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Theorem omsuc 6215
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 4189 . . . . . . 7  |-  suc  B  =  ( B  u.  { B } )
2 iuneq1 3738 . . . . . . 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 3804 . . . . . 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 2108 . . . . 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 5642 . . . . . . . 8  |-  ( x  =  B  ->  ( A  .o  x )  =  ( A  .o  B
) )
76oveq1d 5649 . . . . . . 7  |-  ( x  =  B  ->  (
( A  .o  x
)  +o  A )  =  ( ( A  .o  B )  +o  A ) )
87iunxsng 3800 . . . . . 6  |-  ( B  e.  On  ->  U_ x  e.  { B }  (
( A  .o  x
)  +o  A )  =  ( ( A  .o  B )  +o  A ) )
98uneq2d 3152 . . . . 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 2132 . . . 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 4308 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  On )
13 omv2 6208 . . . 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 6208 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  U_ x  e.  B  ( ( A  .o  x )  +o  A ) )
1615uneq1d 3151 . . 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 2130 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc  B )  =  ( ( A  .o  B )  u.  ( ( A  .o  B )  +o  A ) ) )
18 omcl 6204 . . 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 6214 . . . 4  |-  ( ( ( A  .o  B
)  e.  On  /\  A  e.  On )  ->  ( A  .o  B
)  C_  ( ( A  .o  B )  +o  A ) )
21 ssequn1 3168 . . . 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 2120 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 1289    e. wcel 1438    u. cun 2995    C_ wss 2997   {csn 3441   U_ciun 3725   Oncon0 4181   suc csuc 4183  (class class class)co 5634    +o coa 6160    .o comu 6161
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-oadd 6167  df-omul 6168
This theorem is referenced by:  onmsuc  6216
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