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Theorem omsuc 6334
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 4261 . . . . . . 7  |-  suc  B  =  ( B  u.  { B } )
2 iuneq1 3794 . . . . . . 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 3860 . . . . . 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 2136 . . . . 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 5748 . . . . . . . 8  |-  ( x  =  B  ->  ( A  .o  x )  =  ( A  .o  B
) )
76oveq1d 5755 . . . . . . 7  |-  ( x  =  B  ->  (
( A  .o  x
)  +o  A )  =  ( ( A  .o  B )  +o  A ) )
87iunxsng 3856 . . . . . 6  |-  ( B  e.  On  ->  U_ x  e.  { B }  (
( A  .o  x
)  +o  A )  =  ( ( A  .o  B )  +o  A ) )
98uneq2d 3198 . . . . 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 2160 . . . 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 273 . . 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 4385 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  On )
13 omv2 6327 . . . 4  |-  ( ( A  e.  On  /\  suc  B  e.  On )  ->  ( A  .o  suc  B )  =  U_ x  e.  suc  B ( ( A  .o  x
)  +o  A ) )
1412, 13sylan2 282 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc  B )  =  U_ x  e.  suc  B ( ( A  .o  x )  +o  A ) )
15 omv2 6327 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  U_ x  e.  B  ( ( A  .o  x )  +o  A ) )
1615uneq1d 3197 . . 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 2158 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc  B )  =  ( ( A  .o  B )  u.  ( ( A  .o  B )  +o  A ) ) )
18 omcl 6323 . . 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 6333 . . . 4  |-  ( ( ( A  .o  B
)  e.  On  /\  A  e.  On )  ->  ( A  .o  B
)  C_  ( ( A  .o  B )  +o  A ) )
21 ssequn1 3214 . . . 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 406 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  u.  (
( A  .o  B
)  +o  A ) )  =  ( ( A  .o  B )  +o  A ) )
2417, 23eqtrd 2148 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 1314    e. wcel 1463    u. cun 3037    C_ wss 3039   {csn 3495   U_ciun 3781   Oncon0 4253   suc csuc 4255  (class class class)co 5740    +o coa 6276    .o comu 6277
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-oadd 6283  df-omul 6284
This theorem is referenced by:  onmsuc  6335
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