Theorem omwordri 8403

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 7283	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
2		oveq2 7283	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
3	1, 2	sseq12d 3954	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o ∅) ⊆ (𝐵 ·o ∅)))
4		oveq2 7283	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
5		oveq2 7283	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
6	4, 5	sseq12d 3954	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦)))
7		oveq2 7283	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
8		oveq2 7283	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
9	7, 8	sseq12d 3954	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o suc 𝑦) ⊆ (𝐵 ·o suc 𝑦)))
10		oveq2 7283	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐶))
11		oveq2 7283	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐶))
12	10, 11	sseq12d 3954	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))
13		om0 8347	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
14		0ss 4330	. . . . . . 7 ⊢ ∅ ⊆ (𝐵 ·o ∅)
15	13, 14	eqsstrdi 3975	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) ⊆ (𝐵 ·o ∅))
16	15	ad2antrr 723	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ·o ∅) ⊆ (𝐵 ·o ∅))
17		omcl 8366	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑦) ∈ On)
18	17	3adant2 1130	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑦) ∈ On)
19		omcl 8366	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
20	19	3adant1 1129	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
21		simp1 1135	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
22		oawordri 8381	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 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 407	. . . . . . . . . . 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 713	. . . . . . . . . 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 8380	. . . . . . . . . . . . 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 1408	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ((𝐵 ·o 𝑦) +o 𝐴) ⊆ ((𝐵 ·o 𝑦) +o 𝐵)))
28	27	biimpa 477	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐵 ·o 𝑦) +o 𝐴) ⊆ ((𝐵 ·o 𝑦) +o 𝐵))
29	28	adantrr 714	. . . . . . . . . 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 3931	. . . . . . . . 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 8356	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
32	31	3adant2 1130	. . . . . . . . . 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 8356	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
35	34	3adant1 1129	. . . . . . . . . 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 3968	. . . . . . . 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 1356	. . . . . . 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 424	. . . . 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 3436	. . . . . . . 8 ⊢ 𝑥 ∈ V
42		ss2iun 4942	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦))
43		omlim 8363	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦))
44	43	ad2ant2rl 746	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → (𝐴 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦))
45		omlim 8363	. . . . . . . . . . . 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 3954	. . . . . . . . . 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 245	. . . . . . . . 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 676	. . . . . . . 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 700	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥)))
51	50	expcom 414	. . . . . 6 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥))))
52	51	adantrd 492	. . . . 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 7711	. . . 4 ⊢ (𝐶 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))
54	53	expd 416	. . 3 ⊢ (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶))))
55	54	3impib 1115	. 2 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))
56	55	3coml 1126	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ⊆ wss 3887  ∅c0 4256  ∪ ciun 4924  Oncon0 6266  Lim wlim 6267  suc csuc 6268  (class class class)co 7275   +_o coa 8294   ·_o comu 8295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-oadd 8301  df-omul 8302

This theorem is referenced by:  omword2  8405  oewordri  8423  oeordsuc  8425