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Theorem ennnfonelemp1 12231
Description: Lemma for ennnfone 12250. Value of  H at a successor. (Contributed by Jim Kingdon, 23-Jul-2023.)
Hypotheses
Ref Expression
ennnfonelemh.dceq  |-  ( ph  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )
ennnfonelemh.f  |-  ( ph  ->  F : om -onto-> A
)
ennnfonelemh.ne  |-  ( ph  ->  A. n  e.  om  E. k  e.  om  A. j  e.  suc  n ( F `  k )  =/=  ( F `  j ) )
ennnfonelemh.g  |-  G  =  ( x  e.  ( A  ^pm  om ) ,  y  e.  om  |->  if ( ( F `  y )  e.  ( F " y ) ,  x ,  ( x  u.  { <. dom  x ,  ( F `
 y ) >. } ) ) )
ennnfonelemh.n  |-  N  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
ennnfonelemh.j  |-  J  =  ( x  e.  NN0  |->  if ( x  =  0 ,  (/) ,  ( `' N `  ( x  -  1 ) ) ) )
ennnfonelemh.h  |-  H  =  seq 0 ( G ,  J )
ennnfonelemp1.p  |-  ( ph  ->  P  e.  NN0 )
Assertion
Ref Expression
ennnfonelemp1  |-  ( ph  ->  ( H `  ( P  +  1 ) )  =  if ( ( F `  ( `' N `  P ) )  e.  ( F
" ( `' N `  P ) ) ,  ( H `  P
) ,  ( ( H `  P )  u.  { <. dom  ( H `  P ) ,  ( F `  ( `' N `  P ) ) >. } ) ) )
Distinct variable groups:    A, j, x, y    x, F, y   
j, G    x, H, y    j, J    x, N, y    P, j, x, y    ph, j, x, y
Allowed substitution hints:    ph( k, n)    A( k, n)    P( k, n)    F( j, k, n)    G( x, y, k, n)    H( j, k, n)    J( x, y, k, n)    N( j,
k, n)

Proof of Theorem ennnfonelemp1
Dummy variables  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ennnfonelemp1.p . . . . 5  |-  ( ph  ->  P  e.  NN0 )
2 nn0uz 9479 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
31, 2eleqtrdi 2250 . . . 4  |-  ( ph  ->  P  e.  ( ZZ>= ` 
0 ) )
4 ennnfonelemh.dceq . . . . 5  |-  ( ph  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )
5 ennnfonelemh.f . . . . 5  |-  ( ph  ->  F : om -onto-> A
)
6 ennnfonelemh.ne . . . . 5  |-  ( ph  ->  A. n  e.  om  E. k  e.  om  A. j  e.  suc  n ( F `  k )  =/=  ( F `  j ) )
7 ennnfonelemh.g . . . . 5  |-  G  =  ( x  e.  ( A  ^pm  om ) ,  y  e.  om  |->  if ( ( F `  y )  e.  ( F " y ) ,  x ,  ( x  u.  { <. dom  x ,  ( F `
 y ) >. } ) ) )
8 ennnfonelemh.n . . . . 5  |-  N  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
9 ennnfonelemh.j . . . . 5  |-  J  =  ( x  e.  NN0  |->  if ( x  =  0 ,  (/) ,  ( `' N `  ( x  -  1 ) ) ) )
10 ennnfonelemh.h . . . . 5  |-  H  =  seq 0 ( G ,  J )
114, 5, 6, 7, 8, 9, 10ennnfonelemj0 12226 . . . 4  |-  ( ph  ->  ( J `  0
)  e.  { g  e.  ( A  ^pm  om )  |  dom  g  e.  om } )
124, 5, 6, 7, 8, 9, 10ennnfonelemg 12228 . . . 4  |-  ( (
ph  /\  ( f  e.  { g  e.  ( A  ^pm  om )  |  dom  g  e.  om }  /\  j  e.  om ) )  ->  (
f G j )  e.  { g  e.  ( A  ^pm  om )  |  dom  g  e.  om } )
134, 5, 6, 7, 8, 9, 10ennnfonelemjn 12227 . . . 4  |-  ( (
ph  /\  f  e.  ( ZZ>= `  ( 0  +  1 ) ) )  ->  ( J `  f )  e.  om )
143, 11, 12, 13seqp1cd 10375 . . 3  |-  ( ph  ->  (  seq 0 ( G ,  J ) `
 ( P  + 
1 ) )  =  ( (  seq 0
( G ,  J
) `  P ) G ( J `  ( P  +  1
) ) ) )
1510fveq1i 5472 . . . 4  |-  ( H `
 ( P  + 
1 ) )  =  (  seq 0 ( G ,  J ) `
 ( P  + 
1 ) )
1615a1i 9 . . 3  |-  ( ph  ->  ( H `  ( P  +  1 ) )  =  (  seq 0 ( G ,  J ) `  ( P  +  1 ) ) )
1710fveq1i 5472 . . . . 5  |-  ( H `
 P )  =  (  seq 0 ( G ,  J ) `
 P )
1817a1i 9 . . . 4  |-  ( ph  ->  ( H `  P
)  =  (  seq 0 ( G ,  J ) `  P
) )
19 eqeq1 2164 . . . . . . 7  |-  ( x  =  ( P  + 
1 )  ->  (
x  =  0  <->  ( P  +  1 )  =  0 ) )
20 fvoveq1 5850 . . . . . . 7  |-  ( x  =  ( P  + 
1 )  ->  ( `' N `  ( x  -  1 ) )  =  ( `' N `  ( ( P  + 
1 )  -  1 ) ) )
2119, 20ifbieq2d 3530 . . . . . 6  |-  ( x  =  ( P  + 
1 )  ->  if ( x  =  0 ,  (/) ,  ( `' N `  ( x  -  1 ) ) )  =  if ( ( P  +  1 )  =  0 ,  (/) ,  ( `' N `  ( ( P  + 
1 )  -  1 ) ) ) )
22 peano2nn0 9136 . . . . . . 7  |-  ( P  e.  NN0  ->  ( P  +  1 )  e. 
NN0 )
231, 22syl 14 . . . . . 6  |-  ( ph  ->  ( P  +  1 )  e.  NN0 )
24 nn0p1gt0 9125 . . . . . . . . . . . 12  |-  ( P  e.  NN0  ->  0  < 
( P  +  1 ) )
2524gt0ne0d 8392 . . . . . . . . . . 11  |-  ( P  e.  NN0  ->  ( P  +  1 )  =/=  0 )
2625neneqd 2348 . . . . . . . . . 10  |-  ( P  e.  NN0  ->  -.  ( P  +  1 )  =  0 )
2726iffalsed 3516 . . . . . . . . 9  |-  ( P  e.  NN0  ->  if ( ( P  +  1 )  =  0 ,  (/) ,  ( `' N `  ( ( P  + 
1 )  -  1 ) ) )  =  ( `' N `  ( ( P  + 
1 )  -  1 ) ) )
28 nn0cn 9106 . . . . . . . . . . 11  |-  ( P  e.  NN0  ->  P  e.  CC )
29 1cnd 7897 . . . . . . . . . . 11  |-  ( P  e.  NN0  ->  1  e.  CC )
3028, 29pncand 8192 . . . . . . . . . 10  |-  ( P  e.  NN0  ->  ( ( P  +  1 )  -  1 )  =  P )
3130fveq2d 5475 . . . . . . . . 9  |-  ( P  e.  NN0  ->  ( `' N `  ( ( P  +  1 )  -  1 ) )  =  ( `' N `  P ) )
3227, 31eqtrd 2190 . . . . . . . 8  |-  ( P  e.  NN0  ->  if ( ( P  +  1 )  =  0 ,  (/) ,  ( `' N `  ( ( P  + 
1 )  -  1 ) ) )  =  ( `' N `  P ) )
338frechashgf1o 10337 . . . . . . . . . . 11  |-  N : om
-1-1-onto-> NN0
34 f1ocnv 5430 . . . . . . . . . . 11  |-  ( N : om -1-1-onto-> NN0  ->  `' N : NN0
-1-1-onto-> om )
3533, 34ax-mp 5 . . . . . . . . . 10  |-  `' N : NN0
-1-1-onto-> om
36 f1of 5417 . . . . . . . . . 10  |-  ( `' N : NN0 -1-1-onto-> om  ->  `' N : NN0 --> om )
3735, 36mp1i 10 . . . . . . . . 9  |-  ( P  e.  NN0  ->  `' N : NN0 --> om )
38 id 19 . . . . . . . . 9  |-  ( P  e.  NN0  ->  P  e. 
NN0 )
3937, 38ffvelrnd 5606 . . . . . . . 8  |-  ( P  e.  NN0  ->  ( `' N `  P )  e.  om )
4032, 39eqeltrd 2234 . . . . . . 7  |-  ( P  e.  NN0  ->  if ( ( P  +  1 )  =  0 ,  (/) ,  ( `' N `  ( ( P  + 
1 )  -  1 ) ) )  e. 
om )
411, 40syl 14 . . . . . 6  |-  ( ph  ->  if ( ( P  +  1 )  =  0 ,  (/) ,  ( `' N `  ( ( P  +  1 )  -  1 ) ) )  e.  om )
429, 21, 23, 41fvmptd3 5564 . . . . 5  |-  ( ph  ->  ( J `  ( P  +  1 ) )  =  if ( ( P  +  1 )  =  0 ,  (/) ,  ( `' N `  ( ( P  + 
1 )  -  1 ) ) ) )
431, 32syl 14 . . . . 5  |-  ( ph  ->  if ( ( P  +  1 )  =  0 ,  (/) ,  ( `' N `  ( ( P  +  1 )  -  1 ) ) )  =  ( `' N `  P ) )
4442, 43eqtr2d 2191 . . . 4  |-  ( ph  ->  ( `' N `  P )  =  ( J `  ( P  +  1 ) ) )
4518, 44oveq12d 5845 . . 3  |-  ( ph  ->  ( ( H `  P ) G ( `' N `  P ) )  =  ( (  seq 0 ( G ,  J ) `  P ) G ( J `  ( P  +  1 ) ) ) )
4614, 16, 453eqtr4d 2200 . 2  |-  ( ph  ->  ( H `  ( P  +  1 ) )  =  ( ( H `  P ) G ( `' N `  P ) ) )
474, 5, 6, 7, 8, 9, 10ennnfonelemh 12229 . . . 4  |-  ( ph  ->  H : NN0 --> ( A 
^pm  om ) )
4847, 1ffvelrnd 5606 . . 3  |-  ( ph  ->  ( H `  P
)  e.  ( A 
^pm  om ) )
491, 39syl 14 . . 3  |-  ( ph  ->  ( `' N `  P )  e.  om )
5048elexd 2725 . . . 4  |-  ( ph  ->  ( H `  P
)  e.  _V )
51 dmexg 4853 . . . . . . . 8  |-  ( ( H `  P )  e.  _V  ->  dom  ( H `  P )  e.  _V )
5250, 51syl 14 . . . . . . 7  |-  ( ph  ->  dom  ( H `  P )  e.  _V )
53 fof 5395 . . . . . . . . 9  |-  ( F : om -onto-> A  ->  F : om --> A )
545, 53syl 14 . . . . . . . 8  |-  ( ph  ->  F : om --> A )
5554, 49ffvelrnd 5606 . . . . . . 7  |-  ( ph  ->  ( F `  ( `' N `  P ) )  e.  A )
56 opexg 4191 . . . . . . 7  |-  ( ( dom  ( H `  P )  e.  _V  /\  ( F `  ( `' N `  P ) )  e.  A )  ->  <. dom  ( H `  P ) ,  ( F `  ( `' N `  P ) ) >.  e.  _V )
5752, 55, 56syl2anc 409 . . . . . 6  |-  ( ph  -> 
<. dom  ( H `  P ) ,  ( F `  ( `' N `  P ) ) >.  e.  _V )
58 snexg 4148 . . . . . 6  |-  ( <. dom  ( H `  P
) ,  ( F `
 ( `' N `  P ) ) >.  e.  _V  ->  { <. dom  ( H `  P ) ,  ( F `  ( `' N `  P ) ) >. }  e.  _V )
5957, 58syl 14 . . . . 5  |-  ( ph  ->  { <. dom  ( H `  P ) ,  ( F `  ( `' N `  P ) ) >. }  e.  _V )
60 unexg 4406 . . . . 5  |-  ( ( ( H `  P
)  e.  _V  /\  {
<. dom  ( H `  P ) ,  ( F `  ( `' N `  P ) ) >. }  e.  _V )  ->  ( ( H `
 P )  u. 
{ <. dom  ( H `  P ) ,  ( F `  ( `' N `  P ) ) >. } )  e. 
_V )
6150, 59, 60syl2anc 409 . . . 4  |-  ( ph  ->  ( ( H `  P )  u.  { <. dom  ( H `  P ) ,  ( F `  ( `' N `  P ) ) >. } )  e. 
_V )
624, 5, 49ennnfonelemdc 12224 . . . 4  |-  ( ph  -> DECID  ( F `  ( `' N `  P ) )  e.  ( F
" ( `' N `  P ) ) )
6350, 61, 62ifcldcd 3541 . . 3  |-  ( ph  ->  if ( ( F `
 ( `' N `  P ) )  e.  ( F " ( `' N `  P ) ) ,  ( H `
 P ) ,  ( ( H `  P )  u.  { <. dom  ( H `  P ) ,  ( F `  ( `' N `  P ) ) >. } ) )  e.  _V )
64 id 19 . . . . 5  |-  ( x  =  ( H `  P )  ->  x  =  ( H `  P ) )
65 dmeq 4789 . . . . . . . 8  |-  ( x  =  ( H `  P )  ->  dom  x  =  dom  ( H `
 P ) )
6665opeq1d 3749 . . . . . . 7  |-  ( x  =  ( H `  P )  ->  <. dom  x ,  ( F `  y ) >.  =  <. dom  ( H `  P
) ,  ( F `
 y ) >.
)
6766sneqd 3574 . . . . . 6  |-  ( x  =  ( H `  P )  ->  { <. dom  x ,  ( F `
 y ) >. }  =  { <. dom  ( H `  P ) ,  ( F `  y ) >. } )
6864, 67uneq12d 3263 . . . . 5  |-  ( x  =  ( H `  P )  ->  (
x  u.  { <. dom  x ,  ( F `
 y ) >. } )  =  ( ( H `  P
)  u.  { <. dom  ( H `  P
) ,  ( F `
 y ) >. } ) )
6964, 68ifeq12d 3525 . . . 4  |-  ( x  =  ( H `  P )  ->  if ( ( F `  y )  e.  ( F " y ) ,  x ,  ( x  u.  { <. dom  x ,  ( F `
 y ) >. } ) )  =  if ( ( F `
 y )  e.  ( F " y
) ,  ( H `
 P ) ,  ( ( H `  P )  u.  { <. dom  ( H `  P ) ,  ( F `  y )
>. } ) ) )
70 fveq2 5471 . . . . . 6  |-  ( y  =  ( `' N `  P )  ->  ( F `  y )  =  ( F `  ( `' N `  P ) ) )
71 imaeq2 4927 . . . . . 6  |-  ( y  =  ( `' N `  P )  ->  ( F " y )  =  ( F " ( `' N `  P ) ) )
7270, 71eleq12d 2228 . . . . 5  |-  ( y  =  ( `' N `  P )  ->  (
( F `  y
)  e.  ( F
" y )  <->  ( F `  ( `' N `  P ) )  e.  ( F " ( `' N `  P ) ) ) )
7370opeq2d 3750 . . . . . . 7  |-  ( y  =  ( `' N `  P )  ->  <. dom  ( H `  P ) ,  ( F `  y ) >.  =  <. dom  ( H `  P
) ,  ( F `
 ( `' N `  P ) ) >.
)
7473sneqd 3574 . . . . . 6  |-  ( y  =  ( `' N `  P )  ->  { <. dom  ( H `  P
) ,  ( F `
 y ) >. }  =  { <. dom  ( H `  P ) ,  ( F `  ( `' N `  P ) ) >. } )
7574uneq2d 3262 . . . . 5  |-  ( y  =  ( `' N `  P )  ->  (
( H `  P
)  u.  { <. dom  ( H `  P
) ,  ( F `
 y ) >. } )  =  ( ( H `  P
)  u.  { <. dom  ( H `  P
) ,  ( F `
 ( `' N `  P ) ) >. } ) )
7672, 75ifbieq2d 3530 . . . 4  |-  ( y  =  ( `' N `  P )  ->  if ( ( F `  y )  e.  ( F " y ) ,  ( H `  P ) ,  ( ( H `  P
)  u.  { <. dom  ( H `  P
) ,  ( F `
 y ) >. } ) )  =  if ( ( F `
 ( `' N `  P ) )  e.  ( F " ( `' N `  P ) ) ,  ( H `
 P ) ,  ( ( H `  P )  u.  { <. dom  ( H `  P ) ,  ( F `  ( `' N `  P ) ) >. } ) ) )
7769, 76, 7ovmpog 5958 . . 3  |-  ( ( ( H `  P
)  e.  ( A 
^pm  om )  /\  ( `' N `  P )  e.  om  /\  if ( ( F `  ( `' N `  P ) )  e.  ( F
" ( `' N `  P ) ) ,  ( H `  P
) ,  ( ( H `  P )  u.  { <. dom  ( H `  P ) ,  ( F `  ( `' N `  P ) ) >. } ) )  e.  _V )  -> 
( ( H `  P ) G ( `' N `  P ) )  =  if ( ( F `  ( `' N `  P ) )  e.  ( F
" ( `' N `  P ) ) ,  ( H `  P
) ,  ( ( H `  P )  u.  { <. dom  ( H `  P ) ,  ( F `  ( `' N `  P ) ) >. } ) ) )
7848, 49, 63, 77syl3anc 1220 . 2  |-  ( ph  ->  ( ( H `  P ) G ( `' N `  P ) )  =  if ( ( F `  ( `' N `  P ) )  e.  ( F
" ( `' N `  P ) ) ,  ( H `  P
) ,  ( ( H `  P )  u.  { <. dom  ( H `  P ) ,  ( F `  ( `' N `  P ) ) >. } ) ) )
7946, 78eqtrd 2190 1  |-  ( ph  ->  ( H `  ( P  +  1 ) )  =  if ( ( F `  ( `' N `  P ) )  e.  ( F
" ( `' N `  P ) ) ,  ( H `  P
) ,  ( ( H `  P )  u.  { <. dom  ( H `  P ) ,  ( F `  ( `' N `  P ) ) >. } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4  DECID wdc 820    = wceq 1335    e. wcel 2128    =/= wne 2327   A.wral 2435   E.wrex 2436   {crab 2439   _Vcvv 2712    u. cun 3100   (/)c0 3395   ifcif 3506   {csn 3561   <.cop 3564    |-> cmpt 4028   suc csuc 4328   omcom 4552   `'ccnv 4588   dom cdm 4589   "cima 4592   -->wf 5169   -onto->wfo 5171   -1-1-onto->wf1o 5172   ` cfv 5173  (class class class)co 5827    e. cmpo 5829  freccfrec 6340    ^pm cpm 6597   0cc0 7735   1c1 7736    + caddc 7738    - cmin 8051   NN0cn0 9096   ZZcz 9173   ZZ>=cuz 9445    seqcseq 10354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-iinf 4550  ax-cnex 7826  ax-resscn 7827  ax-1cn 7828  ax-1re 7829  ax-icn 7830  ax-addcl 7831  ax-addrcl 7832  ax-mulcl 7833  ax-addcom 7835  ax-addass 7837  ax-distr 7839  ax-i2m1 7840  ax-0lt1 7841  ax-0id 7843  ax-rnegex 7844  ax-cnre 7846  ax-pre-ltirr 7847  ax-pre-ltwlin 7848  ax-pre-lttrn 7849  ax-pre-ltadd 7851
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-id 4256  df-iord 4329  df-on 4331  df-ilim 4332  df-suc 4334  df-iom 4553  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-riota 5783  df-ov 5830  df-oprab 5831  df-mpo 5832  df-1st 6091  df-2nd 6092  df-recs 6255  df-frec 6341  df-pm 6599  df-pnf 7917  df-mnf 7918  df-xr 7919  df-ltxr 7920  df-le 7921  df-sub 8053  df-neg 8054  df-inn 8840  df-n0 9097  df-z 9174  df-uz 9446  df-seqfrec 10355
This theorem is referenced by:  ennnfonelem1  12232  ennnfonelemhdmp1  12234  ennnfonelemss  12235  ennnfonelemkh  12237  ennnfonelemhf1o  12238
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