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Theorem iseqf1olemnab 10527
Description: Lemma for seq3f1o 10543. (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 276 . . 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 10526 . . . . . 6  |-  ( ph  ->  ( Q `  A
)  =  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K , 
( J `  ( A  -  1 ) ) ) ,  ( J `  A ) ) )
87adantr 276 . . . . 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 529 . . . . . 6  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  A  e.  ( K ... ( `' J `  K ) ) )
109iftrued 3556 . . . . 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 2222 . . . 4  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( Q `  A
)  =  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) )
12 f1ocnvfv2 5803 . . . . . . . 8  |-  ( ( J : ( M ... N ) -1-1-onto-> ( M ... N )  /\  K  e.  ( M ... N ) )  -> 
( J `  ( `' J `  K ) )  =  K )
134, 3, 12syl2anc 411 . . . . . . 7  |-  ( ph  ->  ( J `  ( `' J `  K ) )  =  K )
1413ad2antrr 488 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  ( J `  ( `' J `  K )
)  =  K )
15 f1ofn 5484 . . . . . . . . 9  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  J  Fn  ( M ... N ) )
164, 15syl 14 . . . . . . . 8  |-  ( ph  ->  J  Fn  ( M ... N ) )
1716ad2antrr 488 . . . . . . 7  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  J  Fn  ( M ... N
) )
18 elfzuz 10057 . . . . . . . . . 10  |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M )
)
19 fzss1 10099 . . . . . . . . . 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 5496 . . . . . . . . . . . 12  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  `' J : ( M ... N ) -1-1-onto-> ( M ... N
) )
22 f1of 5483 . . . . . . . . . . . 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, 3ffvelcdmd 5676 . . . . . . . . . 10  |-  ( ph  ->  ( `' J `  K )  e.  ( M ... N ) )
25 elfzuz3 10058 . . . . . . . . . 10  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  ( `' J `  K )
) )
26 fzss2 10100 . . . . . . . . . 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 3180 . . . . . . . 8  |-  ( ph  ->  ( K ... ( `' J `  K ) )  C_  ( M ... N ) )
2928ad2antrr 488 . . . . . . 7  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  ( K ... ( `' J `  K ) )  C_  ( M ... N ) )
30 elfzubelfz 10072 . . . . . . . . 9  |-  ( A  e.  ( K ... ( `' J `  K ) )  ->  ( `' J `  K )  e.  ( K ... ( `' J `  K ) ) )
3130adantr 276 . . . . . . . 8  |-  ( ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  ( `' J `  K )  e.  ( K ... ( `' J `  K ) ) )
3231ad2antlr 489 . . . . . . 7  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  ( `' J `  K )  e.  ( K ... ( `' J `  K ) ) )
33 fnfvima 5775 . . . . . . 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 1249 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  ( J `  ( `' J `  K )
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
3514, 34eqeltrrd 2267 . . . . 5  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  K  e.  ( J " ( K ... ( `' J `  K ) ) ) )
3616ad2antrr 488 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  J  Fn  ( M ... N
) )
3728ad2antrr 488 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( K ... ( `' J `  K ) )  C_  ( M ... N ) )
383adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  K  e.  ( M ... N ) )
39 elfzelz 10061 . . . . . . . . . 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 276 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  K  e.  ZZ )
4224ad2antrr 488 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( `' J `  K )  e.  ( M ... N ) )
43 elfzelz 10061 . . . . . . . . 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 276 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  A  e.  ( M ... N ) )
46 elfzelz 10061 . . . . . . . . . . 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 276 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  A  e.  ZZ )
49 peano2zm 9326 . . . . . . . . 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 1179 . . . . . . 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 110 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  -.  A  =  K )
53 eqcom 2191 . . . . . . . . . . 11  |-  ( A  =  K  <->  K  =  A )
5452, 53sylnib 677 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  -.  K  =  A )
559adantr 276 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  A  e.  ( K ... ( `' J `  K ) ) )
56 elfzle1 10063 . . . . . . . . . . . 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 9335 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  A  e.  ZZ )  ->  ( K  <_  A  <->  ( K  <  A  \/  K  =  A )
) )
5941, 48, 58syl2anc 411 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( K  <_  A  <->  ( K  <  A  \/  K  =  A ) ) )
6057, 59mpbid 147 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( K  <  A  \/  K  =  A ) )
6154, 60ecased 1360 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  K  <  A )
62 zltlem1 9345 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  A  e.  ZZ )  ->  ( K  <  A  <->  K  <_  ( A  - 
1 ) ) )
6341, 48, 62syl2anc 411 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( K  <  A  <->  K  <_  ( A  -  1 ) ) )
6461, 63mpbid 147 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  K  <_  ( A  -  1 ) )
6550zred 9410 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  RR )
6648zred 9410 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  A  e.  RR )
6744zred 9410 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( `' J `  K )  e.  RR )
6866lem1d 8925 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  <_  A )
69 elfzle2 10064 . . . . . . . . . 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 8116 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  <_  ( `' J `  K ) )
7264, 71jca 306 . . . . . . 7  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( K  <_  ( A  - 
1 )  /\  ( A  -  1 )  <_  ( `' J `  K ) ) )
73 elfz2 10051 . . . . . . 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 417 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  ( K ... ( `' J `  K ) ) )
75 fnfvima 5775 . . . . . 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 1249 . . . . 5  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( J `  ( A  -  1 ) )  e.  ( J "
( K ... ( `' J `  K ) ) ) )
77 zdceq 9363 . . . . . 6  |-  ( ( A  e.  ZZ  /\  K  e.  ZZ )  -> DECID  A  =  K )
7847, 40, 77syl2anc 411 . . . . 5  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> DECID  A  =  K )
7935, 76, 78ifcldadc 3578 . . . 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 2266 . . 3  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( Q `  A
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
812, 80eqeltrrd 2267 . 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 10526 . . . . 5  |-  ( ph  ->  ( Q `  B
)  =  if ( B  e.  ( K ... ( `' J `  K ) ) ,  if ( B  =  K ,  K , 
( J `  ( B  -  1 ) ) ) ,  ( J `  B ) ) )
8483adantr 276 . . . 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 531 . . . . 5  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  -.  B  e.  ( K ... ( `' J `  K ) ) )
8685iffalsed 3559 . . . 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 2222 . . 3  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( Q `  B
)  =  ( J `
 B ) )
88 f1of1 5482 . . . . . . 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 5798 . . . . . 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 1249 . . . . 5  |-  ( ph  ->  ( ( J `  B )  e.  ( J " ( K ... ( `' J `  K ) ) )  <-> 
B  e.  ( K ... ( `' J `  K ) ) ) )
9291adantr 276 . . . 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 674 . . 3  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  -.  ( J `  B
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
9487, 93eqneltrd 2285 . 2  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  -.  ( Q `  B
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
9581, 94pm2.65da 662 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 104    <-> wb 105    \/ wo 709  DECID wdc 835    /\ w3a 980    = wceq 1364    e. wcel 2160    C_ wss 3144   ifcif 3549   class class class wbr 4021    |-> cmpt 4082   `'ccnv 4646   "cima 4650    Fn wfn 5233   -->wf 5234   -1-1->wf1 5235   -1-1-onto->wf1o 5237   ` cfv 5238  (class class class)co 5900   1c1 7847    < clt 8027    <_ cle 8028    - cmin 8163   ZZcz 9288   ZZ>=cuz 9563   ...cfz 10044
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-addcom 7946  ax-addass 7948  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-0id 7954  ax-rnegex 7955  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-ltadd 7962
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-sub 8165  df-neg 8166  df-inn 8955  df-n0 9212  df-z 9289  df-uz 9564  df-fz 10045
This theorem is referenced by:  iseqf1olemmo  10531
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