Theorem omwordri 8594

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 7424	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
2		oveq2 7424	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
3	1, 2	sseq12d 4012	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o ∅) ⊆ (𝐵 ·o ∅)))
4		oveq2 7424	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
5		oveq2 7424	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
6	4, 5	sseq12d 4012	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦)))
7		oveq2 7424	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
8		oveq2 7424	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
9	7, 8	sseq12d 4012	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o suc 𝑦) ⊆ (𝐵 ·o suc 𝑦)))
10		oveq2 7424	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐶))
11		oveq2 7424	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐶))
12	10, 11	sseq12d 4012	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))
13		om0 8539	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
14		0ss 4394	. . . . . . 7 ⊢ ∅ ⊆ (𝐵 ·o ∅)
15	13, 14	eqsstrdi 4033	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) ⊆ (𝐵 ·o ∅))
16	15	ad2antrr 724	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ·o ∅) ⊆ (𝐵 ·o ∅))
17		omcl 8558	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑦) ∈ On)
18	17	3adant2 1128	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑦) ∈ On)
19		omcl 8558	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
20	19	3adant1 1127	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
21		simp1 1133	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
22		oawordri 8572	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 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 405	. . . . . . . . . . 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 714	. . . . . . . . . 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 8571	. . . . . . . . . . . . 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 1406	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ((𝐵 ·o 𝑦) +o 𝐴) ⊆ ((𝐵 ·o 𝑦) +o 𝐵)))
28	27	biimpa 475	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐵 ·o 𝑦) +o 𝐴) ⊆ ((𝐵 ·o 𝑦) +o 𝐵))
29	28	adantrr 715	. . . . . . . . . 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 3989	. . . . . . . . 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 8548	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
32	31	3adant2 1128	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
33	32	adantr 479	. . . . . . . . 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 8548	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
35	34	3adant1 1127	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
36	35	adantr 479	. . . . . . . . 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 4026	. . . . . . . 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 1354	. . . . . . 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 422	. . . . 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 3466	. . . . . . . 8 ⊢ 𝑥 ∈ V
42		ss2iun 5011	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦))
43		omlim 8555	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦))
44	43	ad2ant2rl 747	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → (𝐴 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦))
45		omlim 8555	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦))
46	45	adantl 480	. . . . . . . . . . 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 4012	. . . . . . . . . 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 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 677	. . . . . . . 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 701	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥)))
51	50	expcom 412	. . . . . 6 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥))))
52	51	adantrd 490	. . . . 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 7867	. . . 4 ⊢ (𝐶 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))
54	53	expd 414	. . 3 ⊢ (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶))))
55	54	3impib 1113	. 2 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))
56	55	3coml 1124	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  ∀wral 3051  Vcvv 3462   ⊆ wss 3946  ∅c0 4322  ∪ ciun 4993  Oncon0 6368  Lim wlim 6369  suc csuc 6370  (class class class)co 7416   +_o coa 8485   ·_o comu 8486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-oadd 8492  df-omul 8493

This theorem is referenced by:  omword2  8596  oewordri  8614  oeordsuc  8616  omabs2  43035