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Theorem omcl 6622
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 5953 . . . 4  |-  ( x  =  (/)  ->  ( A  .o  x )  =  ( A  .o  (/) ) )
21eleq1d 2424 . . 3  |-  ( x  =  (/)  ->  ( ( A  .o  x )  e.  On  <->  ( A  .o  (/) )  e.  On ) )
3 oveq2 5953 . . . 4  |-  ( x  =  y  ->  ( A  .o  x )  =  ( A  .o  y
) )
43eleq1d 2424 . . 3  |-  ( x  =  y  ->  (
( A  .o  x
)  e.  On  <->  ( A  .o  y )  e.  On ) )
5 oveq2 5953 . . . 4  |-  ( x  =  suc  y  -> 
( A  .o  x
)  =  ( A  .o  suc  y ) )
65eleq1d 2424 . . 3  |-  ( x  =  suc  y  -> 
( ( A  .o  x )  e.  On  <->  ( A  .o  suc  y
)  e.  On ) )
7 oveq2 5953 . . . 4  |-  ( x  =  B  ->  ( A  .o  x )  =  ( A  .o  B
) )
87eleq1d 2424 . . 3  |-  ( x  =  B  ->  (
( A  .o  x
)  e.  On  <->  ( A  .o  B )  e.  On ) )
9 om0 6603 . . . 4  |-  ( A  e.  On  ->  ( A  .o  (/) )  =  (/) )
10 0elon 4527 . . . 4  |-  (/)  e.  On
119, 10syl6eqel 2446 . . 3  |-  ( A  e.  On  ->  ( A  .o  (/) )  e.  On )
12 oacl 6621 . . . . . . 7  |-  ( ( ( A  .o  y
)  e.  On  /\  A  e.  On )  ->  ( ( A  .o  y )  +o  A
)  e.  On )
1312expcom 424 . . . . . 6  |-  ( A  e.  On  ->  (
( A  .o  y
)  e.  On  ->  ( ( A  .o  y
)  +o  A )  e.  On ) )
1413adantr 451 . . . . 5  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  .o  y )  e.  On  ->  ( ( A  .o  y )  +o  A
)  e.  On ) )
15 omsuc 6612 . . . . . 6  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( A  .o  suc  y )  =  ( ( A  .o  y
)  +o  A ) )
1615eleq1d 2424 . . . . 5  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  .o  suc  y )  e.  On  <->  ( ( A  .o  y
)  +o  A )  e.  On ) )
1714, 16sylibrd 225 . . . 4  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  .o  y )  e.  On  ->  ( A  .o  suc  y )  e.  On ) )
1817expcom 424 . . 3  |-  ( y  e.  On  ->  ( A  e.  On  ->  ( ( A  .o  y
)  e.  On  ->  ( A  .o  suc  y
)  e.  On ) ) )
19 vex 2867 . . . . . 6  |-  x  e. 
_V
20 iunon 6442 . . . . . 6  |-  ( ( x  e.  _V  /\  A. y  e.  x  ( A  .o  y )  e.  On )  ->  U_ y  e.  x  ( A  .o  y
)  e.  On )
2119, 20mpan 651 . . . . 5  |-  ( A. y  e.  x  ( A  .o  y )  e.  On  ->  U_ y  e.  x  ( A  .o  y )  e.  On )
22 omlim 6619 . . . . . . 7  |-  ( ( A  e.  On  /\  ( x  e.  _V  /\ 
Lim  x ) )  ->  ( A  .o  x )  =  U_ y  e.  x  ( A  .o  y ) )
2319, 22mpanr1 664 . . . . . 6  |-  ( ( A  e.  On  /\  Lim  x )  ->  ( A  .o  x )  = 
U_ y  e.  x  ( A  .o  y
) )
2423eleq1d 2424 . . . . 5  |-  ( ( A  e.  On  /\  Lim  x )  ->  (
( A  .o  x
)  e.  On  <->  U_ y  e.  x  ( A  .o  y )  e.  On ) )
2521, 24syl5ibr 212 . . . 4  |-  ( ( A  e.  On  /\  Lim  x )  ->  ( A. y  e.  x  ( A  .o  y
)  e.  On  ->  ( A  .o  x )  e.  On ) )
2625expcom 424 . . 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 4737 . 2  |-  ( B  e.  On  ->  ( A  e.  On  ->  ( A  .o  B )  e.  On ) )
2827impcom 419 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864   (/)c0 3531   U_ciun 3986   Oncon0 4474   Lim wlim 4475   suc csuc 4476  (class class class)co 5945    +o coa 6563    .o comu 6564
This theorem is referenced by:  oecl  6623  omordi  6651  omord2  6652  omcan  6654  omword  6655  omwordri  6657  om00  6660  om00el  6661  omlimcl  6663  odi  6664  omass  6665  oneo  6666  omeulem1  6667  omeulem2  6668  omopth2  6669  oeoelem  6683  oeoe  6684  oeeui  6687  oaabs2  6730  omxpenlem  7051  omxpen  7052  cantnfle  7462  cantnflt  7463  cantnflem1d  7480  cantnflem1  7481  cantnflem3  7483  cantnflem4  7484  cnfcomlem  7492  xpnum  7674  infxpenc  7735  dfac12lem2  7860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-recs 6475  df-rdg 6510  df-oadd 6570  df-omul 6571
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