Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  omword2 Structured version   Visualization version   GIF version

Theorem omword2 7699
 Description: An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.)
Assertion
Ref Expression
omword2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐵 ·𝑜 𝐴))

Proof of Theorem omword2
StepHypRef Expression
1 om1r 7668 . . 3 (𝐴 ∈ On → (1𝑜 ·𝑜 𝐴) = 𝐴)
21ad2antrr 762 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (1𝑜 ·𝑜 𝐴) = 𝐴)
3 eloni 5771 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
4 ordgt0ge1 7622 . . . . . 6 (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1𝑜𝐵))
54biimpa 500 . . . . 5 ((Ord 𝐵 ∧ ∅ ∈ 𝐵) → 1𝑜𝐵)
63, 5sylan 487 . . . 4 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → 1𝑜𝐵)
76adantll 750 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 1𝑜𝐵)
8 1on 7612 . . . . . 6 1𝑜 ∈ On
9 omwordri 7697 . . . . . 6 ((1𝑜 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1𝑜𝐵 → (1𝑜 ·𝑜 𝐴) ⊆ (𝐵 ·𝑜 𝐴)))
108, 9mp3an1 1451 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (1𝑜𝐵 → (1𝑜 ·𝑜 𝐴) ⊆ (𝐵 ·𝑜 𝐴)))
1110ancoms 468 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝐵 → (1𝑜 ·𝑜 𝐴) ⊆ (𝐵 ·𝑜 𝐴)))
1211adantr 480 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (1𝑜𝐵 → (1𝑜 ·𝑜 𝐴) ⊆ (𝐵 ·𝑜 𝐴)))
137, 12mpd 15 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (1𝑜 ·𝑜 𝐴) ⊆ (𝐵 ·𝑜 𝐴))
142, 13eqsstr3d 3673 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐵 ·𝑜 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ⊆ wss 3607  ∅c0 3948  Ord word 5760  Oncon0 5761  (class class class)co 6690  1𝑜c1o 7598   ·𝑜 comu 7603 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610 This theorem is referenced by:  omeulem1  7707  omabslem  7771  omabs  7772
 Copyright terms: Public domain W3C validator