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Theorem hsmexlem8 8268
Description: Lemma for hsmex 8276. 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 5709 . 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 4860 . . . . 5  |-  ( b  =  z  ->  ( X  X.  b )  =  ( X  X.  z
) )
43pweqd 3772 . . . 4  |-  ( b  =  z  ->  ~P ( X  X.  b
)  =  ~P ( X  X.  z ) )
54fveq2d 5699 . . 3  |-  ( b  =  z  ->  (har `  ~P ( X  X.  b ) )  =  (har `  ~P ( X  X.  z ) ) )
6 xpeq2 4860 . . . . 5  |-  ( b  =  ( H `  a )  ->  ( X  X.  b )  =  ( X  X.  ( H `  a )
) )
76pweqd 3772 . . . 4  |-  ( b  =  ( H `  a )  ->  ~P ( X  X.  b
)  =  ~P ( X  X.  ( H `  a ) ) )
87fveq2d 5699 . . 3  |-  ( b  =  ( H `  a )  ->  (har `  ~P ( X  X.  b ) )  =  (har `  ~P ( X  X.  ( H `  a ) ) ) )
92, 5, 8frsucmpt2 6664 . 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 1649    e. wcel 1721   _Vcvv 2924   ~Pcpw 3767    e. cmpt 4234   suc csuc 4551   omcom 4812    X. cxp 4843    |` cres 4847   ` cfv 5421   reccrdg 6634  harchar 7488
This theorem is referenced by:  hsmexlem9  8269  hsmexlem4  8273
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 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-recs 6600  df-rdg 6635
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