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Theorem hsmexlem8 8793
Description: Lemma for hsmex 8801. 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 5867 . 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 5007 . . . . 5  |-  ( b  =  z  ->  ( X  X.  b )  =  ( X  X.  z
) )
43pweqd 4008 . . . 4  |-  ( b  =  z  ->  ~P ( X  X.  b
)  =  ~P ( X  X.  z ) )
54fveq2d 5861 . . 3  |-  ( b  =  z  ->  (har `  ~P ( X  X.  b ) )  =  (har `  ~P ( X  X.  z ) ) )
6 xpeq2 5007 . . . . 5  |-  ( b  =  ( H `  a )  ->  ( X  X.  b )  =  ( X  X.  ( H `  a )
) )
76pweqd 4008 . . . 4  |-  ( b  =  ( H `  a )  ->  ~P ( X  X.  b
)  =  ~P ( X  X.  ( H `  a ) ) )
87fveq2d 5861 . . 3  |-  ( b  =  ( H `  a )  ->  (har `  ~P ( X  X.  b ) )  =  (har `  ~P ( X  X.  ( H `  a ) ) ) )
92, 5, 8frsucmpt2 7095 . 2  |-  ( ( a  e.  om  /\  (har `  ~P ( X  X.  ( H `  a ) ) )  e.  _V )  -> 
( H `  suc  a )  =  (har
`  ~P ( X  X.  ( H `  a ) ) ) )
101, 9mpan2 671 1  |-  ( a  e.  om  ->  ( H `  suc  a )  =  (har `  ~P ( X  X.  ( H `  a )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   _Vcvv 3106   ~Pcpw 4003    |-> cmpt 4498   suc csuc 4873    X. cxp 4990    |` cres 4994   ` cfv 5579   omcom 6671   reccrdg 7065  harchar 7971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-recs 7032  df-rdg 7066
This theorem is referenced by:  hsmexlem9  8794  hsmexlem4  8798
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