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Theorem omcl 6743
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 6052 . . . 4  |-  ( x  =  (/)  ->  ( A  .o  x )  =  ( A  .o  (/) ) )
21eleq1d 2474 . . 3  |-  ( x  =  (/)  ->  ( ( A  .o  x )  e.  On  <->  ( A  .o  (/) )  e.  On ) )
3 oveq2 6052 . . . 4  |-  ( x  =  y  ->  ( A  .o  x )  =  ( A  .o  y
) )
43eleq1d 2474 . . 3  |-  ( x  =  y  ->  (
( A  .o  x
)  e.  On  <->  ( A  .o  y )  e.  On ) )
5 oveq2 6052 . . . 4  |-  ( x  =  suc  y  -> 
( A  .o  x
)  =  ( A  .o  suc  y ) )
65eleq1d 2474 . . 3  |-  ( x  =  suc  y  -> 
( ( A  .o  x )  e.  On  <->  ( A  .o  suc  y
)  e.  On ) )
7 oveq2 6052 . . . 4  |-  ( x  =  B  ->  ( A  .o  x )  =  ( A  .o  B
) )
87eleq1d 2474 . . 3  |-  ( x  =  B  ->  (
( A  .o  x
)  e.  On  <->  ( A  .o  B )  e.  On ) )
9 om0 6724 . . . 4  |-  ( A  e.  On  ->  ( A  .o  (/) )  =  (/) )
10 0elon 4598 . . . 4  |-  (/)  e.  On
119, 10syl6eqel 2496 . . 3  |-  ( A  e.  On  ->  ( A  .o  (/) )  e.  On )
12 oacl 6742 . . . . . . 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 6733 . . . . . 6  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( A  .o  suc  y )  =  ( ( A  .o  y
)  +o  A ) )
1615eleq1d 2474 . . . . 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 2923 . . . . . 6  |-  x  e. 
_V
20 iunon 6563 . . . . . 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 6740 . . . . . . 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 2474 . . . . 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 4807 . 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 1649    e. wcel 1721   A.wral 2670   _Vcvv 2920   (/)c0 3592   U_ciun 4057   Oncon0 4545   Lim wlim 4546   suc csuc 4547  (class class class)co 6044    +o coa 6684    .o comu 6685
This theorem is referenced by:  oecl  6744  omordi  6772  omord2  6773  omcan  6775  omword  6776  omwordri  6778  om00  6781  om00el  6782  omlimcl  6784  odi  6785  omass  6786  oneo  6787  omeulem1  6788  omeulem2  6789  omopth2  6790  oeoelem  6804  oeoe  6805  oeeui  6808  oaabs2  6851  omxpenlem  7172  omxpen  7173  cantnfle  7586  cantnflt  7587  cantnflem1d  7604  cantnflem1  7605  cantnflem3  7607  cantnflem4  7608  cnfcomlem  7616  xpnum  7798  infxpenc  7859  dfac12lem2  7984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-recs 6596  df-rdg 6631  df-oadd 6691  df-omul 6692
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