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Theorem omsuc 6116
Description: Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
omsuc  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc  B )  =  ( ( A  .o  B )  +o  A ) )

Proof of Theorem omsuc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-suc 4134 . . . . . . 7  |-  suc  B  =  ( B  u.  { B } )
2 iuneq1 3699 . . . . . . 7  |-  ( suc 
B  =  ( B  u.  { B }
)  ->  U_ x  e. 
suc  B ( ( A  .o  x )  +o  A )  = 
U_ x  e.  ( B  u.  { B } ) ( ( A  .o  x )  +o  A ) )
31, 2ax-mp 7 . . . . . 6  |-  U_ x  e.  suc  B ( ( A  .o  x )  +o  A )  = 
U_ x  e.  ( B  u.  { B } ) ( ( A  .o  x )  +o  A )
4 iunxun 3764 . . . . . 6  |-  U_ x  e.  ( B  u.  { B } ) ( ( A  .o  x )  +o  A )  =  ( U_ x  e.  B  ( ( A  .o  x )  +o  A )  u.  U_ x  e.  { B }  ( ( A  .o  x )  +o  A ) )
53, 4eqtri 2102 . . . . 5  |-  U_ x  e.  suc  B ( ( A  .o  x )  +o  A )  =  ( U_ x  e.  B  ( ( A  .o  x )  +o  A )  u.  U_ x  e.  { B }  ( ( A  .o  x )  +o  A ) )
6 oveq2 5551 . . . . . . . 8  |-  ( x  =  B  ->  ( A  .o  x )  =  ( A  .o  B
) )
76oveq1d 5558 . . . . . . 7  |-  ( x  =  B  ->  (
( A  .o  x
)  +o  A )  =  ( ( A  .o  B )  +o  A ) )
87iunxsng 3761 . . . . . 6  |-  ( B  e.  On  ->  U_ x  e.  { B }  (
( A  .o  x
)  +o  A )  =  ( ( A  .o  B )  +o  A ) )
98uneq2d 3127 . . . . 5  |-  ( B  e.  On  ->  ( U_ x  e.  B  ( ( A  .o  x )  +o  A
)  u.  U_ x  e.  { B }  (
( A  .o  x
)  +o  A ) )  =  ( U_ x  e.  B  (
( A  .o  x
)  +o  A )  u.  ( ( A  .o  B )  +o  A ) ) )
105, 9syl5eq 2126 . . . 4  |-  ( B  e.  On  ->  U_ x  e.  suc  B ( ( A  .o  x )  +o  A )  =  ( U_ x  e.  B  ( ( A  .o  x )  +o  A )  u.  (
( A  .o  B
)  +o  A ) ) )
1110adantl 271 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  U_ x  e.  suc  B ( ( A  .o  x )  +o  A
)  =  ( U_ x  e.  B  (
( A  .o  x
)  +o  A )  u.  ( ( A  .o  B )  +o  A ) ) )
12 suceloni 4253 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  On )
13 omv2 6109 . . . 4  |-  ( ( A  e.  On  /\  suc  B  e.  On )  ->  ( A  .o  suc  B )  =  U_ x  e.  suc  B ( ( A  .o  x
)  +o  A ) )
1412, 13sylan2 280 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc  B )  =  U_ x  e.  suc  B ( ( A  .o  x )  +o  A ) )
15 omv2 6109 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  U_ x  e.  B  ( ( A  .o  x )  +o  A ) )
1615uneq1d 3126 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  u.  (
( A  .o  B
)  +o  A ) )  =  ( U_ x  e.  B  (
( A  .o  x
)  +o  A )  u.  ( ( A  .o  B )  +o  A ) ) )
1711, 14, 163eqtr4d 2124 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc  B )  =  ( ( A  .o  B )  u.  ( ( A  .o  B )  +o  A ) ) )
18 omcl 6105 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
19 simpl 107 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  e.  On )
20 oaword1 6115 . . . 4  |-  ( ( ( A  .o  B
)  e.  On  /\  A  e.  On )  ->  ( A  .o  B
)  C_  ( ( A  .o  B )  +o  A ) )
21 ssequn1 3143 . . . 4  |-  ( ( A  .o  B ) 
C_  ( ( A  .o  B )  +o  A )  <->  ( ( A  .o  B )  u.  ( ( A  .o  B )  +o  A
) )  =  ( ( A  .o  B
)  +o  A ) )
2220, 21sylib 120 . . 3  |-  ( ( ( A  .o  B
)  e.  On  /\  A  e.  On )  ->  ( ( A  .o  B )  u.  (
( A  .o  B
)  +o  A ) )  =  ( ( A  .o  B )  +o  A ) )
2318, 19, 22syl2anc 403 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  u.  (
( A  .o  B
)  +o  A ) )  =  ( ( A  .o  B )  +o  A ) )
2417, 23eqtrd 2114 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc  B )  =  ( ( A  .o  B )  +o  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434    u. cun 2972    C_ wss 2974   {csn 3406   U_ciun 3686   Oncon0 4126   suc csuc 4128  (class class class)co 5543    +o coa 6062    .o comu 6063
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-oadd 6069  df-omul 6070
This theorem is referenced by:  onmsuc  6117
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