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Theorem omsuc 6336
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 4263 . . . . . . 7  |-  suc  B  =  ( B  u.  { B } )
2 iuneq1 3796 . . . . . . 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 5 . . . . . 6  |-  U_ x  e.  suc  B ( ( A  .o  x )  +o  A )  = 
U_ x  e.  ( B  u.  { B } ) ( ( A  .o  x )  +o  A )
4 iunxun 3862 . . . . . 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 2138 . . . . 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 5750 . . . . . . . 8  |-  ( x  =  B  ->  ( A  .o  x )  =  ( A  .o  B
) )
76oveq1d 5757 . . . . . . 7  |-  ( x  =  B  ->  (
( A  .o  x
)  +o  A )  =  ( ( A  .o  B )  +o  A ) )
87iunxsng 3858 . . . . . 6  |-  ( B  e.  On  ->  U_ x  e.  { B }  (
( A  .o  x
)  +o  A )  =  ( ( A  .o  B )  +o  A ) )
98uneq2d 3200 . . . . 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 2162 . . . 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 275 . . 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 4387 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  On )
13 omv2 6329 . . . 4  |-  ( ( A  e.  On  /\  suc  B  e.  On )  ->  ( A  .o  suc  B )  =  U_ x  e.  suc  B ( ( A  .o  x
)  +o  A ) )
1412, 13sylan2 284 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc  B )  =  U_ x  e.  suc  B ( ( A  .o  x )  +o  A ) )
15 omv2 6329 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  U_ x  e.  B  ( ( A  .o  x )  +o  A ) )
1615uneq1d 3199 . . 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 2160 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc  B )  =  ( ( A  .o  B )  u.  ( ( A  .o  B )  +o  A ) ) )
18 omcl 6325 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
19 simpl 108 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  e.  On )
20 oaword1 6335 . . . 4  |-  ( ( ( A  .o  B
)  e.  On  /\  A  e.  On )  ->  ( A  .o  B
)  C_  ( ( A  .o  B )  +o  A ) )
21 ssequn1 3216 . . . 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 121 . . 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 408 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  u.  (
( A  .o  B
)  +o  A ) )  =  ( ( A  .o  B )  +o  A ) )
2417, 23eqtrd 2150 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 103    = wceq 1316    e. wcel 1465    u. cun 3039    C_ wss 3041   {csn 3497   U_ciun 3783   Oncon0 4255   suc csuc 4257  (class class class)co 5742    +o coa 6278    .o comu 6279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-oadd 6285  df-omul 6286
This theorem is referenced by:  onmsuc  6337
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