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Theorem iseqf1olemqval 10501
Description: Lemma for seq3f1o 10518. Value of the function  Q. (Contributed by Jim Kingdon, 28-Aug-2022.)
Hypotheses
Ref Expression
iseqf1olemqcl.k  |-  ( ph  ->  K  e.  ( M ... N ) )
iseqf1olemqcl.j  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
iseqf1olemqcl.a  |-  ( ph  ->  A  e.  ( M ... N ) )
iseqf1olemqval.q  |-  Q  =  ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )
Assertion
Ref Expression
iseqf1olemqval  |-  ( ph  ->  ( Q `  A
)  =  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K , 
( J `  ( A  -  1 ) ) ) ,  ( J `  A ) ) )
Distinct variable groups:    u, A    u, J    u, K    u, M    u, N
Allowed substitution hints:    ph( u)    Q( u)

Proof of Theorem iseqf1olemqval
StepHypRef Expression
1 iseqf1olemqcl.a . 2  |-  ( ph  ->  A  e.  ( M ... N ) )
2 iseqf1olemqcl.k . . 3  |-  ( ph  ->  K  e.  ( M ... N ) )
3 iseqf1olemqcl.j . . 3  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
42, 3, 1iseqf1olemqcl 10500 . 2  |-  ( ph  ->  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) ,  ( J `  A
) )  e.  ( M ... N ) )
5 eleq1 2250 . . . 4  |-  ( u  =  A  ->  (
u  e.  ( K ... ( `' J `  K ) )  <->  A  e.  ( K ... ( `' J `  K ) ) ) )
6 eqeq1 2194 . . . . 5  |-  ( u  =  A  ->  (
u  =  K  <->  A  =  K ) )
7 oveq1 5895 . . . . . 6  |-  ( u  =  A  ->  (
u  -  1 )  =  ( A  - 
1 ) )
87fveq2d 5531 . . . . 5  |-  ( u  =  A  ->  ( J `  ( u  -  1 ) )  =  ( J `  ( A  -  1
) ) )
96, 8ifbieq2d 3570 . . . 4  |-  ( u  =  A  ->  if ( u  =  K ,  K ,  ( J `
 ( u  - 
1 ) ) )  =  if ( A  =  K ,  K ,  ( J `  ( A  -  1
) ) ) )
10 fveq2 5527 . . . 4  |-  ( u  =  A  ->  ( J `  u )  =  ( J `  A ) )
115, 9, 10ifbieq12d 3572 . . 3  |-  ( u  =  A  ->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K , 
( J `  (
u  -  1 ) ) ) ,  ( J `  u ) )  =  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K , 
( J `  ( A  -  1 ) ) ) ,  ( J `  A ) ) )
12 iseqf1olemqval.q . . 3  |-  Q  =  ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )
1311, 12fvmptg 5605 . 2  |-  ( ( A  e.  ( M ... N )  /\  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) ,  ( J `  A
) )  e.  ( M ... N ) )  ->  ( Q `  A )  =  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) ,  ( J `  A
) ) )
141, 4, 13syl2anc 411 1  |-  ( ph  ->  ( Q `  A
)  =  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K , 
( J `  ( A  -  1 ) ) ) ,  ( J `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363    e. wcel 2158   ifcif 3546    |-> cmpt 4076   `'ccnv 4637   -1-1-onto->wf1o 5227   ` cfv 5228  (class class class)co 5888   1c1 7826    - cmin 8142   ...cfz 10022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-0id 7933  ax-rnegex 7934  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-ltadd 7941
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-inn 8934  df-n0 9191  df-z 9268  df-uz 9543  df-fz 10023
This theorem is referenced by:  iseqf1olemnab  10502  iseqf1olemab  10503  iseqf1olemnanb  10504  iseqf1olemqk  10508  seq3f1olemqsumkj  10512  seq3f1olemqsumk  10513
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