Theorem omwordri 8587

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 7432	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
2		oveq2 7432	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
3	1, 2	sseq12d 3942	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o ∅) ⊆ (𝐵 ·o ∅)))
4		oveq2 7432	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
5		oveq2 7432	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
6	4, 5	sseq12d 3942	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦)))
7		oveq2 7432	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
8		oveq2 7432	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
9	7, 8	sseq12d 3942	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o suc 𝑦) ⊆ (𝐵 ·o suc 𝑦)))
10		oveq2 7432	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐶))
11		oveq2 7432	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐶))
12	10, 11	sseq12d 3942	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥) ↔ (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))
13		om0 8532	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
14		0ss 4349	. . . . . . 7 ⊢ ∅ ⊆ (𝐵 ·o ∅)
15	13, 14	eqsstrdi 3953	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) ⊆ (𝐵 ·o ∅))
16	15	ad2antrr 739	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ·o ∅) ⊆ (𝐵 ·o ∅))
17		omcl 8551	. . . . . . . . . . . . . 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 8551	. . . . . . . . . . . . . 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 8565	. . . . . . . . . . . . 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 412	. . . . . . . . . . 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 729	. . . . . . . . . 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 8564	. . . . . . . . . . . . 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 482	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐵 ·o 𝑦) +o 𝐴) ⊆ ((𝐵 ·o 𝑦) +o 𝐵))
29	28	adantrr 730	. . . . . . . . . 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 3919	. . . . . . . . 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 8541	. . . . . . . . . . 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 486	. . . . . . . . 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 8541	. . . . . . . . . . 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 486	. . . . . . . . 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 3964	. . . . . . . 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 429	. . . . 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 3454	. . . . . . . 8 ⊢ 𝑥 ∈ V
42		ss2iun 4969	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦))
43		omlim 8548	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦))
44	43	ad2ant2rl 762	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → (𝐴 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦))
45		omlim 8548	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦))
46	45	adantl 487	. . . . . . . . . . 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 3942	. . . . . . . . . 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 692	. . . . . . . 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 716	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥)))
51	50	expcom 419	. . . . . 6 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ⊆ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑥) ⊆ (𝐵 ·o 𝑥))))
52	51	adantrd 497	. . . . 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 7880	. . . 4 ⊢ (𝐶 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))
54	53	expd 421	. . 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3896  ∅c0 4277  ∪ ciun 4950  Oncon0 6365  Lim wlim 6366  suc csuc 6367  (class class class)co 7424   +_o coa 8480   ·_o comu 8481

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-oadd 8487  df-omul 8488

This theorem is used by:  omword2  8589  oewordri  8608  oeordsuc  8610  omabs2  44443