)) → (A ∈ suc y → (C ·o A) ∈ (C ·o suc y))))))
46	45	com12 13	. . . . . . . . . . 11 ⊢ (y ∈ On → (C ∈ On → (∅ ∈ C → ((A ∈ y → (C ·o A) ∈ (C ·o y)) → (A ∈ suc y → (C ·o A) ∈ (C ·o suc y))))))
47	46	adantld 307	. . . . . . . . . 10 ⊢ (y ∈ On → ((A ∈ On ∧ C ∈ On) → (∅ ∈ C → ((A ∈ y → (C ·o A) ∈ (C ·o y)) → (A ∈ suc y → (C ·o A) ∈ (C ·o suc y))))))
48	47	imp3a 279	. . . . . . . . 9 ⊢ (y ∈ On → (((A ∈ On ∧ C ∈ On) ∧ ∅ ∈ C) → ((A ∈ y → (C ·o A) ∈ (C ·o y)) → (A ∈ suc y → (C ·o A) ∈ (C ·o suc y)))))
49		limsuc 2361	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Lim x → (A ∈ x ↔ suc A ∈ x))
50	49	biimpa 324	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Lim x ∧ A ∈ x) → suc A ∈ x)
51		opreq2 3007	. . . . . . . . . . . . . . . . . . . 20 ⊢ (y = suc A → (C ·o y) = (C ·o suc A))
52	51	ssiun2s 2020	. . . . . . . . . . . . . . . . . . 19 ⊢ (suc A ∈ x → (C ·o suc A) ⊆ ∪y ∈ x (C ·o y))
53	50, 52	syl 12	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim x ∧ A ∈ x) → (C ·o suc A) ⊆ ∪y ∈ x (C ·o y))
54	53	adantll 309	. . . . . . . . . . . . . . . . 17 ⊢ (((C ∈ On ∧ Lim x) ∧ A ∈ x) → (C ·o suc A) ⊆ ∪y ∈ x (C ·o y))
55		visset 1350	. . . . . . . . . . . . . . . . . . 19 ⊢ x ∈ V
56		omlim 3136	. . . . . . . . . . . . . . . . . . 19 ⊢ ((C ∈ On ∧ (x ∈ V ∧ Lim x)) → (C ·o x) = ∪y ∈ x (C ·o y))
57	55, 56	mpan21 531	. . . . . . . . . . . . . . . . . 18 ⊢ ((C ∈ On ∧ Lim x) → (C ·o x) = ∪y ∈ x (C ·o y))
58	57	adantr 306	. . . . . . . . . . . . . . . . 17 ⊢ (((C ∈ On ∧ Lim x) ∧ A ∈ x) → (C ·o x) = ∪y ∈ x (C ·o y))
59	54, 58	sseqtr4d 1537	. . . . . . . . . . . . . . . 16 ⊢ (((C ∈ On ∧ Lim x) ∧ A ∈ x) → (C ·o suc A) ⊆ (C ·o x))
60		id 9	. . . . . . . . . . . . . . . . . 18 ⊢ ((C ∈ On ∧ Lim x) → (C ∈ On ∧ Lim x))
61	60	adantrr 312	. . . . . . . . . . . . . . . . 17 ⊢ ((C ∈ On ∧ (Lim x ∧ ∅ ∈ C)) → (C ∈ On ∧ Lim x))
62	61	adantlr 310	. . . . . . . . . . . . . . . 16 ⊢ (((C ∈ On ∧ A ∈ On) ∧ (Lim x ∧ ∅ ∈ C)) → (C ∈ On ∧ Lim x))
63	59, 62	sylan 343	. . . . . . . . . . . . . . 15 ⊢ ((((C ∈ On ∧ A ∈ On) ∧ (Lim x ∧ ∅ ∈ C)) ∧ A ∈ x) → (C ·o suc A) ⊆ (C ·o x))
64		oaord1 3153	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((C ·o A) ∈ On ∧ C ∈ On) → (∅ ∈ C ↔ (C ·o A) ∈ ((C ·o A) +o C)))
65		omcl 3139	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((C ∈ On ∧ A ∈ On) → (C ·o A) ∈ On)
66	64, 65	sylan 343	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((C ∈ On ∧ A ∈ On) ∧ C ∈ On) → (∅ ∈ C ↔ (C ·o A) ∈ ((C ·o A) +o C)))
67	66	anabss1 381	. . . . . . . . . . . . . . . . . . 19 ⊢ ((C ∈ On ∧ A ∈ On) → (∅ ∈ C ↔ (C ·o A) ∈ ((C ·o A) +o C)))
68	67	biimpa 324	. . . . . . . . . . . . . . . . . 18 ⊢ (((C ∈ On ∧ A ∈ On) ∧ ∅ ∈ C) → (C ·o A) ∈ ((C ·o A) +o C))
69		omsuc 3133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((C ∈ On ∧ A ∈ On) → (C ·o suc A) = ((C ·o A) +o C))
70	69	adantr 306	. . . . . . . . . . . . . . . . . 18 ⊢ (((C ∈ On ∧ A ∈ On) ∧ ∅ ∈ C) → (C ·o suc A) = ((C ·o A) +o C))
71	68, 70	eleqtrrd 1166	. . . . . . . . . . . . . . . . 17 ⊢ (((C ∈ On ∧ A ∈ On) ∧ ∅ ∈ C) → (C ·o A) ∈ (C ·o suc A))
72	71	adantrl 311	. . . . . . . . . . . . . . . 16 ⊢ (((C ∈ On ∧ A ∈ On) ∧ (Lim x ∧ ∅ ∈ C)) → (C ·o A) ∈ (C ·o suc A))
73	72	adantr 306	. . . . . . . . . . . . . . 15 ⊢ ((((C ∈ On ∧ A ∈ On) ∧ (Lim x ∧ ∅ ∈ C)) ∧ A ∈ x) → (C ·o A) ∈ (C ·o suc A))
74	63, 73	sseldd 1507	. . . . . . . . . . . . . 14 ⊢ ((((C ∈ On ∧ A ∈ On) ∧ (Lim x ∧ ∅ ∈ C)) ∧ A ∈ x) → (C ·o A) ∈ (C ·o x))
75	74	exp 291	. . . . . . . . . . . . 13 ⊢ (((C ∈ On ∧ A ∈ On) ∧ (Lim x ∧ ∅ ∈ C)) → (A ∈ x → (C ·o A) ∈ (C ·o x)))
76	75	exp43 301	. . . . . . . . . . . 12 ⊢ (C ∈ On → (A ∈ On → (Lim x → (∅ ∈ C → (A ∈ x → (C ·o A) ∈ (C ·o x))))))
77	76	com13 33	. . . . . . . . . . 11 ⊢ (Lim x → (A ∈ On → (C ∈ On → (∅ ∈ C → (A ∈ x → (C ·o A) ∈ (C ·o x))))))
78	77	imp4c 284	. . . . . . . . . 10 ⊢ (Lim x → (((A ∈ On ∧ C ∈ On) ∧ ∅ ∈ C) → (A ∈ x → (C ·o A) ∈ (C ·o x))))
79	78	a1dd 42	. . . . . . . . 9 ⊢ (Lim x → (((A ∈ On ∧ C ∈ On) ∧ ∅ ∈ C) → (∀y ∈ x (A ∈ y → (C ·o A) ∈ (C ·o y)) → (A ∈ x → (C ·o A) ∈ (C ·o x)))))
80	6, 10, 14, 18, 21, 48, 79	tfinds3 2406	. . . . . . . 8 ⊢ (B ∈ On → (((A ∈ On ∧ C ∈ On) ∧ ∅ ∈ C) → (A ∈ B → (C ·o A) ∈ (C ·o B))))
81	80	com23 32	. . . . . . 7 ⊢ (B ∈ On → (A ∈ B → (((A ∈ On ∧ C ∈ On) ∧ ∅ ∈ C) → (C ·o A) ∈ (C ·o B))))
82	81	exp4a 295	. . . . . 6 ⊢ (B ∈ On → (A ∈ B → ((A ∈ On ∧ C ∈ On) → (∅ ∈ C → (C ·o A) ∈ (C ·o B)))))
83	82	exp4a 295	. . . . 5 ⊢ (B ∈ On → (A ∈ B → (A ∈ On → (C ∈ On → (∅ ∈ C → (C ·o A) ∈ (C ·o B))))))
84	2, 83	mpdd 47	. . . 4 ⊢ (B ∈ On → (A ∈ B → (C ∈ On → (∅ ∈ C → (C ·o A) ∈ (C ·o B)))))
85	84	com34 36	. . 3 ⊢ (B ∈ On → (A ∈ B → (∅ ∈ C → (C ∈ On → (C ·o A) ∈ (C ·o B)))))
86	85	com24 37	. 2 ⊢ (B ∈ On → (C ∈ On → (∅ ∈ C → (A ∈ B → (C ·o A) ∈ (C ·o B)))))
87	86	imp31 280	1 ⊢ (((B ∈ On ∧ C ∈ On) ∧ ∅ ∈ C) → (A ∈ B → (C ·o A) ∈ (C ·o B)))

Syntax hints:   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  ∪ciun 1994  Oncon0 2199  Lim wlim 2200  suc csuc 2201  (class class class)co 3001   +_o coa 3101   ·_o comu 3102

This theorem is referenced by:  oen0 3165

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-oadd 3106  df-omul 3107

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