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Theorem ackbij1lem5 7850
Description: Lemma for ackbij2 7869. (Contributed by Stefan O'Rear, 19-Nov-2014.)
Assertion
Ref Expression
ackbij1lem5  |-  ( A  e.  om  ->  ( card `  ~P suc  A
)  =  ( (
card `  ~P A )  +o  ( card `  ~P A ) ) )

Proof of Theorem ackbij1lem5
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 suceq 4457 . . . . 5  |-  ( a  =  A  ->  suc  a  =  suc  A )
21pweqd 3630 . . . 4  |-  ( a  =  A  ->  ~P suc  a  =  ~P suc  A )
32fveq2d 5529 . . 3  |-  ( a  =  A  ->  ( card `  ~P suc  a
)  =  ( card `  ~P suc  A ) )
4 pweq 3628 . . . . 5  |-  ( a  =  A  ->  ~P a  =  ~P A
)
54fveq2d 5529 . . . 4  |-  ( a  =  A  ->  ( card `  ~P a )  =  ( card `  ~P A ) )
65, 5oveq12d 5876 . . 3  |-  ( a  =  A  ->  (
( card `  ~P a
)  +o  ( card `  ~P a ) )  =  ( ( card `  ~P A )  +o  ( card `  ~P A ) ) )
73, 6eqeq12d 2297 . 2  |-  ( a  =  A  ->  (
( card `  ~P suc  a
)  =  ( (
card `  ~P a
)  +o  ( card `  ~P a ) )  <-> 
( card `  ~P suc  A
)  =  ( (
card `  ~P A )  +o  ( card `  ~P A ) ) ) )
8 vex 2791 . . . . . . . . 9  |-  a  e. 
_V
98sucex 4602 . . . . . . . 8  |-  suc  a  e.  _V
109pw2en 6969 . . . . . . 7  |-  ~P suc  a  ~~  ( 2o  ^m  suc  a )
11 df-suc 4398 . . . . . . . . . 10  |-  suc  a  =  ( a  u. 
{ a } )
1211oveq2i 5869 . . . . . . . . 9  |-  ( 2o 
^m  suc  a )  =  ( 2o  ^m  ( a  u.  {
a } ) )
13 nnord 4664 . . . . . . . . . . 11  |-  ( a  e.  om  ->  Ord  a )
14 orddisj 4430 . . . . . . . . . . 11  |-  ( Ord  a  ->  ( a  i^i  { a } )  =  (/) )
15 snex 4216 . . . . . . . . . . . 12  |-  { a }  e.  _V
16 2onn 6638 . . . . . . . . . . . . 13  |-  2o  e.  om
1716elexi 2797 . . . . . . . . . . . 12  |-  2o  e.  _V
18 mapunen 7030 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  _V  /\ 
{ a }  e.  _V  /\  2o  e.  _V )  /\  ( a  i^i 
{ a } )  =  (/) )  ->  ( 2o  ^m  ( a  u. 
{ a } ) )  ~~  ( ( 2o  ^m  a )  X.  ( 2o  ^m  { a } ) ) )
1918ex 423 . . . . . . . . . . . 12  |-  ( ( a  e.  _V  /\  { a }  e.  _V  /\  2o  e.  _V )  ->  ( ( a  i^i 
{ a } )  =  (/)  ->  ( 2o 
^m  ( a  u. 
{ a } ) )  ~~  ( ( 2o  ^m  a )  X.  ( 2o  ^m  { a } ) ) ) )
208, 15, 17, 19mp3an 1277 . . . . . . . . . . 11  |-  ( ( a  i^i  { a } )  =  (/)  ->  ( 2o  ^m  (
a  u.  { a } ) )  ~~  ( ( 2o  ^m  a )  X.  ( 2o  ^m  { a } ) ) )
2113, 14, 203syl 18 . . . . . . . . . 10  |-  ( a  e.  om  ->  ( 2o  ^m  ( a  u. 
{ a } ) )  ~~  ( ( 2o  ^m  a )  X.  ( 2o  ^m  { a } ) ) )
22 ovex 5883 . . . . . . . . . . . 12  |-  ( 2o 
^m  a )  e. 
_V
2322enref 6894 . . . . . . . . . . 11  |-  ( 2o 
^m  a )  ~~  ( 2o  ^m  a
)
2417, 8mapsnen 6938 . . . . . . . . . . 11  |-  ( 2o 
^m  { a } )  ~~  2o
25 xpen 7024 . . . . . . . . . . 11  |-  ( ( ( 2o  ^m  a
)  ~~  ( 2o  ^m  a )  /\  ( 2o  ^m  { a } )  ~~  2o )  ->  ( ( 2o 
^m  a )  X.  ( 2o  ^m  {
a } ) ) 
~~  ( ( 2o 
^m  a )  X.  2o ) )
2623, 24, 25mp2an 653 . . . . . . . . . 10  |-  ( ( 2o  ^m  a )  X.  ( 2o  ^m  { a } ) ) 
~~  ( ( 2o 
^m  a )  X.  2o )
27 entr 6913 . . . . . . . . . 10  |-  ( ( ( 2o  ^m  (
a  u.  { a } ) )  ~~  ( ( 2o  ^m  a )  X.  ( 2o  ^m  { a } ) )  /\  (
( 2o  ^m  a
)  X.  ( 2o 
^m  { a } ) )  ~~  (
( 2o  ^m  a
)  X.  2o ) )  ->  ( 2o  ^m  ( a  u.  {
a } ) ) 
~~  ( ( 2o 
^m  a )  X.  2o ) )
2821, 26, 27sylancl 643 . . . . . . . . 9  |-  ( a  e.  om  ->  ( 2o  ^m  ( a  u. 
{ a } ) )  ~~  ( ( 2o  ^m  a )  X.  2o ) )
2912, 28syl5eqbr 4056 . . . . . . . 8  |-  ( a  e.  om  ->  ( 2o  ^m  suc  a ) 
~~  ( ( 2o 
^m  a )  X.  2o ) )
308pw2en 6969 . . . . . . . . . 10  |-  ~P a  ~~  ( 2o  ^m  a
)
3117enref 6894 . . . . . . . . . 10  |-  2o  ~~  2o
32 xpen 7024 . . . . . . . . . 10  |-  ( ( ~P a  ~~  ( 2o  ^m  a )  /\  2o  ~~  2o )  -> 
( ~P a  X.  2o )  ~~  (
( 2o  ^m  a
)  X.  2o ) )
3330, 31, 32mp2an 653 . . . . . . . . 9  |-  ( ~P a  X.  2o ) 
~~  ( ( 2o 
^m  a )  X.  2o )
3433ensymi 6911 . . . . . . . 8  |-  ( ( 2o  ^m  a )  X.  2o )  ~~  ( ~P a  X.  2o )
35 entr 6913 . . . . . . . 8  |-  ( ( ( 2o  ^m  suc  a )  ~~  (
( 2o  ^m  a
)  X.  2o )  /\  ( ( 2o 
^m  a )  X.  2o )  ~~  ( ~P a  X.  2o ) )  ->  ( 2o  ^m  suc  a ) 
~~  ( ~P a  X.  2o ) )
3629, 34, 35sylancl 643 . . . . . . 7  |-  ( a  e.  om  ->  ( 2o  ^m  suc  a ) 
~~  ( ~P a  X.  2o ) )
37 entr 6913 . . . . . . 7  |-  ( ( ~P suc  a  ~~  ( 2o  ^m  suc  a
)  /\  ( 2o  ^m 
suc  a )  ~~  ( ~P a  X.  2o ) )  ->  ~P suc  a  ~~  ( ~P a  X.  2o ) )
3810, 36, 37sylancr 644 . . . . . 6  |-  ( a  e.  om  ->  ~P suc  a  ~~  ( ~P a  X.  2o ) )
398pwex 4193 . . . . . . 7  |-  ~P a  e.  _V
40 xp2cda 7806 . . . . . . 7  |-  ( ~P a  e.  _V  ->  ( ~P a  X.  2o )  =  ( ~P a  +c  ~P a ) )
4139, 40ax-mp 8 . . . . . 6  |-  ( ~P a  X.  2o )  =  ( ~P a  +c  ~P a )
4238, 41syl6breq 4062 . . . . 5  |-  ( a  e.  om  ->  ~P suc  a  ~~  ( ~P a  +c  ~P a
) )
43 nnfi 7053 . . . . . . . . 9  |-  ( a  e.  om  ->  a  e.  Fin )
44 pwfi 7151 . . . . . . . . 9  |-  ( a  e.  Fin  <->  ~P a  e.  Fin )
4543, 44sylib 188 . . . . . . . 8  |-  ( a  e.  om  ->  ~P a  e.  Fin )
46 ficardid 7595 . . . . . . . 8  |-  ( ~P a  e.  Fin  ->  (
card `  ~P a
)  ~~  ~P a
)
4745, 46syl 15 . . . . . . 7  |-  ( a  e.  om  ->  ( card `  ~P a ) 
~~  ~P a )
48 cdaen 7799 . . . . . . 7  |-  ( ( ( card `  ~P a )  ~~  ~P a  /\  ( card `  ~P a )  ~~  ~P a )  ->  (
( card `  ~P a
)  +c  ( card `  ~P a ) ) 
~~  ( ~P a  +c  ~P a ) )
4947, 47, 48syl2anc 642 . . . . . 6  |-  ( a  e.  om  ->  (
( card `  ~P a
)  +c  ( card `  ~P a ) ) 
~~  ( ~P a  +c  ~P a ) )
50 ensym 6910 . . . . . 6  |-  ( ( ( card `  ~P a )  +c  ( card `  ~P a ) )  ~~  ( ~P a  +c  ~P a
)  ->  ( ~P a  +c  ~P a ) 
~~  ( ( card `  ~P a )  +c  ( card `  ~P a ) ) )
5149, 50syl 15 . . . . 5  |-  ( a  e.  om  ->  ( ~P a  +c  ~P a
)  ~~  ( ( card `  ~P a )  +c  ( card `  ~P a ) ) )
52 entr 6913 . . . . 5  |-  ( ( ~P suc  a  ~~  ( ~P a  +c  ~P a )  /\  ( ~P a  +c  ~P a
)  ~~  ( ( card `  ~P a )  +c  ( card `  ~P a ) ) )  ->  ~P suc  a  ~~  ( ( card `  ~P a )  +c  ( card `  ~P a ) ) )
5342, 51, 52syl2anc 642 . . . 4  |-  ( a  e.  om  ->  ~P suc  a  ~~  ( (
card `  ~P a
)  +c  ( card `  ~P a ) ) )
54 carden2b 7600 . . . 4  |-  ( ~P
suc  a  ~~  (
( card `  ~P a
)  +c  ( card `  ~P a ) )  ->  ( card `  ~P suc  a )  =  (
card `  ( ( card `  ~P a )  +c  ( card `  ~P a ) ) ) )
5553, 54syl 15 . . 3  |-  ( a  e.  om  ->  ( card `  ~P suc  a
)  =  ( card `  ( ( card `  ~P a )  +c  ( card `  ~P a ) ) ) )
56 ficardom 7594 . . . . 5  |-  ( ~P a  e.  Fin  ->  (
card `  ~P a
)  e.  om )
5745, 56syl 15 . . . 4  |-  ( a  e.  om  ->  ( card `  ~P a )  e.  om )
58 nnacda 7827 . . . 4  |-  ( ( ( card `  ~P a )  e.  om  /\  ( card `  ~P a )  e.  om )  ->  ( card `  (
( card `  ~P a
)  +c  ( card `  ~P a ) ) )  =  ( (
card `  ~P a
)  +o  ( card `  ~P a ) ) )
5957, 57, 58syl2anc 642 . . 3  |-  ( a  e.  om  ->  ( card `  ( ( card `  ~P a )  +c  ( card `  ~P a ) ) )  =  ( ( card `  ~P a )  +o  ( card `  ~P a ) ) )
6055, 59eqtrd 2315 . 2  |-  ( a  e.  om  ->  ( card `  ~P suc  a
)  =  ( (
card `  ~P a
)  +o  ( card `  ~P a ) ) )
617, 60vtoclga 2849 1  |-  ( A  e.  om  ->  ( card `  ~P suc  A
)  =  ( (
card `  ~P A )  +o  ( card `  ~P A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150    i^i cin 3151   (/)c0 3455   ~Pcpw 3625   {csn 3640   class class class wbr 4023   Ord word 4391   suc csuc 4394   omcom 4656    X. cxp 4687   ` cfv 5255  (class class class)co 5858   2oc2o 6473    +o coa 6476    ^m cmap 6772    ~~ cen 6860   Fincfn 6863   cardccrd 7568    +c ccda 7793
This theorem is referenced by:  ackbij1lem14  7859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-cda 7794
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