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Theorem ackbij1lem5 8060
Description: Lemma for ackbij2 8079. (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 4606 . . . . 5  |-  ( a  =  A  ->  suc  a  =  suc  A )
21pweqd 3764 . . . 4  |-  ( a  =  A  ->  ~P suc  a  =  ~P suc  A )
32fveq2d 5691 . . 3  |-  ( a  =  A  ->  ( card `  ~P suc  a
)  =  ( card `  ~P suc  A ) )
4 pweq 3762 . . . . 5  |-  ( a  =  A  ->  ~P a  =  ~P A
)
54fveq2d 5691 . . . 4  |-  ( a  =  A  ->  ( card `  ~P a )  =  ( card `  ~P A ) )
65, 5oveq12d 6058 . . 3  |-  ( a  =  A  ->  (
( card `  ~P a
)  +o  ( card `  ~P a ) )  =  ( ( card `  ~P A )  +o  ( card `  ~P A ) ) )
73, 6eqeq12d 2418 . 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 2919 . . . . . . . . 9  |-  a  e. 
_V
98sucex 4750 . . . . . . . 8  |-  suc  a  e.  _V
109pw2en 7174 . . . . . . 7  |-  ~P suc  a  ~~  ( 2o  ^m  suc  a )
11 df-suc 4547 . . . . . . . . . 10  |-  suc  a  =  ( a  u. 
{ a } )
1211oveq2i 6051 . . . . . . . . 9  |-  ( 2o 
^m  suc  a )  =  ( 2o  ^m  ( a  u.  {
a } ) )
13 nnord 4812 . . . . . . . . . . 11  |-  ( a  e.  om  ->  Ord  a )
14 orddisj 4579 . . . . . . . . . . 11  |-  ( Ord  a  ->  ( a  i^i  { a } )  =  (/) )
15 snex 4365 . . . . . . . . . . . 12  |-  { a }  e.  _V
16 2onn 6842 . . . . . . . . . . . . 13  |-  2o  e.  om
1716elexi 2925 . . . . . . . . . . . 12  |-  2o  e.  _V
18 mapunen 7235 . . . . . . . . . . . . 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 424 . . . . . . . . . . . 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 1279 . . . . . . . . . . 11  |-  ( ( a  i^i  { a } )  =  (/)  ->  ( 2o  ^m  (
a  u.  { a } ) )  ~~  ( ( 2o  ^m  a )  X.  ( 2o  ^m  { a } ) ) )
2113, 14, 203syl 19 . . . . . . . . . 10  |-  ( a  e.  om  ->  ( 2o  ^m  ( a  u. 
{ a } ) )  ~~  ( ( 2o  ^m  a )  X.  ( 2o  ^m  { a } ) ) )
22 ovex 6065 . . . . . . . . . . . 12  |-  ( 2o 
^m  a )  e. 
_V
2322enref 7099 . . . . . . . . . . 11  |-  ( 2o 
^m  a )  ~~  ( 2o  ^m  a
)
2417, 8mapsnen 7143 . . . . . . . . . . 11  |-  ( 2o 
^m  { a } )  ~~  2o
25 xpen 7229 . . . . . . . . . . 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 654 . . . . . . . . . 10  |-  ( ( 2o  ^m  a )  X.  ( 2o  ^m  { a } ) ) 
~~  ( ( 2o 
^m  a )  X.  2o )
27 entr 7118 . . . . . . . . . 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 644 . . . . . . . . 9  |-  ( a  e.  om  ->  ( 2o  ^m  ( a  u. 
{ a } ) )  ~~  ( ( 2o  ^m  a )  X.  2o ) )
2912, 28syl5eqbr 4205 . . . . . . . 8  |-  ( a  e.  om  ->  ( 2o  ^m  suc  a ) 
~~  ( ( 2o 
^m  a )  X.  2o ) )
308pw2en 7174 . . . . . . . . . 10  |-  ~P a  ~~  ( 2o  ^m  a
)
3117enref 7099 . . . . . . . . . 10  |-  2o  ~~  2o
32 xpen 7229 . . . . . . . . . 10  |-  ( ( ~P a  ~~  ( 2o  ^m  a )  /\  2o  ~~  2o )  -> 
( ~P a  X.  2o )  ~~  (
( 2o  ^m  a
)  X.  2o ) )
3330, 31, 32mp2an 654 . . . . . . . . 9  |-  ( ~P a  X.  2o ) 
~~  ( ( 2o 
^m  a )  X.  2o )
3433ensymi 7116 . . . . . . . 8  |-  ( ( 2o  ^m  a )  X.  2o )  ~~  ( ~P a  X.  2o )
35 entr 7118 . . . . . . . 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 644 . . . . . . 7  |-  ( a  e.  om  ->  ( 2o  ^m  suc  a ) 
~~  ( ~P a  X.  2o ) )
37 entr 7118 . . . . . . 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 645 . . . . . 6  |-  ( a  e.  om  ->  ~P suc  a  ~~  ( ~P a  X.  2o ) )
398pwex 4342 . . . . . . 7  |-  ~P a  e.  _V
40 xp2cda 8016 . . . . . . 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 4211 . . . . 5  |-  ( a  e.  om  ->  ~P suc  a  ~~  ( ~P a  +c  ~P a
) )
43 nnfi 7258 . . . . . . . . 9  |-  ( a  e.  om  ->  a  e.  Fin )
44 pwfi 7360 . . . . . . . . 9  |-  ( a  e.  Fin  <->  ~P a  e.  Fin )
4543, 44sylib 189 . . . . . . . 8  |-  ( a  e.  om  ->  ~P a  e.  Fin )
46 ficardid 7805 . . . . . . . 8  |-  ( ~P a  e.  Fin  ->  (
card `  ~P a
)  ~~  ~P a
)
4745, 46syl 16 . . . . . . 7  |-  ( a  e.  om  ->  ( card `  ~P a ) 
~~  ~P a )
48 cdaen 8009 . . . . . . 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 643 . . . . . 6  |-  ( a  e.  om  ->  (
( card `  ~P a
)  +c  ( card `  ~P a ) ) 
~~  ( ~P a  +c  ~P a ) )
5049ensymd 7117 . . . . 5  |-  ( a  e.  om  ->  ( ~P a  +c  ~P a
)  ~~  ( ( card `  ~P a )  +c  ( card `  ~P a ) ) )
51 entr 7118 . . . . 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 ) ) )
5242, 50, 51syl2anc 643 . . . 4  |-  ( a  e.  om  ->  ~P suc  a  ~~  ( (
card `  ~P a
)  +c  ( card `  ~P a ) ) )
53 carden2b 7810 . . . 4  |-  ( ~P
suc  a  ~~  (
( card `  ~P a
)  +c  ( card `  ~P a ) )  ->  ( card `  ~P suc  a )  =  (
card `  ( ( card `  ~P a )  +c  ( card `  ~P a ) ) ) )
5452, 53syl 16 . . 3  |-  ( a  e.  om  ->  ( card `  ~P suc  a
)  =  ( card `  ( ( card `  ~P a )  +c  ( card `  ~P a ) ) ) )
55 ficardom 7804 . . . . 5  |-  ( ~P a  e.  Fin  ->  (
card `  ~P a
)  e.  om )
5645, 55syl 16 . . . 4  |-  ( a  e.  om  ->  ( card `  ~P a )  e.  om )
57 nnacda 8037 . . . 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 ) ) )
5856, 56, 57syl2anc 643 . . 3  |-  ( a  e.  om  ->  ( card `  ( ( card `  ~P a )  +c  ( card `  ~P a ) ) )  =  ( ( card `  ~P a )  +o  ( card `  ~P a ) ) )
5954, 58eqtrd 2436 . 2  |-  ( a  e.  om  ->  ( card `  ~P suc  a
)  =  ( (
card `  ~P a
)  +o  ( card `  ~P a ) ) )
607, 59vtoclga 2977 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 936    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278    i^i cin 3279   (/)c0 3588   ~Pcpw 3759   {csn 3774   class class class wbr 4172   Ord word 4540   suc csuc 4543   omcom 4804    X. cxp 4835   ` cfv 5413  (class class class)co 6040   2oc2o 6677    +o coa 6680    ^m cmap 6977    ~~ cen 7065   Fincfn 7068   cardccrd 7778    +c ccda 8003
This theorem is referenced by:  ackbij1lem14  8069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-cda 8004
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