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Theorem omcl 6772
Description: Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. (Contributed by NM, 3-Aug-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
omcl  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )

Proof of Theorem omcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6081 . . . 4  |-  ( x  =  (/)  ->  ( A  .o  x )  =  ( A  .o  (/) ) )
21eleq1d 2501 . . 3  |-  ( x  =  (/)  ->  ( ( A  .o  x )  e.  On  <->  ( A  .o  (/) )  e.  On ) )
3 oveq2 6081 . . . 4  |-  ( x  =  y  ->  ( A  .o  x )  =  ( A  .o  y
) )
43eleq1d 2501 . . 3  |-  ( x  =  y  ->  (
( A  .o  x
)  e.  On  <->  ( A  .o  y )  e.  On ) )
5 oveq2 6081 . . . 4  |-  ( x  =  suc  y  -> 
( A  .o  x
)  =  ( A  .o  suc  y ) )
65eleq1d 2501 . . 3  |-  ( x  =  suc  y  -> 
( ( A  .o  x )  e.  On  <->  ( A  .o  suc  y
)  e.  On ) )
7 oveq2 6081 . . . 4  |-  ( x  =  B  ->  ( A  .o  x )  =  ( A  .o  B
) )
87eleq1d 2501 . . 3  |-  ( x  =  B  ->  (
( A  .o  x
)  e.  On  <->  ( A  .o  B )  e.  On ) )
9 om0 6753 . . . 4  |-  ( A  e.  On  ->  ( A  .o  (/) )  =  (/) )
10 0elon 4626 . . . 4  |-  (/)  e.  On
119, 10syl6eqel 2523 . . 3  |-  ( A  e.  On  ->  ( A  .o  (/) )  e.  On )
12 oacl 6771 . . . . . . 7  |-  ( ( ( A  .o  y
)  e.  On  /\  A  e.  On )  ->  ( ( A  .o  y )  +o  A
)  e.  On )
1312expcom 425 . . . . . 6  |-  ( A  e.  On  ->  (
( A  .o  y
)  e.  On  ->  ( ( A  .o  y
)  +o  A )  e.  On ) )
1413adantr 452 . . . . 5  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  .o  y )  e.  On  ->  ( ( A  .o  y )  +o  A
)  e.  On ) )
15 omsuc 6762 . . . . . 6  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( A  .o  suc  y )  =  ( ( A  .o  y
)  +o  A ) )
1615eleq1d 2501 . . . . 5  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  .o  suc  y )  e.  On  <->  ( ( A  .o  y
)  +o  A )  e.  On ) )
1714, 16sylibrd 226 . . . 4  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  .o  y )  e.  On  ->  ( A  .o  suc  y )  e.  On ) )
1817expcom 425 . . 3  |-  ( y  e.  On  ->  ( A  e.  On  ->  ( ( A  .o  y
)  e.  On  ->  ( A  .o  suc  y
)  e.  On ) ) )
19 vex 2951 . . . . . 6  |-  x  e. 
_V
20 iunon 6592 . . . . . 6  |-  ( ( x  e.  _V  /\  A. y  e.  x  ( A  .o  y )  e.  On )  ->  U_ y  e.  x  ( A  .o  y
)  e.  On )
2119, 20mpan 652 . . . . 5  |-  ( A. y  e.  x  ( A  .o  y )  e.  On  ->  U_ y  e.  x  ( A  .o  y )  e.  On )
22 omlim 6769 . . . . . . 7  |-  ( ( A  e.  On  /\  ( x  e.  _V  /\ 
Lim  x ) )  ->  ( A  .o  x )  =  U_ y  e.  x  ( A  .o  y ) )
2319, 22mpanr1 665 . . . . . 6  |-  ( ( A  e.  On  /\  Lim  x )  ->  ( A  .o  x )  = 
U_ y  e.  x  ( A  .o  y
) )
2423eleq1d 2501 . . . . 5  |-  ( ( A  e.  On  /\  Lim  x )  ->  (
( A  .o  x
)  e.  On  <->  U_ y  e.  x  ( A  .o  y )  e.  On ) )
2521, 24syl5ibr 213 . . . 4  |-  ( ( A  e.  On  /\  Lim  x )  ->  ( A. y  e.  x  ( A  .o  y
)  e.  On  ->  ( A  .o  x )  e.  On ) )
2625expcom 425 . . 3  |-  ( Lim  x  ->  ( A  e.  On  ->  ( A. y  e.  x  ( A  .o  y )  e.  On  ->  ( A  .o  x )  e.  On ) ) )
272, 4, 6, 8, 11, 18, 26tfinds3 4836 . 2  |-  ( B  e.  On  ->  ( A  e.  On  ->  ( A  .o  B )  e.  On ) )
2827impcom 420 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   (/)c0 3620   U_ciun 4085   Oncon0 4573   Lim wlim 4574   suc csuc 4575  (class class class)co 6073    +o coa 6713    .o comu 6714
This theorem is referenced by:  oecl  6773  omordi  6801  omord2  6802  omcan  6804  omword  6805  omwordri  6807  om00  6810  om00el  6811  omlimcl  6813  odi  6814  omass  6815  oneo  6816  omeulem1  6817  omeulem2  6818  omopth2  6819  oeoelem  6833  oeoe  6834  oeeui  6837  oaabs2  6880  omxpenlem  7201  omxpen  7202  cantnfle  7618  cantnflt  7619  cantnflem1d  7636  cantnflem1  7637  cantnflem3  7639  cantnflem4  7640  cnfcomlem  7648  xpnum  7830  infxpenc  7891  dfac12lem2  8016
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-recs 6625  df-rdg 6660  df-oadd 6720  df-omul 6721
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