MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ackbij1lem5 Structured version   Unicode version

Theorem ackbij1lem5 8109
Description: Lemma for ackbij2 8128. (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 4649 . . . . 5  |-  ( a  =  A  ->  suc  a  =  suc  A )
21pweqd 3806 . . . 4  |-  ( a  =  A  ->  ~P suc  a  =  ~P suc  A )
32fveq2d 5735 . . 3  |-  ( a  =  A  ->  ( card `  ~P suc  a
)  =  ( card `  ~P suc  A ) )
4 pweq 3804 . . . . 5  |-  ( a  =  A  ->  ~P a  =  ~P A
)
54fveq2d 5735 . . . 4  |-  ( a  =  A  ->  ( card `  ~P a )  =  ( card `  ~P A ) )
65, 5oveq12d 6102 . . 3  |-  ( a  =  A  ->  (
( card `  ~P a
)  +o  ( card `  ~P a ) )  =  ( ( card `  ~P A )  +o  ( card `  ~P A ) ) )
73, 6eqeq12d 2452 . 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 2961 . . . . . . . . 9  |-  a  e. 
_V
98sucex 4794 . . . . . . . 8  |-  suc  a  e.  _V
109pw2en 7218 . . . . . . 7  |-  ~P suc  a  ~~  ( 2o  ^m  suc  a )
11 df-suc 4590 . . . . . . . . . 10  |-  suc  a  =  ( a  u. 
{ a } )
1211oveq2i 6095 . . . . . . . . 9  |-  ( 2o 
^m  suc  a )  =  ( 2o  ^m  ( a  u.  {
a } ) )
13 nnord 4856 . . . . . . . . . . 11  |-  ( a  e.  om  ->  Ord  a )
14 orddisj 4622 . . . . . . . . . . 11  |-  ( Ord  a  ->  ( a  i^i  { a } )  =  (/) )
15 snex 4408 . . . . . . . . . . . 12  |-  { a }  e.  _V
16 2onn 6886 . . . . . . . . . . . . 13  |-  2o  e.  om
1716elexi 2967 . . . . . . . . . . . 12  |-  2o  e.  _V
18 mapunen 7279 . . . . . . . . . . . . 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 425 . . . . . . . . . . . 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 1280 . . . . . . . . . . 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 6109 . . . . . . . . . . . 12  |-  ( 2o 
^m  a )  e. 
_V
2322enref 7143 . . . . . . . . . . 11  |-  ( 2o 
^m  a )  ~~  ( 2o  ^m  a
)
2417, 8mapsnen 7187 . . . . . . . . . . 11  |-  ( 2o 
^m  { a } )  ~~  2o
25 xpen 7273 . . . . . . . . . . 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 655 . . . . . . . . . 10  |-  ( ( 2o  ^m  a )  X.  ( 2o  ^m  { a } ) ) 
~~  ( ( 2o 
^m  a )  X.  2o )
27 entr 7162 . . . . . . . . . 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 645 . . . . . . . . 9  |-  ( a  e.  om  ->  ( 2o  ^m  ( a  u. 
{ a } ) )  ~~  ( ( 2o  ^m  a )  X.  2o ) )
2912, 28syl5eqbr 4248 . . . . . . . 8  |-  ( a  e.  om  ->  ( 2o  ^m  suc  a ) 
~~  ( ( 2o 
^m  a )  X.  2o ) )
308pw2en 7218 . . . . . . . . . 10  |-  ~P a  ~~  ( 2o  ^m  a
)
3117enref 7143 . . . . . . . . . 10  |-  2o  ~~  2o
32 xpen 7273 . . . . . . . . . 10  |-  ( ( ~P a  ~~  ( 2o  ^m  a )  /\  2o  ~~  2o )  -> 
( ~P a  X.  2o )  ~~  (
( 2o  ^m  a
)  X.  2o ) )
3330, 31, 32mp2an 655 . . . . . . . . 9  |-  ( ~P a  X.  2o ) 
~~  ( ( 2o 
^m  a )  X.  2o )
3433ensymi 7160 . . . . . . . 8  |-  ( ( 2o  ^m  a )  X.  2o )  ~~  ( ~P a  X.  2o )
35 entr 7162 . . . . . . . 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 645 . . . . . . 7  |-  ( a  e.  om  ->  ( 2o  ^m  suc  a ) 
~~  ( ~P a  X.  2o ) )
37 entr 7162 . . . . . . 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 646 . . . . . 6  |-  ( a  e.  om  ->  ~P suc  a  ~~  ( ~P a  X.  2o ) )
398pwex 4385 . . . . . . 7  |-  ~P a  e.  _V
40 xp2cda 8065 . . . . . . 7  |-  ( ~P a  e.  _V  ->  ( ~P a  X.  2o )  =  ( ~P a  +c  ~P a ) )
4139, 40ax-mp 5 . . . . . 6  |-  ( ~P a  X.  2o )  =  ( ~P a  +c  ~P a )
4238, 41syl6breq 4254 . . . . 5  |-  ( a  e.  om  ->  ~P suc  a  ~~  ( ~P a  +c  ~P a
) )
43 nnfi 7302 . . . . . . . . 9  |-  ( a  e.  om  ->  a  e.  Fin )
44 pwfi 7405 . . . . . . . . 9  |-  ( a  e.  Fin  <->  ~P a  e.  Fin )
4543, 44sylib 190 . . . . . . . 8  |-  ( a  e.  om  ->  ~P a  e.  Fin )
46 ficardid 7854 . . . . . . . 8  |-  ( ~P a  e.  Fin  ->  (
card `  ~P a
)  ~~  ~P a
)
4745, 46syl 16 . . . . . . 7  |-  ( a  e.  om  ->  ( card `  ~P a ) 
~~  ~P a )
48 cdaen 8058 . . . . . . 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 644 . . . . . 6  |-  ( a  e.  om  ->  (
( card `  ~P a
)  +c  ( card `  ~P a ) ) 
~~  ( ~P a  +c  ~P a ) )
5049ensymd 7161 . . . . 5  |-  ( a  e.  om  ->  ( ~P a  +c  ~P a
)  ~~  ( ( card `  ~P a )  +c  ( card `  ~P a ) ) )
51 entr 7162 . . . . 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 644 . . . 4  |-  ( a  e.  om  ->  ~P suc  a  ~~  ( (
card `  ~P a
)  +c  ( card `  ~P a ) ) )
53 carden2b 7859 . . . 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 7853 . . . . 5  |-  ( ~P a  e.  Fin  ->  (
card `  ~P a
)  e.  om )
5645, 55syl 16 . . . 4  |-  ( a  e.  om  ->  ( card `  ~P a )  e.  om )
57 nnacda 8086 . . . 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 644 . . 3  |-  ( a  e.  om  ->  ( card `  ( ( card `  ~P a )  +c  ( card `  ~P a ) ) )  =  ( ( card `  ~P a )  +o  ( card `  ~P a ) ) )
5954, 58eqtrd 2470 . 2  |-  ( a  e.  om  ->  ( card `  ~P suc  a
)  =  ( (
card `  ~P a
)  +o  ( card `  ~P a ) ) )
607, 59vtoclga 3019 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 937    = wceq 1653    e. wcel 1726   _Vcvv 2958    u. cun 3320    i^i cin 3321   (/)c0 3630   ~Pcpw 3801   {csn 3816   class class class wbr 4215   Ord word 4583   suc csuc 4586   omcom 4848    X. cxp 4879   ` cfv 5457  (class class class)co 6084   2oc2o 6721    +o coa 6724    ^m cmap 7021    ~~ cen 7109   Fincfn 7112   cardccrd 7827    +c ccda 8052
This theorem is referenced by:  ackbij1lem14  8118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-cda 8053
  Copyright terms: Public domain W3C validator