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Theorem ackbij1lem5 7940
Description: Lemma for ackbij2 7959. (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 4539 . . . . 5  |-  ( a  =  A  ->  suc  a  =  suc  A )
21pweqd 3706 . . . 4  |-  ( a  =  A  ->  ~P suc  a  =  ~P suc  A )
32fveq2d 5612 . . 3  |-  ( a  =  A  ->  ( card `  ~P suc  a
)  =  ( card `  ~P suc  A ) )
4 pweq 3704 . . . . 5  |-  ( a  =  A  ->  ~P a  =  ~P A
)
54fveq2d 5612 . . . 4  |-  ( a  =  A  ->  ( card `  ~P a )  =  ( card `  ~P A ) )
65, 5oveq12d 5963 . . 3  |-  ( a  =  A  ->  (
( card `  ~P a
)  +o  ( card `  ~P a ) )  =  ( ( card `  ~P A )  +o  ( card `  ~P A ) ) )
73, 6eqeq12d 2372 . 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 2867 . . . . . . . . 9  |-  a  e. 
_V
98sucex 4684 . . . . . . . 8  |-  suc  a  e.  _V
109pw2en 7057 . . . . . . 7  |-  ~P suc  a  ~~  ( 2o  ^m  suc  a )
11 df-suc 4480 . . . . . . . . . 10  |-  suc  a  =  ( a  u. 
{ a } )
1211oveq2i 5956 . . . . . . . . 9  |-  ( 2o 
^m  suc  a )  =  ( 2o  ^m  ( a  u.  {
a } ) )
13 nnord 4746 . . . . . . . . . . 11  |-  ( a  e.  om  ->  Ord  a )
14 orddisj 4512 . . . . . . . . . . 11  |-  ( Ord  a  ->  ( a  i^i  { a } )  =  (/) )
15 snex 4297 . . . . . . . . . . . 12  |-  { a }  e.  _V
16 2onn 6725 . . . . . . . . . . . . 13  |-  2o  e.  om
1716elexi 2873 . . . . . . . . . . . 12  |-  2o  e.  _V
18 mapunen 7118 . . . . . . . . . . . . 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 5970 . . . . . . . . . . . 12  |-  ( 2o 
^m  a )  e. 
_V
2322enref 6982 . . . . . . . . . . 11  |-  ( 2o 
^m  a )  ~~  ( 2o  ^m  a
)
2417, 8mapsnen 7026 . . . . . . . . . . 11  |-  ( 2o 
^m  { a } )  ~~  2o
25 xpen 7112 . . . . . . . . . . 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 7001 . . . . . . . . . 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 4137 . . . . . . . 8  |-  ( a  e.  om  ->  ( 2o  ^m  suc  a ) 
~~  ( ( 2o 
^m  a )  X.  2o ) )
308pw2en 7057 . . . . . . . . . 10  |-  ~P a  ~~  ( 2o  ^m  a
)
3117enref 6982 . . . . . . . . . 10  |-  2o  ~~  2o
32 xpen 7112 . . . . . . . . . 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 6999 . . . . . . . 8  |-  ( ( 2o  ^m  a )  X.  2o )  ~~  ( ~P a  X.  2o )
35 entr 7001 . . . . . . . 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 7001 . . . . . . 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 4274 . . . . . . 7  |-  ~P a  e.  _V
40 xp2cda 7896 . . . . . . 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 4143 . . . . 5  |-  ( a  e.  om  ->  ~P suc  a  ~~  ( ~P a  +c  ~P a
) )
43 nnfi 7141 . . . . . . . . 9  |-  ( a  e.  om  ->  a  e.  Fin )
44 pwfi 7241 . . . . . . . . 9  |-  ( a  e.  Fin  <->  ~P a  e.  Fin )
4543, 44sylib 188 . . . . . . . 8  |-  ( a  e.  om  ->  ~P a  e.  Fin )
46 ficardid 7685 . . . . . . . 8  |-  ( ~P a  e.  Fin  ->  (
card `  ~P a
)  ~~  ~P a
)
4745, 46syl 15 . . . . . . 7  |-  ( a  e.  om  ->  ( card `  ~P a ) 
~~  ~P a )
48 cdaen 7889 . . . . . . 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 6998 . . . . . 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 7001 . . . . 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 7690 . . . 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 7684 . . . . 5  |-  ( ~P a  e.  Fin  ->  (
card `  ~P a
)  e.  om )
5745, 56syl 15 . . . 4  |-  ( a  e.  om  ->  ( card `  ~P a )  e.  om )
58 nnacda 7917 . . . 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 2390 . 2  |-  ( a  e.  om  ->  ( card `  ~P suc  a
)  =  ( (
card `  ~P a
)  +o  ( card `  ~P a ) ) )
617, 60vtoclga 2925 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 1642    e. wcel 1710   _Vcvv 2864    u. cun 3226    i^i cin 3227   (/)c0 3531   ~Pcpw 3701   {csn 3716   class class class wbr 4104   Ord word 4473   suc csuc 4476   omcom 4738    X. cxp 4769   ` cfv 5337  (class class class)co 5945   2oc2o 6560    +o coa 6563    ^m cmap 6860    ~~ cen 6948   Fincfn 6951   cardccrd 7658    +c ccda 7883
This theorem is referenced by:  ackbij1lem14  7949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-card 7662  df-cda 7884
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