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Theorem hsmexlem8 8304
Description: Lemma for hsmex 8312. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
Hypothesis
Ref Expression
hsmexlem7.h  |-  H  =  ( rec ( ( z  e.  _V  |->  (har
`  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om )
Assertion
Ref Expression
hsmexlem8  |-  ( a  e.  om  ->  ( H `  suc  a )  =  (har `  ~P ( X  X.  ( H `  a )
) ) )
Distinct variable groups:    z, X    z, a
Allowed substitution hints:    H( z, a)    X( a)

Proof of Theorem hsmexlem8
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 fvex 5742 . 2  |-  (har `  ~P ( X  X.  ( H `  a )
) )  e.  _V
2 hsmexlem7.h . . 3  |-  H  =  ( rec ( ( z  e.  _V  |->  (har
`  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om )
3 xpeq2 4893 . . . . 5  |-  ( b  =  z  ->  ( X  X.  b )  =  ( X  X.  z
) )
43pweqd 3804 . . . 4  |-  ( b  =  z  ->  ~P ( X  X.  b
)  =  ~P ( X  X.  z ) )
54fveq2d 5732 . . 3  |-  ( b  =  z  ->  (har `  ~P ( X  X.  b ) )  =  (har `  ~P ( X  X.  z ) ) )
6 xpeq2 4893 . . . . 5  |-  ( b  =  ( H `  a )  ->  ( X  X.  b )  =  ( X  X.  ( H `  a )
) )
76pweqd 3804 . . . 4  |-  ( b  =  ( H `  a )  ->  ~P ( X  X.  b
)  =  ~P ( X  X.  ( H `  a ) ) )
87fveq2d 5732 . . 3  |-  ( b  =  ( H `  a )  ->  (har `  ~P ( X  X.  b ) )  =  (har `  ~P ( X  X.  ( H `  a ) ) ) )
92, 5, 8frsucmpt2 6697 . 2  |-  ( ( a  e.  om  /\  (har `  ~P ( X  X.  ( H `  a ) ) )  e.  _V )  -> 
( H `  suc  a )  =  (har
`  ~P ( X  X.  ( H `  a ) ) ) )
101, 9mpan2 653 1  |-  ( a  e.  om  ->  ( H `  suc  a )  =  (har `  ~P ( X  X.  ( H `  a )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2956   ~Pcpw 3799    e. cmpt 4266   suc csuc 4583   omcom 4845    X. cxp 4876    |` cres 4880   ` cfv 5454   reccrdg 6667  harchar 7524
This theorem is referenced by:  hsmexlem9  8305  hsmexlem4  8309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668
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