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Theorem iseqf1olemnab 10216
Description: Lemma for seq3f1o 10232. (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 274 . . 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 10215 . . . . . 6  |-  ( ph  ->  ( Q `  A
)  =  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K , 
( J `  ( A  -  1 ) ) ) ,  ( J `  A ) ) )
87adantr 274 . . . . 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 505 . . . . . 6  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  A  e.  ( K ... ( `' J `  K ) ) )
109iftrued 3451 . . . . 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 2150 . . . 4  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( Q `  A
)  =  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) )
12 f1ocnvfv2 5647 . . . . . . . 8  |-  ( ( J : ( M ... N ) -1-1-onto-> ( M ... N )  /\  K  e.  ( M ... N ) )  -> 
( J `  ( `' J `  K ) )  =  K )
134, 3, 12syl2anc 408 . . . . . . 7  |-  ( ph  ->  ( J `  ( `' J `  K ) )  =  K )
1413ad2antrr 479 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  ( J `  ( `' J `  K )
)  =  K )
15 f1ofn 5336 . . . . . . . . 9  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  J  Fn  ( M ... N ) )
164, 15syl 14 . . . . . . . 8  |-  ( ph  ->  J  Fn  ( M ... N ) )
1716ad2antrr 479 . . . . . . 7  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  J  Fn  ( M ... N
) )
18 elfzuz 9757 . . . . . . . . . 10  |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M )
)
19 fzss1 9798 . . . . . . . . . 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 5348 . . . . . . . . . . . 12  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  `' J : ( M ... N ) -1-1-onto-> ( M ... N
) )
22 f1of 5335 . . . . . . . . . . . 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 5524 . . . . . . . . . 10  |-  ( ph  ->  ( `' J `  K )  e.  ( M ... N ) )
25 elfzuz3 9758 . . . . . . . . . 10  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  ( `' J `  K )
) )
26 fzss2 9799 . . . . . . . . . 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 3077 . . . . . . . 8  |-  ( ph  ->  ( K ... ( `' J `  K ) )  C_  ( M ... N ) )
2928ad2antrr 479 . . . . . . 7  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  ( K ... ( `' J `  K ) )  C_  ( M ... N ) )
30 elfzubelfz 9771 . . . . . . . . 9  |-  ( A  e.  ( K ... ( `' J `  K ) )  ->  ( `' J `  K )  e.  ( K ... ( `' J `  K ) ) )
3130adantr 274 . . . . . . . 8  |-  ( ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  ( `' J `  K )  e.  ( K ... ( `' J `  K ) ) )
3231ad2antlr 480 . . . . . . 7  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  ( `' J `  K )  e.  ( K ... ( `' J `  K ) ) )
33 fnfvima 5620 . . . . . . 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 1201 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  ( J `  ( `' J `  K )
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
3514, 34eqeltrrd 2195 . . . . 5  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  A  =  K )  ->  K  e.  ( J " ( K ... ( `' J `  K ) ) ) )
3616ad2antrr 479 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  J  Fn  ( M ... N
) )
3728ad2antrr 479 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( K ... ( `' J `  K ) )  C_  ( M ... N ) )
383adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  K  e.  ( M ... N ) )
39 elfzelz 9761 . . . . . . . . . 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 274 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  K  e.  ZZ )
4224ad2antrr 479 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( `' J `  K )  e.  ( M ... N ) )
43 elfzelz 9761 . . . . . . . . 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 274 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  A  e.  ( M ... N ) )
46 elfzelz 9761 . . . . . . . . . . 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 274 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  A  e.  ZZ )
49 peano2zm 9050 . . . . . . . . 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 1146 . . . . . . 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 2119 . . . . . . . . . . 11  |-  ( A  =  K  <->  K  =  A )
5452, 53sylnib 650 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  -.  K  =  A )
559adantr 274 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  A  e.  ( K ... ( `' J `  K ) ) )
56 elfzle1 9762 . . . . . . . . . . . 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 9059 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  A  e.  ZZ )  ->  ( K  <_  A  <->  ( K  <  A  \/  K  =  A )
) )
5941, 48, 58syl2anc 408 . . . . . . . . . . 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 1312 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  K  <  A )
62 zltlem1 9069 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  A  e.  ZZ )  ->  ( K  <  A  <->  K  <_  ( A  - 
1 ) ) )
6341, 48, 62syl2anc 408 . . . . . . . . 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 9131 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  RR )
6648zred 9131 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  A  e.  RR )
6744zred 9131 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( `' J `  K )  e.  RR )
6866lem1d 8655 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  <_  A )
69 elfzle2 9763 . . . . . . . . . 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 7854 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  <_  ( `' J `  K ) )
7264, 71jca 304 . . . . . . 7  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( K  <_  ( A  - 
1 )  /\  ( A  -  1 )  <_  ( `' J `  K ) ) )
73 elfz2 9752 . . . . . . 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 413 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  ( K ... ( `' J `  K ) ) )
75 fnfvima 5620 . . . . . 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 1201 . . . . 5  |-  ( ( ( ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  /\  -.  A  =  K )  ->  ( J `  ( A  -  1 ) )  e.  ( J "
( K ... ( `' J `  K ) ) ) )
77 zdceq 9084 . . . . . 6  |-  ( ( A  e.  ZZ  /\  K  e.  ZZ )  -> DECID  A  =  K )
7847, 40, 77syl2anc 408 . . . . 5  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> DECID  A  =  K )
7935, 76, 78ifcldadc 3471 . . . 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 2194 . . 3  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( Q `  A
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
812, 80eqeltrrd 2195 . 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 10215 . . . . 5  |-  ( ph  ->  ( Q `  B
)  =  if ( B  e.  ( K ... ( `' J `  K ) ) ,  if ( B  =  K ,  K , 
( J `  ( B  -  1 ) ) ) ,  ( J `  B ) ) )
8483adantr 274 . . . 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 506 . . . . 5  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  -.  B  e.  ( K ... ( `' J `  K ) ) )
8685iffalsed 3454 . . . 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 2150 . . 3  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  -> 
( Q `  B
)  =  ( J `
 B ) )
88 f1of1 5334 . . . . . . 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 5642 . . . . . 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 1201 . . . . 5  |-  ( ph  ->  ( ( J `  B )  e.  ( J " ( K ... ( `' J `  K ) ) )  <-> 
B  e.  ( K ... ( `' J `  K ) ) ) )
9291adantr 274 . . . 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 647 . . 3  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  -.  ( J `  B
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
9487, 93eqneltrd 2213 . 2  |-  ( (
ph  /\  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )  ->  -.  ( Q `  B
)  e.  ( J
" ( K ... ( `' J `  K ) ) ) )
9581, 94pm2.65da 635 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 682  DECID wdc 804    /\ w3a 947    = wceq 1316    e. wcel 1465    C_ wss 3041   ifcif 3444   class class class wbr 3899    |-> cmpt 3959   `'ccnv 4508   "cima 4512    Fn wfn 5088   -->wf 5089   -1-1->wf1 5090   -1-1-onto->wf1o 5092   ` cfv 5093  (class class class)co 5742   1c1 7589    < clt 7768    <_ cle 7769    - cmin 7901   ZZcz 9012   ZZ>=cuz 9282   ...cfz 9745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8685  df-n0 8936  df-z 9013  df-uz 9283  df-fz 9746
This theorem is referenced by:  iseqf1olemmo  10220
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