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Theorem seq3f1olemstep 9918
Description: Lemma for seq3f1o 9921. Given a permutation which is constant up to a point, supply a new one which is constant for one more position. (Contributed by Jim Kingdon, 19-Aug-2022.)
Hypotheses
Ref Expression
iseqf1o.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
iseqf1o.2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( y 
.+  x ) )
iseqf1o.3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
iseqf1o.4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
iseqf1o.6  |-  ( ph  ->  F : ( M ... N ) -1-1-onto-> ( M ... N ) )
iseqf1o.7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( G `  x )  e.  S
)
iseqf1olemstep.k  |-  ( ph  ->  K  e.  ( M ... N ) )
iseqf1olemstep.j  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
iseqf1olemstep.const  |-  ( ph  ->  A. x  e.  ( M..^ K ) ( J `  x )  =  x )
seq3f1olemstep.jp  |-  ( ph  ->  (  seq M ( 
.+  ,  [_ J  /  f ]_ P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N ) )
seq3f1olemstep.p  |-  P  =  ( x  e.  (
ZZ>= `  M )  |->  if ( x  <_  N ,  ( G `  ( f `  x
) ) ,  ( G `  M ) ) )
Assertion
Ref Expression
seq3f1olemstep  |-  ( ph  ->  E. f ( f : ( M ... N ) -1-1-onto-> ( M ... N
)  /\  A. x  e.  ( M ... K
) ( f `  x )  =  x  /\  (  seq M
(  .+  ,  P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N ) ) )
Distinct variable groups:    .+ , f, x, y, z    f, J, x, y, z    f, K, x, y, z    f, L    f, M, x, y, z    f, N, x, y, z    S, f, x, y, z    ph, x, y, z    x, P, y, z    f, G, x
Allowed substitution hints:    ph( f)    P( f)    F( x, y, z, f)    G( y, z)    L( x, y, z)

Proof of Theorem seq3f1olemstep
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 iseqf1olemstep.j . . . . . 6  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
2 f1of 5247 . . . . . 6  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  J :
( M ... N
) --> ( M ... N ) )
31, 2syl 14 . . . . 5  |-  ( ph  ->  J : ( M ... N ) --> ( M ... N ) )
4 iseqf1olemstep.k . . . . . . 7  |-  ( ph  ->  K  e.  ( M ... N ) )
5 elfzel1 9429 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  M  e.  ZZ )
64, 5syl 14 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
7 elfzel2 9428 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  N  e.  ZZ )
84, 7syl 14 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
96, 8fzfigd 9826 . . . . 5  |-  ( ph  ->  ( M ... N
)  e.  Fin )
10 fex 5516 . . . . 5  |-  ( ( J : ( M ... N ) --> ( M ... N )  /\  ( M ... N )  e.  Fin )  ->  J  e.  _V )
113, 9, 10syl2anc 403 . . . 4  |-  ( ph  ->  J  e.  _V )
1211adantr 270 . . 3  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  J  e.  _V )
131adantr 270 . . . 4  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  J :
( M ... N
)
-1-1-onto-> ( M ... N ) )
14 iseqf1olemstep.const . . . . . . 7  |-  ( ph  ->  A. x  e.  ( M..^ K ) ( J `  x )  =  x )
1514adantr 270 . . . . . 6  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  A. x  e.  ( M..^ K ) ( J `  x
)  =  x )
16 eqcom 2090 . . . . . . . . . 10  |-  ( K  =  ( `' J `  K )  <->  ( `' J `  K )  =  K )
1716biimpi 118 . . . . . . . . 9  |-  ( K  =  ( `' J `  K )  ->  ( `' J `  K )  =  K )
1817adantl 271 . . . . . . . 8  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  ( `' J `  K )  =  K )
19 f1ocnvfvb 5551 . . . . . . . . . 10  |-  ( ( J : ( M ... N ) -1-1-onto-> ( M ... N )  /\  K  e.  ( M ... N )  /\  K  e.  ( M ... N
) )  ->  (
( J `  K
)  =  K  <->  ( `' J `  K )  =  K ) )
201, 4, 4, 19syl3anc 1174 . . . . . . . . 9  |-  ( ph  ->  ( ( J `  K )  =  K  <-> 
( `' J `  K )  =  K ) )
2120adantr 270 . . . . . . . 8  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  ( ( J `  K )  =  K  <->  ( `' J `  K )  =  K ) )
2218, 21mpbird 165 . . . . . . 7  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  ( J `  K )  =  K )
23 elfzelz 9430 . . . . . . . . . 10  |-  ( K  e.  ( M ... N )  ->  K  e.  ZZ )
244, 23syl 14 . . . . . . . . 9  |-  ( ph  ->  K  e.  ZZ )
2524adantr 270 . . . . . . . 8  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  K  e.  ZZ )
26 fveq2 5299 . . . . . . . . . 10  |-  ( x  =  K  ->  ( J `  x )  =  ( J `  K ) )
27 id 19 . . . . . . . . . 10  |-  ( x  =  K  ->  x  =  K )
2826, 27eqeq12d 2102 . . . . . . . . 9  |-  ( x  =  K  ->  (
( J `  x
)  =  x  <->  ( J `  K )  =  K ) )
2928ralsng 3481 . . . . . . . 8  |-  ( K  e.  ZZ  ->  ( A. x  e.  { K }  ( J `  x )  =  x  <-> 
( J `  K
)  =  K ) )
3025, 29syl 14 . . . . . . 7  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  ( A. x  e.  { K }  ( J `  x )  =  x  <-> 
( J `  K
)  =  K ) )
3122, 30mpbird 165 . . . . . 6  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  A. x  e.  { K }  ( J `  x )  =  x )
32 ralun 3182 . . . . . 6  |-  ( ( A. x  e.  ( M..^ K ) ( J `  x )  =  x  /\  A. x  e.  { K }  ( J `  x )  =  x )  ->  A. x  e.  ( ( M..^ K
)  u.  { K } ) ( J `
 x )  =  x )
3315, 31, 32syl2anc 403 . . . . 5  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  A. x  e.  ( ( M..^ K
)  u.  { K } ) ( J `
 x )  =  x )
34 elfzuz 9426 . . . . . . . 8  |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M )
)
35 fzisfzounsn 9635 . . . . . . . 8  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( M ... K )  =  ( ( M..^ K )  u.  { K }
) )
364, 34, 353syl 17 . . . . . . 7  |-  ( ph  ->  ( M ... K
)  =  ( ( M..^ K )  u. 
{ K } ) )
3736raleqdv 2568 . . . . . 6  |-  ( ph  ->  ( A. x  e.  ( M ... K
) ( J `  x )  =  x  <->  A. x  e.  (
( M..^ K )  u.  { K }
) ( J `  x )  =  x ) )
3837adantr 270 . . . . 5  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  ( A. x  e.  ( M ... K ) ( J `
 x )  =  x  <->  A. x  e.  ( ( M..^ K )  u.  { K }
) ( J `  x )  =  x ) )
3933, 38mpbird 165 . . . 4  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  A. x  e.  ( M ... K
) ( J `  x )  =  x )
40 seq3f1olemstep.jp . . . . 5  |-  ( ph  ->  (  seq M ( 
.+  ,  [_ J  /  f ]_ P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N ) )
4140adantr 270 . . . 4  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  (  seq M (  .+  ,  [_ J  /  f ]_ P ) `  N
)  =  (  seq M (  .+  ,  L ) `  N
) )
4213, 39, 413jca 1123 . . 3  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  /\  A. x  e.  ( M ... K
) ( J `  x )  =  x  /\  (  seq M
(  .+  ,  [_ J  /  f ]_ P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N ) ) )
43 nfcv 2228 . . . 4  |-  F/_ f J
44 nfv 1466 . . . . 5  |-  F/ f  J : ( M ... N ) -1-1-onto-> ( M ... N )
45 nfv 1466 . . . . 5  |-  F/ f A. x  e.  ( M ... K ) ( J `  x
)  =  x
46 nfcv 2228 . . . . . . . 8  |-  F/_ f M
47 nfcv 2228 . . . . . . . 8  |-  F/_ f  .+
48 nfcsb1v 2963 . . . . . . . 8  |-  F/_ f [_ J  /  f ]_ P
4946, 47, 48nfseq 9857 . . . . . . 7  |-  F/_ f  seq M (  .+  ,  [_ J  /  f ]_ P )
50 nfcv 2228 . . . . . . 7  |-  F/_ f N
5149, 50nffv 5309 . . . . . 6  |-  F/_ f
(  seq M (  .+  ,  [_ J  /  f ]_ P ) `  N
)
5251nfeq1 2238 . . . . 5  |-  F/ f (  seq M ( 
.+  ,  [_ J  /  f ]_ P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N )
5344, 45, 52nf3an 1503 . . . 4  |-  F/ f ( J : ( M ... N ) -1-1-onto-> ( M ... N )  /\  A. x  e.  ( M ... K
) ( J `  x )  =  x  /\  (  seq M
(  .+  ,  [_ J  /  f ]_ P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N ) )
54 f1oeq1 5238 . . . . 5  |-  ( f  =  J  ->  (
f : ( M ... N ) -1-1-onto-> ( M ... N )  <->  J :
( M ... N
)
-1-1-onto-> ( M ... N ) ) )
55 fveq1 5298 . . . . . . 7  |-  ( f  =  J  ->  (
f `  x )  =  ( J `  x ) )
5655eqeq1d 2096 . . . . . 6  |-  ( f  =  J  ->  (
( f `  x
)  =  x  <->  ( J `  x )  =  x ) )
5756ralbidv 2380 . . . . 5  |-  ( f  =  J  ->  ( A. x  e.  ( M ... K ) ( f `  x )  =  x  <->  A. x  e.  ( M ... K
) ( J `  x )  =  x ) )
58 csbeq1a 2941 . . . . . . . 8  |-  ( f  =  J  ->  P  =  [_ J  /  f ]_ P )
5958seqeq3d 9854 . . . . . . 7  |-  ( f  =  J  ->  seq M (  .+  ,  P )  =  seq M (  .+  ,  [_ J  /  f ]_ P ) )
6059fveq1d 5301 . . . . . 6  |-  ( f  =  J  ->  (  seq M (  .+  ,  P ) `  N
)  =  (  seq M (  .+  ,  [_ J  /  f ]_ P ) `  N
) )
6160eqeq1d 2096 . . . . 5  |-  ( f  =  J  ->  (
(  seq M (  .+  ,  P ) `  N
)  =  (  seq M (  .+  ,  L ) `  N
)  <->  (  seq M
(  .+  ,  [_ J  /  f ]_ P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N ) ) )
6254, 57, 613anbi123d 1248 . . . 4  |-  ( f  =  J  ->  (
( f : ( M ... N ) -1-1-onto-> ( M ... N )  /\  A. x  e.  ( M ... K
) ( f `  x )  =  x  /\  (  seq M
(  .+  ,  P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N ) )  <-> 
( J : ( M ... N ) -1-1-onto-> ( M ... N )  /\  A. x  e.  ( M ... K
) ( J `  x )  =  x  /\  (  seq M
(  .+  ,  [_ J  /  f ]_ P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N ) ) ) )
6343, 53, 62spcegf 2702 . . 3  |-  ( J  e.  _V  ->  (
( J : ( M ... N ) -1-1-onto-> ( M ... N )  /\  A. x  e.  ( M ... K
) ( J `  x )  =  x  /\  (  seq M
(  .+  ,  [_ J  /  f ]_ P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N ) )  ->  E. f ( f : ( M ... N ) -1-1-onto-> ( M ... N
)  /\  A. x  e.  ( M ... K
) ( f `  x )  =  x  /\  (  seq M
(  .+  ,  P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N ) ) ) )
6412, 42, 63sylc 61 . 2  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  E. f
( f : ( M ... N ) -1-1-onto-> ( M ... N )  /\  A. x  e.  ( M ... K
) ( f `  x )  =  x  /\  (  seq M
(  .+  ,  P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N ) ) )
654adantr 270 . . . 4  |-  ( (
ph  /\  -.  K  =  ( `' J `  K ) )  ->  K  e.  ( M ... N ) )
661adantr 270 . . . 4  |-  ( (
ph  /\  -.  K  =  ( `' J `  K ) )  ->  J : ( M ... N ) -1-1-onto-> ( M ... N
) )
67 eqid 2088 . . . 4  |-  ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K , 
( J `  (
u  -  1 ) ) ) ,  ( J `  u ) ) )  =  ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )
6865, 66, 67iseqf1olemqf1o 9910 . . 3  |-  ( (
ph  /\  -.  K  =  ( `' J `  K ) )  -> 
( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) ) : ( M ... N
)
-1-1-onto-> ( M ... N ) )
6914adantr 270 . . . 4  |-  ( (
ph  /\  -.  K  =  ( `' J `  K ) )  ->  A. x  e.  ( M..^ K ) ( J `
 x )  =  x )
7065, 66, 67, 69iseqf1olemqk 9911 . . 3  |-  ( (
ph  /\  -.  K  =  ( `' J `  K ) )  ->  A. x  e.  ( M ... K ) ( ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) ) `  x )  =  x )
71 iseqf1o.1 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
7271adantlr 461 . . . . 5  |-  ( ( ( ph  /\  -.  K  =  ( `' J `  K )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
73 iseqf1o.2 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( y 
.+  x ) )
7473adantlr 461 . . . . 5  |-  ( ( ( ph  /\  -.  K  =  ( `' J `  K )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( y 
.+  x ) )
75 iseqf1o.3 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
7675adantlr 461 . . . . 5  |-  ( ( ( ph  /\  -.  K  =  ( `' J `  K )
)  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
77 iseqf1o.4 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
7877adantr 270 . . . . 5  |-  ( (
ph  /\  -.  K  =  ( `' J `  K ) )  ->  N  e.  ( ZZ>= `  M ) )
79 iseqf1o.6 . . . . . 6  |-  ( ph  ->  F : ( M ... N ) -1-1-onto-> ( M ... N ) )
8079adantr 270 . . . . 5  |-  ( (
ph  /\  -.  K  =  ( `' J `  K ) )  ->  F : ( M ... N ) -1-1-onto-> ( M ... N
) )
81 iseqf1o.7 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( G `  x )  e.  S
)
8281adantlr 461 . . . . 5  |-  ( ( ( ph  /\  -.  K  =  ( `' J `  K )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( G `  x )  e.  S
)
83 neqne 2263 . . . . . 6  |-  ( -.  K  =  ( `' J `  K )  ->  K  =/=  ( `' J `  K ) )
8483adantl 271 . . . . 5  |-  ( (
ph  /\  -.  K  =  ( `' J `  K ) )  ->  K  =/=  ( `' J `  K ) )
85 seq3f1olemstep.p . . . . 5  |-  P  =  ( x  e.  (
ZZ>= `  M )  |->  if ( x  <_  N ,  ( G `  ( f `  x
) ) ,  ( G `  M ) ) )
8672, 74, 76, 78, 80, 82, 65, 66, 69, 84, 67, 85seq3f1olemqsum 9917 . . . 4  |-  ( (
ph  /\  -.  K  =  ( `' J `  K ) )  -> 
(  seq M (  .+  ,  [_ J  /  f ]_ P ) `  N
)  =  (  seq M (  .+  ,  [_ ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  / 
f ]_ P ) `  N ) )
8740adantr 270 . . . 4  |-  ( (
ph  /\  -.  K  =  ( `' J `  K ) )  -> 
(  seq M (  .+  ,  [_ J  /  f ]_ P ) `  N
)  =  (  seq M (  .+  ,  L ) `  N
) )
8886, 87eqtr3d 2122 . . 3  |-  ( (
ph  /\  -.  K  =  ( `' J `  K ) )  -> 
(  seq M (  .+  ,  [_ ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  / 
f ]_ P ) `  N )  =  (  seq M (  .+  ,  L ) `  N
) )
8965, 5syl 14 . . . . 5  |-  ( (
ph  /\  -.  K  =  ( `' J `  K ) )  ->  M  e.  ZZ )
9065, 7syl 14 . . . . 5  |-  ( (
ph  /\  -.  K  =  ( `' J `  K ) )  ->  N  e.  ZZ )
9189, 90fzfigd 9826 . . . 4  |-  ( (
ph  /\  -.  K  =  ( `' J `  K ) )  -> 
( M ... N
)  e.  Fin )
92 mptexg 5514 . . . 4  |-  ( ( M ... N )  e.  Fin  ->  (
u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  e. 
_V )
93 nfcv 2228 . . . . 5  |-  F/_ f
( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )
94 nfv 1466 . . . . . 6  |-  F/ f ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) ) : ( M ... N
)
-1-1-onto-> ( M ... N )
95 nfv 1466 . . . . . 6  |-  F/ f A. x  e.  ( M ... K ) ( ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) ) `  x )  =  x
96 nfcsb1v 2963 . . . . . . . . 9  |-  F/_ f [_ ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  / 
f ]_ P
9746, 47, 96nfseq 9857 . . . . . . . 8  |-  F/_ f  seq M (  .+  ,  [_ ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  / 
f ]_ P )
9897, 50nffv 5309 . . . . . . 7  |-  F/_ f
(  seq M (  .+  ,  [_ ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  / 
f ]_ P ) `  N )
9998nfeq1 2238 . . . . . 6  |-  F/ f (  seq M ( 
.+  ,  [_ (
u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  / 
f ]_ P ) `  N )  =  (  seq M (  .+  ,  L ) `  N
)
10094, 95, 99nf3an 1503 . . . . 5  |-  F/ f ( ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) ) : ( M ... N
)
-1-1-onto-> ( M ... N )  /\  A. x  e.  ( M ... K
) ( ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K , 
( J `  (
u  -  1 ) ) ) ,  ( J `  u ) ) ) `  x
)  =  x  /\  (  seq M (  .+  ,  [_ ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  / 
f ]_ P ) `  N )  =  (  seq M (  .+  ,  L ) `  N
) )
101 f1oeq1 5238 . . . . . 6  |-  ( f  =  ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  -> 
( f : ( M ... N ) -1-1-onto-> ( M ... N )  <-> 
( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) ) : ( M ... N
)
-1-1-onto-> ( M ... N ) ) )
102 fveq1 5298 . . . . . . . 8  |-  ( f  =  ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  -> 
( f `  x
)  =  ( ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) ) `  x ) )
103102eqeq1d 2096 . . . . . . 7  |-  ( f  =  ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  -> 
( ( f `  x )  =  x  <-> 
( ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) ) `  x )  =  x ) )
104103ralbidv 2380 . . . . . 6  |-  ( f  =  ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  -> 
( A. x  e.  ( M ... K
) ( f `  x )  =  x  <->  A. x  e.  ( M ... K ) ( ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) ) `  x )  =  x ) )
105 csbeq1a 2941 . . . . . . . . 9  |-  ( f  =  ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  ->  P  =  [_ ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K , 
( J `  (
u  -  1 ) ) ) ,  ( J `  u ) ) )  /  f ]_ P )
106105seqeq3d 9854 . . . . . . . 8  |-  ( f  =  ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  ->  seq M (  .+  ,  P )  =  seq M (  .+  ,  [_ ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  / 
f ]_ P ) )
107106fveq1d 5301 . . . . . . 7  |-  ( f  =  ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  -> 
(  seq M (  .+  ,  P ) `  N
)  =  (  seq M (  .+  ,  [_ ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  / 
f ]_ P ) `  N ) )
108107eqeq1d 2096 . . . . . 6  |-  ( f  =  ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  -> 
( (  seq M
(  .+  ,  P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N )  <->  (  seq M (  .+  ,  [_ ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  / 
f ]_ P ) `  N )  =  (  seq M (  .+  ,  L ) `  N
) ) )
109101, 104, 1083anbi123d 1248 . . . . 5  |-  ( f  =  ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  -> 
( ( f : ( M ... N
)
-1-1-onto-> ( M ... N )  /\  A. x  e.  ( M ... K
) ( f `  x )  =  x  /\  (  seq M
(  .+  ,  P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N ) )  <-> 
( ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) ) : ( M ... N
)
-1-1-onto-> ( M ... N )  /\  A. x  e.  ( M ... K
) ( ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K , 
( J `  (
u  -  1 ) ) ) ,  ( J `  u ) ) ) `  x
)  =  x  /\  (  seq M (  .+  ,  [_ ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  / 
f ]_ P ) `  N )  =  (  seq M (  .+  ,  L ) `  N
) ) ) )
11093, 100, 109spcegf 2702 . . . 4  |-  ( ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  e. 
_V  ->  ( ( ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) ) : ( M ... N
)
-1-1-onto-> ( M ... N )  /\  A. x  e.  ( M ... K
) ( ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K , 
( J `  (
u  -  1 ) ) ) ,  ( J `  u ) ) ) `  x
)  =  x  /\  (  seq M (  .+  ,  [_ ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  / 
f ]_ P ) `  N )  =  (  seq M (  .+  ,  L ) `  N
) )  ->  E. f
( f : ( M ... N ) -1-1-onto-> ( M ... N )  /\  A. x  e.  ( M ... K
) ( f `  x )  =  x  /\  (  seq M
(  .+  ,  P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N ) ) ) )
11191, 92, 1103syl 17 . . 3  |-  ( (
ph  /\  -.  K  =  ( `' J `  K ) )  -> 
( ( ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K , 
( J `  (
u  -  1 ) ) ) ,  ( J `  u ) ) ) : ( M ... N ) -1-1-onto-> ( M ... N )  /\  A. x  e.  ( M ... K
) ( ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K , 
( J `  (
u  -  1 ) ) ) ,  ( J `  u ) ) ) `  x
)  =  x  /\  (  seq M (  .+  ,  [_ ( u  e.  ( M ... N
)  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )  / 
f ]_ P ) `  N )  =  (  seq M (  .+  ,  L ) `  N
) )  ->  E. f
( f : ( M ... N ) -1-1-onto-> ( M ... N )  /\  A. x  e.  ( M ... K
) ( f `  x )  =  x  /\  (  seq M
(  .+  ,  P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N ) ) ) )
11268, 70, 88, 111mp3and 1276 . 2  |-  ( (
ph  /\  -.  K  =  ( `' J `  K ) )  ->  E. f ( f : ( M ... N
)
-1-1-onto-> ( M ... N )  /\  A. x  e.  ( M ... K
) ( f `  x )  =  x  /\  (  seq M
(  .+  ,  P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N ) ) )
113 f1ocnv 5260 . . . . . . 7  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  `' J : ( M ... N ) -1-1-onto-> ( M ... N
) )
114 f1of 5247 . . . . . . 7  |-  ( `' J : ( M ... N ) -1-1-onto-> ( M ... N )  ->  `' J : ( M ... N ) --> ( M ... N ) )
1151, 113, 1143syl 17 . . . . . 6  |-  ( ph  ->  `' J : ( M ... N ) --> ( M ... N ) )
116115, 4ffvelrnd 5429 . . . . 5  |-  ( ph  ->  ( `' J `  K )  e.  ( M ... N ) )
117 elfzelz 9430 . . . . 5  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  ( `' J `  K )  e.  ZZ )
118116, 117syl 14 . . . 4  |-  ( ph  ->  ( `' J `  K )  e.  ZZ )
119 zdceq 8812 . . . 4  |-  ( ( K  e.  ZZ  /\  ( `' J `  K )  e.  ZZ )  -> DECID  K  =  ( `' J `  K ) )
12024, 118, 119syl2anc 403 . . 3  |-  ( ph  -> DECID  K  =  ( `' J `  K ) )
121 exmiddc 782 . . 3  |-  (DECID  K  =  ( `' J `  K )  ->  ( K  =  ( `' J `  K )  \/  -.  K  =  ( `' J `  K ) ) )
122120, 121syl 14 . 2  |-  ( ph  ->  ( K  =  ( `' J `  K )  \/  -.  K  =  ( `' J `  K ) ) )
12364, 112, 122mpjaodan 747 1  |-  ( ph  ->  E. f ( f : ( M ... N ) -1-1-onto-> ( M ... N
)  /\  A. x  e.  ( M ... K
) ( f `  x )  =  x  /\  (  seq M
(  .+  ,  P
) `  N )  =  (  seq M ( 
.+  ,  L ) `
 N ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664  DECID wdc 780    /\ w3a 924    = wceq 1289   E.wex 1426    e. wcel 1438    =/= wne 2255   A.wral 2359   _Vcvv 2619   [_csb 2933    u. cun 2997   ifcif 3391   {csn 3444   class class class wbr 3843    |-> cmpt 3897   `'ccnv 4435   -->wf 5006   -1-1-onto->wf1o 5009   ` cfv 5010  (class class class)co 5644   Fincfn 6447   1c1 7341    <_ cle 7513    - cmin 7643   ZZcz 8740   ZZ>=cuz 9009   ...cfz 9414  ..^cfzo 9541    seqcseq 9840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-iinf 4401  ax-cnex 7426  ax-resscn 7427  ax-1cn 7428  ax-1re 7429  ax-icn 7430  ax-addcl 7431  ax-addrcl 7432  ax-mulcl 7433  ax-addcom 7435  ax-addass 7437  ax-distr 7439  ax-i2m1 7440  ax-0lt1 7441  ax-0id 7443  ax-rnegex 7444  ax-cnre 7446  ax-pre-ltirr 7447  ax-pre-ltwlin 7448  ax-pre-lttrn 7449  ax-pre-apti 7450  ax-pre-ltadd 7451
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3392  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-id 4118  df-iord 4191  df-on 4193  df-ilim 4194  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-riota 5600  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-1st 5903  df-2nd 5904  df-recs 6062  df-frec 6148  df-1o 6173  df-er 6282  df-en 6448  df-fin 6450  df-pnf 7514  df-mnf 7515  df-xr 7516  df-ltxr 7517  df-le 7518  df-sub 7645  df-neg 7646  df-inn 8413  df-n0 8664  df-z 8741  df-uz 9010  df-fz 9415  df-fzo 9542  df-iseq 9841  df-seq3 9842
This theorem is referenced by:  seq3f1olemp  9919
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