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Theorem omword2 7260
Description: An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.)
Assertion
Ref Expression
omword2  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  A  C_  ( B  .o  A ) )

Proof of Theorem omword2
StepHypRef Expression
1 om1r 7229 . . 3  |-  ( A  e.  On  ->  ( 1o  .o  A )  =  A )
21ad2antrr 724 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  ( 1o  .o  A )  =  A )
3 eloni 5420 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
4 ordgt0ge1 7184 . . . . . 6  |-  ( Ord 
B  ->  ( (/)  e.  B  <->  1o  C_  B ) )
54biimpa 482 . . . . 5  |-  ( ( Ord  B  /\  (/)  e.  B
)  ->  1o  C_  B
)
63, 5sylan 469 . . . 4  |-  ( ( B  e.  On  /\  (/) 
e.  B )  ->  1o  C_  B )
76adantll 712 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  1o  C_  B
)
8 1on 7174 . . . . . 6  |-  1o  e.  On
9 omwordri 7258 . . . . . 6  |-  ( ( 1o  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( 1o  C_  B  ->  ( 1o  .o  A )  C_  ( B  .o  A
) ) )
108, 9mp3an1 1313 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( 1o  C_  B  ->  ( 1o  .o  A
)  C_  ( B  .o  A ) ) )
1110ancoms 451 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  C_  B  ->  ( 1o  .o  A
)  C_  ( B  .o  A ) ) )
1211adantr 463 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  ( 1o  C_  B  ->  ( 1o  .o  A )  C_  ( B  .o  A ) ) )
137, 12mpd 15 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  ( 1o  .o  A )  C_  ( B  .o  A ) )
142, 13eqsstr3d 3477 1  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  A  C_  ( B  .o  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    C_ wss 3414   (/)c0 3738   Ord word 5409   Oncon0 5410  (class class class)co 6278   1oc1o 7160    .o comu 7165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-omul 7172
This theorem is referenced by:  omeulem1  7268  omabslem  7332  omabs  7333
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