Theorem omwordri 8556

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 7419	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
2		oveq2 7419	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
3	1, 2	sseq12d 3978	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o ∅) ⊆ (𝐵 ·o ∅)))
4		oveq2 7419	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
5		oveq2 7419	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
6	4, 5	sseq12d 3978	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦)))
7		oveq2 7419	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
8		oveq2 7419	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
9	7, 8	sseq12d 3978	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o suc 𝑦) ⊆ (𝐵 ·o suc 𝑦)))
10		oveq2 7419	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐶))
11		oveq2 7419	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐶))
12	10, 11	sseq12d 3978	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))
13		om0 8501	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
14		0ss 4364	. . . . . . 7 ⊢ ∅ ⊆ (𝐵 ·o ∅)
15	13, 14	eqsstrdi 3989	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) ⊆ (𝐵 ·o ∅))
16	15	ad2antrr 738	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ·o ∅) ⊆ (𝐵 ·o ∅))
17		omcl 8520	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑦) ∈ On)
18	17	3adant2 1147	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑦) ∈ On)
19		omcl 8520	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
20	19	3adant1 1146	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
21		simp1 1152	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
22		oawordri 8534	. . . . . . . . . . . . 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 1396	. . . . . . . . . . . 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 411	. . . . . . . . . . 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 728	. . . . . . . . . 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 8533	. . . . . . . . . . . . 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 1434	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ((𝐵 ·o 𝑦) +o 𝐴) ⊆ ((𝐵 ·o 𝑦) +o 𝐵)))
28	27	biimpa 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐵 ·o 𝑦) +o 𝐴) ⊆ ((𝐵 ·o 𝑦) +o 𝐵))
29	28	adantrr 729	. . . . . . . . . 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 3955	. . . . . . . . 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 8510	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
32	31	3adant2 1147	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
33	32	adantr 485	. . . . . . . . 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 8510	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
35	34	3adant1 1146	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
36	35	adantr 485	. . . . . . . . 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 4000	. . . . . . . 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 1374	. . . . . . 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 88	. . . . . 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 428	. . . . 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 3467	. . . . . . . 8 ⊢ 𝑥 ∈ V
42		ss2iun 4979	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦))
43		omlim 8517	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦))
44	43	ad2ant2rl 761	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → (𝐴 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦))
45		omlim 8517	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦))
46	45	adantl 486	. . . . . . . . . . 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 3978	. . . . . . . . . 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	imbitrrid 249	. . . . . . . . 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 691	. . . . . . . 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 418	. . . . . 6 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥))))
52	51	adantrd 496	. . . . 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 7860	. . . 4 ⊢ (𝐶 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))
54	53	expd 420	. . 3 ⊢ (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶))))
55	54	3impib 1132	. 2 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))
56	55	3coml 1143	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  ∪ ciun 4960  Oncon0 6361  Lim wlim 6362  suc csuc 6363  (class class class)co 7411   +_o coa 8449   ·_o comu 8450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-oadd 8456  df-omul 8457

This theorem is referenced by:  omword2  8558  oewordri  8577  oeordsuc  8579  omabs2  43950