Theorem omwordri 8553

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 7418	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
2		oveq2 7418	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
3	1, 2	sseq12d 3970	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o ∅) ⊆ (𝐵 ·o ∅)))
4		oveq2 7418	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
5		oveq2 7418	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
6	4, 5	sseq12d 3970	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦)))
7		oveq2 7418	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
8		oveq2 7418	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
9	7, 8	sseq12d 3970	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o suc 𝑦) ⊆ (𝐵 ·o suc 𝑦)))
10		oveq2 7418	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐶))
11		oveq2 7418	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐶))
12	10, 11	sseq12d 3970	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))
13		om0 8498	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
14		0ss 4357	. . . . . . 7 ⊢ ∅ ⊆ (𝐵 ·o ∅)
15	13, 14	eqsstrdi 3981	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) ⊆ (𝐵 ·o ∅))
16	15	ad2antrr 738	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ·o ∅) ⊆ (𝐵 ·o ∅))
17		omcl 8517	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑦) ∈ On)
18	17	3adant2 1149	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑦) ∈ On)
19		omcl 8517	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
20	19	3adant1 1148	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
21		simp1 1154	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
22		oawordri 8531	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 8530	. . . . . . . . . . . . 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 1436	. . . . . . . . . . . 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 3947	. . . . . . . . 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 8507	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
32	31	3adant2 1149	. . . . . . . . . 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 8507	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
35	34	3adant1 1148	. . . . . . . . . 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 3992	. . . . . . . 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 1376	. . . . . . 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 3459	. . . . . . . 8 ⊢ 𝑥 ∈ V
42		ss2iun 4975	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦))
43		omlim 8514	. . . . . . . . . . . 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 8514	. . . . . . . . . . . 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 3970	. . . . . . . . . 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 7857	. . . 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 1134	. 2 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))
56	55	3coml 1145	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ⊆ wss 3905  ∅c0 4286  ∪ ciun 4956  Oncon0 6360  Lim wlim 6361  suc csuc 6362  (class class class)co 7410   +_o coa 8446   ·_o comu 8447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-oadd 8453  df-omul 8454

This theorem is referenced by:  omword2  8555  oewordri  8574  oeordsuc  8576  omabs2  44059