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Theorem omlim 7181
Description: Ordinal multiplication with a limit ordinal. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 3-Aug-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
omlim  |-  ( ( A  e.  On  /\  ( B  e.  C  /\  Lim  B ) )  ->  ( A  .o  B )  =  U_ x  e.  B  ( A  .o  x ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem omlim
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 limelon 4927 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  B  e.  On )
2 simpr 461 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  Lim  B )
31, 2jca 532 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( B  e.  On  /\  Lim  B ) )
4 rdglim2a 7097 . . . 4  |-  ( ( B  e.  On  /\  Lim  B )  ->  ( rec ( ( y  e. 
_V  |->  ( y  +o  A ) ) ,  (/) ) `  B )  =  U_ x  e.  B  ( rec (
( y  e.  _V  |->  ( y  +o  A
) ) ,  (/) ) `  x )
)
54adantl 466 . . 3  |-  ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  ->  ( rec (
( y  e.  _V  |->  ( y  +o  A
) ) ,  (/) ) `  B )  =  U_ x  e.  B  ( rec ( ( y  e.  _V  |->  ( y  +o  A ) ) ,  (/) ) `  x
) )
6 omv 7160 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  ( rec ( ( y  e. 
_V  |->  ( y  +o  A ) ) ,  (/) ) `  B ) )
7 onelon 4889 . . . . . . . 8  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
8 omv 7160 . . . . . . . 8  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  .o  x
)  =  ( rec ( ( y  e. 
_V  |->  ( y  +o  A ) ) ,  (/) ) `  x ) )
97, 8sylan2 474 . . . . . . 7  |-  ( ( A  e.  On  /\  ( B  e.  On  /\  x  e.  B ) )  ->  ( A  .o  x )  =  ( rec ( ( y  e.  _V  |->  ( y  +o  A ) ) ,  (/) ) `  x
) )
109anassrs 648 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  x  e.  B
)  ->  ( A  .o  x )  =  ( rec ( ( y  e.  _V  |->  ( y  +o  A ) ) ,  (/) ) `  x
) )
1110iuneq2dv 4333 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  U_ x  e.  B  ( A  .o  x
)  =  U_ x  e.  B  ( rec ( ( y  e. 
_V  |->  ( y  +o  A ) ) ,  (/) ) `  x ) )
126, 11eqeq12d 2463 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  =  U_ x  e.  B  ( A  .o  x )  <->  ( rec ( ( y  e. 
_V  |->  ( y  +o  A ) ) ,  (/) ) `  B )  =  U_ x  e.  B  ( rec (
( y  e.  _V  |->  ( y  +o  A
) ) ,  (/) ) `  x )
) )
1312adantrr 716 . . 3  |-  ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  ->  ( ( A  .o  B )  = 
U_ x  e.  B  ( A  .o  x
)  <->  ( rec (
( y  e.  _V  |->  ( y  +o  A
) ) ,  (/) ) `  B )  =  U_ x  e.  B  ( rec ( ( y  e.  _V  |->  ( y  +o  A ) ) ,  (/) ) `  x
) ) )
145, 13mpbird 232 . 2  |-  ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  ->  ( A  .o  B )  =  U_ x  e.  B  ( A  .o  x ) )
153, 14sylan2 474 1  |-  ( ( A  e.  On  /\  ( B  e.  C  /\  Lim  B ) )  ->  ( A  .o  B )  =  U_ x  e.  B  ( A  .o  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   _Vcvv 3093   (/)c0 3767   U_ciun 4311    |-> cmpt 4491   Oncon0 4864   Lim wlim 4865   ` cfv 5574  (class class class)co 6277   reccrdg 7073    +o coa 7125    .o comu 7126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-recs 7040  df-rdg 7074  df-omul 7133
This theorem is referenced by:  omcl  7184  om0r  7187  om1r  7190  omordi  7213  omwordri  7219  omordlim  7224  omlimcl  7225  odi  7226  omass  7227  omeulem1  7229  oeoalem  7243  oeoelem  7245  omabslem  7293  omabs  7294
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