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Theorem ackbij1lem5 8030
Description: Lemma for ackbij2 8049. (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 4580 . . . . 5  |-  ( a  =  A  ->  suc  a  =  suc  A )
21pweqd 3740 . . . 4  |-  ( a  =  A  ->  ~P suc  a  =  ~P suc  A )
32fveq2d 5665 . . 3  |-  ( a  =  A  ->  ( card `  ~P suc  a
)  =  ( card `  ~P suc  A ) )
4 pweq 3738 . . . . 5  |-  ( a  =  A  ->  ~P a  =  ~P A
)
54fveq2d 5665 . . . 4  |-  ( a  =  A  ->  ( card `  ~P a )  =  ( card `  ~P A ) )
65, 5oveq12d 6031 . . 3  |-  ( a  =  A  ->  (
( card `  ~P a
)  +o  ( card `  ~P a ) )  =  ( ( card `  ~P A )  +o  ( card `  ~P A ) ) )
73, 6eqeq12d 2394 . 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 2895 . . . . . . . . 9  |-  a  e. 
_V
98sucex 4724 . . . . . . . 8  |-  suc  a  e.  _V
109pw2en 7144 . . . . . . 7  |-  ~P suc  a  ~~  ( 2o  ^m  suc  a )
11 df-suc 4521 . . . . . . . . . 10  |-  suc  a  =  ( a  u. 
{ a } )
1211oveq2i 6024 . . . . . . . . 9  |-  ( 2o 
^m  suc  a )  =  ( 2o  ^m  ( a  u.  {
a } ) )
13 nnord 4786 . . . . . . . . . . 11  |-  ( a  e.  om  ->  Ord  a )
14 orddisj 4553 . . . . . . . . . . 11  |-  ( Ord  a  ->  ( a  i^i  { a } )  =  (/) )
15 snex 4339 . . . . . . . . . . . 12  |-  { a }  e.  _V
16 2onn 6812 . . . . . . . . . . . . 13  |-  2o  e.  om
1716elexi 2901 . . . . . . . . . . . 12  |-  2o  e.  _V
18 mapunen 7205 . . . . . . . . . . . . 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 6038 . . . . . . . . . . . 12  |-  ( 2o 
^m  a )  e. 
_V
2322enref 7069 . . . . . . . . . . 11  |-  ( 2o 
^m  a )  ~~  ( 2o  ^m  a
)
2417, 8mapsnen 7113 . . . . . . . . . . 11  |-  ( 2o 
^m  { a } )  ~~  2o
25 xpen 7199 . . . . . . . . . . 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 7088 . . . . . . . . . 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 4179 . . . . . . . 8  |-  ( a  e.  om  ->  ( 2o  ^m  suc  a ) 
~~  ( ( 2o 
^m  a )  X.  2o ) )
308pw2en 7144 . . . . . . . . . 10  |-  ~P a  ~~  ( 2o  ^m  a
)
3117enref 7069 . . . . . . . . . 10  |-  2o  ~~  2o
32 xpen 7199 . . . . . . . . . 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 7086 . . . . . . . 8  |-  ( ( 2o  ^m  a )  X.  2o )  ~~  ( ~P a  X.  2o )
35 entr 7088 . . . . . . . 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 7088 . . . . . . 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 4316 . . . . . . 7  |-  ~P a  e.  _V
40 xp2cda 7986 . . . . . . 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 4185 . . . . 5  |-  ( a  e.  om  ->  ~P suc  a  ~~  ( ~P a  +c  ~P a
) )
43 nnfi 7228 . . . . . . . . 9  |-  ( a  e.  om  ->  a  e.  Fin )
44 pwfi 7330 . . . . . . . . 9  |-  ( a  e.  Fin  <->  ~P a  e.  Fin )
4543, 44sylib 189 . . . . . . . 8  |-  ( a  e.  om  ->  ~P a  e.  Fin )
46 ficardid 7775 . . . . . . . 8  |-  ( ~P a  e.  Fin  ->  (
card `  ~P a
)  ~~  ~P a
)
4745, 46syl 16 . . . . . . 7  |-  ( a  e.  om  ->  ( card `  ~P a ) 
~~  ~P a )
48 cdaen 7979 . . . . . . 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 7087 . . . . 5  |-  ( a  e.  om  ->  ( ~P a  +c  ~P a
)  ~~  ( ( card `  ~P a )  +c  ( card `  ~P a ) ) )
51 entr 7088 . . . . 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 7780 . . . 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 7774 . . . . 5  |-  ( ~P a  e.  Fin  ->  (
card `  ~P a
)  e.  om )
5645, 55syl 16 . . . 4  |-  ( a  e.  om  ->  ( card `  ~P a )  e.  om )
57 nnacda 8007 . . . 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 2412 . 2  |-  ( a  e.  om  ->  ( card `  ~P suc  a
)  =  ( (
card `  ~P a
)  +o  ( card `  ~P a ) ) )
607, 59vtoclga 2953 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 1717   _Vcvv 2892    u. cun 3254    i^i cin 3255   (/)c0 3564   ~Pcpw 3735   {csn 3750   class class class wbr 4146   Ord word 4514   suc csuc 4517   omcom 4778    X. cxp 4809   ` cfv 5387  (class class class)co 6013   2oc2o 6647    +o coa 6650    ^m cmap 6947    ~~ cen 7035   Fincfn 7038   cardccrd 7748    +c ccda 7973
This theorem is referenced by:  ackbij1lem14  8039
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-card 7752  df-cda 7974
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