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Theorem omword2 6809
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 6778 . . 3  |-  ( A  e.  On  ->  ( 1o  .o  A )  =  A )
21ad2antrr 707 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  ( 1o  .o  A )  =  A )
3 eloni 4583 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
4 ordgt0ge1 6733 . . . . . 6  |-  ( Ord 
B  ->  ( (/)  e.  B  <->  1o  C_  B ) )
54biimpa 471 . . . . 5  |-  ( ( Ord  B  /\  (/)  e.  B
)  ->  1o  C_  B
)
63, 5sylan 458 . . . 4  |-  ( ( B  e.  On  /\  (/) 
e.  B )  ->  1o  C_  B )
76adantll 695 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  1o  C_  B
)
8 1on 6723 . . . . . 6  |-  1o  e.  On
9 omwordri 6807 . . . . . 6  |-  ( ( 1o  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( 1o  C_  B  ->  ( 1o  .o  A )  C_  ( B  .o  A
) ) )
108, 9mp3an1 1266 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( 1o  C_  B  ->  ( 1o  .o  A
)  C_  ( B  .o  A ) ) )
1110ancoms 440 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  C_  B  ->  ( 1o  .o  A
)  C_  ( B  .o  A ) ) )
1211adantr 452 . . 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 3375 1  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  A  C_  ( B  .o  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312   (/)c0 3620   Ord word 4572   Oncon0 4573  (class class class)co 6073   1oc1o 6709    .o comu 6714
This theorem is referenced by:  omeulem1  6817  omabslem  6881  omabs  6882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-omul 6721
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