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Theorem omlim 6779
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 4646 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  B  e.  On )
2 simpr 449 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  Lim  B )
31, 2jca 520 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( B  e.  On  /\  Lim  B ) )
4 rdglim2a 6693 . . . 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 454 . . 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 6758 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  ( rec ( ( y  e. 
_V  |->  ( y  +o  A ) ) ,  (/) ) `  B ) )
7 onelon 4608 . . . . . . . 8  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
8 omv 6758 . . . . . . . 8  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  .o  x
)  =  ( rec ( ( y  e. 
_V  |->  ( y  +o  A ) ) ,  (/) ) `  x ) )
97, 8sylan2 462 . . . . . . 7  |-  ( ( A  e.  On  /\  ( B  e.  On  /\  x  e.  B ) )  ->  ( A  .o  x )  =  ( rec ( ( y  e.  _V  |->  ( y  +o  A ) ) ,  (/) ) `  x
) )
109anassrs 631 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  x  e.  B
)  ->  ( A  .o  x )  =  ( rec ( ( y  e.  _V  |->  ( y  +o  A ) ) ,  (/) ) `  x
) )
1110iuneq2dv 4116 . . . . 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 2452 . . . 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 699 . . 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 225 . 2  |-  ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  ->  ( A  .o  B )  =  U_ x  e.  B  ( A  .o  x ) )
153, 14sylan2 462 1  |-  ( ( A  e.  On  /\  ( B  e.  C  /\  Lim  B ) )  ->  ( A  .o  B )  =  U_ x  e.  B  ( A  .o  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   U_ciun 4095    e. cmpt 4268   Oncon0 4583   Lim wlim 4584   ` cfv 5456  (class class class)co 6083   reccrdg 6669    +o coa 6723    .o comu 6724
This theorem is referenced by:  omcl  6782  om0r  6785  om1r  6788  omordi  6811  omwordri  6817  omordlim  6822  omlimcl  6823  odi  6824  omass  6825  omeulem1  6827  oeoalem  6841  oeoelem  6843  omabslem  6891  omabs  6892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-recs 6635  df-rdg 6670  df-omul 6731
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