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Theorem omlim 6736
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 4604 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  B  e.  On )
2 simpr 448 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  Lim  B )
31, 2jca 519 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( B  e.  On  /\  Lim  B ) )
4 rdglim2a 6650 . . . 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 453 . . 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 6715 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  ( rec ( ( y  e. 
_V  |->  ( y  +o  A ) ) ,  (/) ) `  B ) )
7 onelon 4566 . . . . . . . 8  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
8 omv 6715 . . . . . . . 8  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  .o  x
)  =  ( rec ( ( y  e. 
_V  |->  ( y  +o  A ) ) ,  (/) ) `  x ) )
97, 8sylan2 461 . . . . . . 7  |-  ( ( A  e.  On  /\  ( B  e.  On  /\  x  e.  B ) )  ->  ( A  .o  x )  =  ( rec ( ( y  e.  _V  |->  ( y  +o  A ) ) ,  (/) ) `  x
) )
109anassrs 630 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  x  e.  B
)  ->  ( A  .o  x )  =  ( rec ( ( y  e.  _V  |->  ( y  +o  A ) ) ,  (/) ) `  x
) )
1110iuneq2dv 4074 . . . . 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 2418 . . . 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 698 . . 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 224 . 2  |-  ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  ->  ( A  .o  B )  =  U_ x  e.  B  ( A  .o  x ) )
153, 14sylan2 461 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   (/)c0 3588   U_ciun 4053    e. cmpt 4226   Oncon0 4541   Lim wlim 4542   ` cfv 5413  (class class class)co 6040   reccrdg 6626    +o coa 6680    .o comu 6681
This theorem is referenced by:  omcl  6739  om0r  6742  om1r  6745  omordi  6768  omwordri  6774  omordlim  6779  omlimcl  6780  odi  6781  omass  6782  omeulem1  6784  oeoalem  6798  oeoelem  6800  omabslem  6848  omabs  6849
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-recs 6592  df-rdg 6627  df-omul 6688
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