Theorem omwordi 7856

Description: Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)

Assertion

Ref	Expression
omwordi	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))

Proof of Theorem omwordi

Step	Hyp	Ref	Expression
1		omword 7855	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
2	1	biimpd 220	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
3	2	ex 401	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))))
4		eloni 5918	. . . . . 6 ⊢ (𝐶 ∈ On → Ord 𝐶)
5		ord0eln0 5962	. . . . . . 7 ⊢ (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
6	5	necon2bbid 2980	. . . . . 6 ⊢ (Ord 𝐶 → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
7	4, 6	syl 17	. . . . 5 ⊢ (𝐶 ∈ On → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
8	7	3ad2ant3 1165	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
9		ssid 3783	. . . . . . 7 ⊢ ∅ ⊆ ∅
10		om0r 7824	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (∅ ·𝑜 𝐴) = ∅)
11	10	adantr 472	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·𝑜 𝐴) = ∅)
12		om0r 7824	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ·𝑜 𝐵) = ∅)
13	12	adantl 473	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·𝑜 𝐵) = ∅)
14	11, 13	sseq12d 3794	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((∅ ·𝑜 𝐴) ⊆ (∅ ·𝑜 𝐵) ↔ ∅ ⊆ ∅))
15	9, 14	mpbiri 249	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·𝑜 𝐴) ⊆ (∅ ·𝑜 𝐵))
16		oveq1 6849	. . . . . . 7 ⊢ (𝐶 = ∅ → (𝐶 ·𝑜 𝐴) = (∅ ·𝑜 𝐴))
17		oveq1 6849	. . . . . . 7 ⊢ (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
18	16, 17	sseq12d 3794	. . . . . 6 ⊢ (𝐶 = ∅ → ((𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵) ↔ (∅ ·𝑜 𝐴) ⊆ (∅ ·𝑜 𝐵)))
19	15, 18	syl5ibrcom 238	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 = ∅ → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
20	19	3adant3 1162	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 = ∅ → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
21	8, 20	sylbird 251	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ ∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
22	21	a1dd 50	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ ∅ ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))))
23	3, 22	pm2.61d 171	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1107   = wceq 1652   ∈ wcel 2155   ⊆ wss 3732  ∅c0 4079  Ord word 5907  Oncon0 5908  (class class class)co 6842   ·_𝑜 comu 7762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-oadd 7768  df-omul 7769

This theorem is referenced by:  omword1  7858  omass  7865  omeulem1  7867  oewordri  7877  oeoalem  7881  oeeui  7887  oaabs2  7930  omxpenlem  8268  cantnflt  8784  cantnflem1d  8800