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Theorem iseqf1olemnab 10474
Description: Lemma for seq3f1o 10490. (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 10473 . . . . . 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 3541 . . . . 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 2210 . . . 4  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( Q `  A
)  =  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) )
12 f1ocnvfv2 5773 . . . . . . . 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 5458 . . . . . . . . 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 10007 . . . . . . . . . 10  |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M )
)
19 fzss1 10049 . . . . . . . . . 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 5470 . . . . . . . . . . . 12  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  `' J : ( M ... N ) -1-1-onto-> ( M ... N
) )
22 f1of 5457 . . . . . . . . . . . 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 5648 . . . . . . . . . 10  |-  ( ph  ->  ( `' J `  K )  e.  ( M ... N ) )
25 elfzuz3 10008 . . . . . . . . . 10  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  ( `' J `  K )
) )
26 fzss2 10050 . . . . . . . . . 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 3165 . . . . . . . 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 10022 . . . . . . . . 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 5746 . . . . . . 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 1238 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  ( J `  ( `' J `  K )
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
3514, 34eqeltrrd 2255 . . . . 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 10011 . . . . . . . . . 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 10011 . . . . . . . . 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 10011 . . . . . . . . . . 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 9280 . . . . . . . . 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 1177 . . . . . . 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 2179 . . . . . . . . . . 11  |-  ( A  =  K  <->  K  =  A )
5452, 53sylnib 676 . . . . . . . . . 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 10013 . . . . . . . . . . . 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 9289 . . . . . . . . . . . 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 1349 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  K  <  A )
62 zltlem1 9299 . . . . . . . . . 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 9364 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  RR )
6648zred 9364 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  A  e.  RR )
6744zred 9364 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( `' J `  K )  e.  RR )
6866lem1d 8879 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  <_  A )
69 elfzle2 10014 . . . . . . . . . 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 8071 . . . . . . . 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 10002 . . . . . . 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 5746 . . . . . 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 1238 . . . . 5  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( J `  ( A  -  1 ) )  e.  ( J "
( K ... ( `' J `  K ) ) ) )
77 zdceq 9317 . . . . . 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 3563 . . . 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 2254 . . 3  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( Q `  A
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
812, 80eqeltrrd 2255 . 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 10473 . . . . 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 3544 . . . 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 2210 . . 3  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( Q `  B
)  =  ( J `
 B ) )
88 f1of1 5456 . . . . . . 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 5768 . . . . . 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 1238 . . . . 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 673 . . 3  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  -.  ( J `  B
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
9487, 93eqneltrd 2273 . 2  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  -.  ( Q `  B
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
9581, 94pm2.65da 661 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 708  DECID wdc 834    /\ w3a 978    = wceq 1353    e. wcel 2148    C_ wss 3129   ifcif 3534   class class class wbr 4000    |-> cmpt 4061   `'ccnv 4622   "cima 4626    Fn wfn 5207   -->wf 5208   -1-1->wf1 5209   -1-1-onto->wf1o 5211   ` cfv 5212  (class class class)co 5869   1c1 7803    < clt 7982    <_ cle 7983    - cmin 8118   ZZcz 9242   ZZ>=cuz 9517   ...cfz 9995
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996
This theorem is referenced by:  iseqf1olemmo  10478
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