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Theorem iseqf1olemnab 10102
Description: Lemma for seq3f1o 10118. (Contributed by Jim Kingdon, 27-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 ) )
iseqf1olemnab.b  |-  ( ph  ->  B  e.  ( M ... N ) )
iseqf1olemnab.eq  |-  ( ph  ->  ( Q `  A
)  =  ( Q `
 B ) )
iseqf1olemnab.q  |-  Q  =  ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )
Assertion
Ref Expression
iseqf1olemnab  |-  ( ph  ->  -.  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )
Distinct variable groups:    u, A    u, B    u, J    u, K    u, M    u, N
Allowed substitution hints:    ph( u)    Q( u)

Proof of Theorem iseqf1olemnab
StepHypRef Expression
1 iseqf1olemnab.eq . . . 4  |-  ( ph  ->  ( Q `  A
)  =  ( Q `
 B ) )
21adantr 272 . . 3  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( Q `  A
)  =  ( Q `
 B ) )
3 iseqf1olemqcl.k . . . . . . 7  |-  ( ph  ->  K  e.  ( M ... N ) )
4 iseqf1olemqcl.j . . . . . . 7  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
5 iseqf1olemqcl.a . . . . . . 7  |-  ( ph  ->  A  e.  ( M ... N ) )
6 iseqf1olemnab.q . . . . . . 7  |-  Q  =  ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )
73, 4, 5, 6iseqf1olemqval 10101 . . . . . 6  |-  ( ph  ->  ( Q `  A
)  =  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K , 
( J `  ( A  -  1 ) ) ) ,  ( J `  A ) ) )
87adantr 272 . . . . 5  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( Q `  A
)  =  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K , 
( J `  ( A  -  1 ) ) ) ,  ( J `  A ) ) )
9 simprl 501 . . . . . 6  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  A  e.  ( K ... ( `' J `  K ) ) )
109iftrued 3428 . . . . 5  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) ,  ( J `  A
) )  =  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) )
118, 10eqtrd 2132 . . . 4  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( Q `  A
)  =  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) )
12 f1ocnvfv2 5611 . . . . . . . 8  |-  ( ( J : ( M ... N ) -1-1-onto-> ( M ... N )  /\  K  e.  ( M ... N ) )  -> 
( J `  ( `' J `  K ) )  =  K )
134, 3, 12syl2anc 406 . . . . . . 7  |-  ( ph  ->  ( J `  ( `' J `  K ) )  =  K )
1413ad2antrr 475 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  ( J `  ( `' J `  K )
)  =  K )
15 f1ofn 5302 . . . . . . . . 9  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  J  Fn  ( M ... N ) )
164, 15syl 14 . . . . . . . 8  |-  ( ph  ->  J  Fn  ( M ... N ) )
1716ad2antrr 475 . . . . . . 7  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  J  Fn  ( M ... N
) )
18 elfzuz 9643 . . . . . . . . . 10  |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M )
)
19 fzss1 9684 . . . . . . . . . 10  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( K ... ( `' J `  K ) )  C_  ( M ... ( `' J `  K ) ) )
203, 18, 193syl 17 . . . . . . . . 9  |-  ( ph  ->  ( K ... ( `' J `  K ) )  C_  ( M ... ( `' J `  K ) ) )
21 f1ocnv 5314 . . . . . . . . . . . 12  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  `' J : ( M ... N ) -1-1-onto-> ( M ... N
) )
22 f1of 5301 . . . . . . . . . . . 12  |-  ( `' J : ( M ... N ) -1-1-onto-> ( M ... N )  ->  `' J : ( M ... N ) --> ( M ... N ) )
234, 21, 223syl 17 . . . . . . . . . . 11  |-  ( ph  ->  `' J : ( M ... N ) --> ( M ... N ) )
2423, 3ffvelrnd 5488 . . . . . . . . . 10  |-  ( ph  ->  ( `' J `  K )  e.  ( M ... N ) )
25 elfzuz3 9644 . . . . . . . . . 10  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  ( `' J `  K )
) )
26 fzss2 9685 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  ( `' J `  K ) )  ->  ( M ... ( `' J `  K ) )  C_  ( M ... N ) )
2724, 25, 263syl 17 . . . . . . . . 9  |-  ( ph  ->  ( M ... ( `' J `  K ) )  C_  ( M ... N ) )
2820, 27sstrd 3057 . . . . . . . 8  |-  ( ph  ->  ( K ... ( `' J `  K ) )  C_  ( M ... N ) )
2928ad2antrr 475 . . . . . . 7  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  ( K ... ( `' J `  K ) )  C_  ( M ... N ) )
30 elfzubelfz 9657 . . . . . . . . 9  |-  ( A  e.  ( K ... ( `' J `  K ) )  ->  ( `' J `  K )  e.  ( K ... ( `' J `  K ) ) )
3130adantr 272 . . . . . . . 8  |-  ( ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  ( `' J `  K )  e.  ( K ... ( `' J `  K ) ) )
3231ad2antlr 476 . . . . . . 7  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  ( `' J `  K )  e.  ( K ... ( `' J `  K ) ) )
33 fnfvima 5584 . . . . . . 7  |-  ( ( J  Fn  ( M ... N )  /\  ( K ... ( `' J `  K ) )  C_  ( M ... N )  /\  ( `' J `  K )  e.  ( K ... ( `' J `  K ) ) )  ->  ( J `  ( `' J `  K )
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
3417, 29, 32, 33syl3anc 1184 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  ( J `  ( `' J `  K )
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
3514, 34eqeltrrd 2177 . . . . 5  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  K  e.  ( J " ( K ... ( `' J `  K ) ) ) )
3616ad2antrr 475 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  J  Fn  ( M ... N
) )
3728ad2antrr 475 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( K ... ( `' J `  K ) )  C_  ( M ... N ) )
383adantr 272 . . . . . . . . . 10  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  K  e.  ( M ... N ) )
39 elfzelz 9647 . . . . . . . . . 10  |-  ( K  e.  ( M ... N )  ->  K  e.  ZZ )
4038, 39syl 14 . . . . . . . . 9  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  K  e.  ZZ )
4140adantr 272 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  K  e.  ZZ )
4224ad2antrr 475 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( `' J `  K )  e.  ( M ... N ) )
43 elfzelz 9647 . . . . . . . . 9  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  ( `' J `  K )  e.  ZZ )
4442, 43syl 14 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( `' J `  K )  e.  ZZ )
455adantr 272 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  A  e.  ( M ... N ) )
46 elfzelz 9647 . . . . . . . . . . 11  |-  ( A  e.  ( M ... N )  ->  A  e.  ZZ )
4745, 46syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  A  e.  ZZ )
4847adantr 272 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  A  e.  ZZ )
49 peano2zm 8944 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( A  -  1 )  e.  ZZ )
5048, 49syl 14 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  ZZ )
5141, 44, 503jca 1129 . . . . . . 7  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( K  e.  ZZ  /\  ( `' J `  K )  e.  ZZ  /\  ( A  -  1 )  e.  ZZ ) )
52 simpr 109 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  -.  A  =  K )
53 eqcom 2102 . . . . . . . . . . 11  |-  ( A  =  K  <->  K  =  A )
5452, 53sylnib 642 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  -.  K  =  A )
559adantr 272 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  A  e.  ( K ... ( `' J `  K ) ) )
56 elfzle1 9648 . . . . . . . . . . . 12  |-  ( A  e.  ( K ... ( `' J `  K ) )  ->  K  <_  A )
5755, 56syl 14 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  K  <_  A )
58 zleloe 8953 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  A  e.  ZZ )  ->  ( K  <_  A  <->  ( K  <  A  \/  K  =  A )
) )
5941, 48, 58syl2anc 406 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( K  <_  A  <->  ( K  <  A  \/  K  =  A ) ) )
6057, 59mpbid 146 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( K  <  A  \/  K  =  A ) )
6154, 60ecased 1295 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  K  <  A )
62 zltlem1 8963 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  A  e.  ZZ )  ->  ( K  <  A  <->  K  <_  ( A  - 
1 ) ) )
6341, 48, 62syl2anc 406 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( K  <  A  <->  K  <_  ( A  -  1 ) ) )
6461, 63mpbid 146 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  K  <_  ( A  -  1 ) )
6550zred 9025 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  RR )
6648zred 9025 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  A  e.  RR )
6744zred 9025 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( `' J `  K )  e.  RR )
6866lem1d 8549 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  <_  A )
69 elfzle2 9649 . . . . . . . . . 10  |-  ( A  e.  ( K ... ( `' J `  K ) )  ->  A  <_  ( `' J `  K ) )
7055, 69syl 14 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  A  <_  ( `' J `  K ) )
7165, 66, 67, 68, 70letrd 7757 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  <_  ( `' J `  K ) )
7264, 71jca 302 . . . . . . 7  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( K  <_  ( A  - 
1 )  /\  ( A  -  1 )  <_  ( `' J `  K ) ) )
73 elfz2 9638 . . . . . . 7  |-  ( ( A  -  1 )  e.  ( K ... ( `' J `  K ) )  <->  ( ( K  e.  ZZ  /\  ( `' J `  K )  e.  ZZ  /\  ( A  -  1 )  e.  ZZ )  /\  ( K  <_  ( A  -  1 )  /\  ( A  -  1
)  <_  ( `' J `  K )
) ) )
7451, 72, 73sylanbrc 411 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  ( K ... ( `' J `  K ) ) )
75 fnfvima 5584 . . . . . 6  |-  ( ( J  Fn  ( M ... N )  /\  ( K ... ( `' J `  K ) )  C_  ( M ... N )  /\  ( A  -  1 )  e.  ( K ... ( `' J `  K ) ) )  ->  ( J `  ( A  -  1 ) )  e.  ( J "
( K ... ( `' J `  K ) ) ) )
7636, 37, 74, 75syl3anc 1184 . . . . 5  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( J `  ( A  -  1 ) )  e.  ( J "
( K ... ( `' J `  K ) ) ) )
77 zdceq 8978 . . . . . 6  |-  ( ( A  e.  ZZ  /\  K  e.  ZZ )  -> DECID  A  =  K )
7847, 40, 77syl2anc 406 . . . . 5  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> DECID  A  =  K )
7935, 76, 78ifcldadc 3448 . . . 4  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) )  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
8011, 79eqeltrd 2176 . . 3  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( Q `  A
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
812, 80eqeltrrd 2177 . 2  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( Q `  B
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
82 iseqf1olemnab.b . . . . . 6  |-  ( ph  ->  B  e.  ( M ... N ) )
833, 4, 82, 6iseqf1olemqval 10101 . . . . 5  |-  ( ph  ->  ( Q `  B
)  =  if ( B  e.  ( K ... ( `' J `  K ) ) ,  if ( B  =  K ,  K , 
( J `  ( B  -  1 ) ) ) ,  ( J `  B ) ) )
8483adantr 272 . . . 4  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( Q `  B
)  =  if ( B  e.  ( K ... ( `' J `  K ) ) ,  if ( B  =  K ,  K , 
( J `  ( B  -  1 ) ) ) ,  ( J `  B ) ) )
85 simprr 502 . . . . 5  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  -.  B  e.  ( K ... ( `' J `  K ) ) )
8685iffalsed 3431 . . . 4  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  if ( B  e.  ( K ... ( `' J `  K ) ) ,  if ( B  =  K ,  K ,  ( J `  ( B  -  1 ) ) ) ,  ( J `  B
) )  =  ( J `  B ) )
8784, 86eqtrd 2132 . . 3  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( Q `  B
)  =  ( J `
 B ) )
88 f1of1 5300 . . . . . . 7  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  J :
( M ... N
) -1-1-> ( M ... N ) )
894, 88syl 14 . . . . . 6  |-  ( ph  ->  J : ( M ... N ) -1-1-> ( M ... N ) )
90 f1elima 5606 . . . . . 6  |-  ( ( J : ( M ... N ) -1-1-> ( M ... N )  /\  B  e.  ( M ... N )  /\  ( K ... ( `' J `  K ) )  C_  ( M ... N ) )  -> 
( ( J `  B )  e.  ( J " ( K ... ( `' J `  K ) ) )  <-> 
B  e.  ( K ... ( `' J `  K ) ) ) )
9189, 82, 28, 90syl3anc 1184 . . . . 5  |-  ( ph  ->  ( ( J `  B )  e.  ( J " ( K ... ( `' J `  K ) ) )  <-> 
B  e.  ( K ... ( `' J `  K ) ) ) )
9291adantr 272 . . . 4  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( ( J `  B )  e.  ( J " ( K ... ( `' J `  K ) ) )  <-> 
B  e.  ( K ... ( `' J `  K ) ) ) )
9385, 92mtbird 639 . . 3  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  -.  ( J `  B
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
9487, 93eqneltrd 2195 . 2  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  -.  ( Q `  B
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
9581, 94pm2.65da 628 1  |-  ( ph  ->  -.  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 670  DECID wdc 786    /\ w3a 930    = wceq 1299    e. wcel 1448    C_ wss 3021   ifcif 3421   class class class wbr 3875    |-> cmpt 3929   `'ccnv 4476   "cima 4480    Fn wfn 5054   -->wf 5055   -1-1->wf1 5056   -1-1-onto->wf1o 5058   ` cfv 5059  (class class class)co 5706   1c1 7501    < clt 7672    <_ cle 7673    - cmin 7804   ZZcz 8906   ZZ>=cuz 9176   ...cfz 9631
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-ltadd 7611
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177  df-fz 9632
This theorem is referenced by:  iseqf1olemmo  10106
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