ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iseqf1olemab Unicode version

Theorem iseqf1olemab 10289
Description: Lemma for seq3f1o 10304. (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
) ) )
iseqf1olemab.a  |-  ( ph  ->  A  e.  ( K ... ( `' J `  K ) ) )
iseqf1olemab.b  |-  ( ph  ->  B  e.  ( K ... ( `' J `  K ) ) )
Assertion
Ref Expression
iseqf1olemab  |-  ( ph  ->  A  =  B )
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 iseqf1olemab
StepHypRef Expression
1 eqtr3 2160 . . . . 5  |-  ( ( B  =  K  /\  A  =  K )  ->  B  =  A )
21eqcomd 2146 . . . 4  |-  ( ( B  =  K  /\  A  =  K )  ->  A  =  B )
32adantll 468 . . 3  |-  ( ( ( ph  /\  B  =  K )  /\  A  =  K )  ->  A  =  B )
4 iseqf1olemqcl.a . . . . . . . 8  |-  ( ph  ->  A  e.  ( M ... N ) )
5 elfzelz 9833 . . . . . . . 8  |-  ( A  e.  ( M ... N )  ->  A  e.  ZZ )
64, 5syl 14 . . . . . . 7  |-  ( ph  ->  A  e.  ZZ )
76zred 9193 . . . . . 6  |-  ( ph  ->  A  e.  RR )
87ltm1d 8710 . . . . 5  |-  ( ph  ->  ( A  -  1 )  <  A )
98ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  ( A  -  1 )  <  A )
107ad2antrr 480 . . . . 5  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  A  e.  RR )
11 peano2rem 8049 . . . . . 6  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
1210, 11syl 14 . . . . 5  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  RR )
13 iseqf1olemab.a . . . . . . . 8  |-  ( ph  ->  A  e.  ( K ... ( `' J `  K ) ) )
14 elfzle2 9835 . . . . . . . 8  |-  ( A  e.  ( K ... ( `' J `  K ) )  ->  A  <_  ( `' J `  K ) )
1513, 14syl 14 . . . . . . 7  |-  ( ph  ->  A  <_  ( `' J `  K )
)
1615ad2antrr 480 . . . . . 6  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  A  <_  ( `' J `  K )
)
17 iseqf1olemqcl.k . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  ( M ... N ) )
18 iseqf1olemqcl.j . . . . . . . . . . . 12  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
19 iseqf1olemnab.q . . . . . . . . . . . 12  |-  Q  =  ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )
2017, 18, 4, 19iseqf1olemqval 10287 . . . . . . . . . . 11  |-  ( ph  ->  ( Q `  A
)  =  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K , 
( J `  ( A  -  1 ) ) ) ,  ( J `  A ) ) )
2113iftrued 3482 . . . . . . . . . . 11  |-  ( ph  ->  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) ,  ( J `  A
) )  =  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) )
2220, 21eqtrd 2173 . . . . . . . . . 10  |-  ( ph  ->  ( Q `  A
)  =  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) )
2322ad2antrr 480 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  ( Q `  A
)  =  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) )
24 iseqf1olemnab.eq . . . . . . . . . . 11  |-  ( ph  ->  ( Q `  A
)  =  ( Q `
 B ) )
2524ad2antrr 480 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  ( Q `  A
)  =  ( Q `
 B ) )
26 iseqf1olemnab.b . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  ( M ... N ) )
2717, 18, 26, 19iseqf1olemqval 10287 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q `  B
)  =  if ( B  e.  ( K ... ( `' J `  K ) ) ,  if ( B  =  K ,  K , 
( J `  ( B  -  1 ) ) ) ,  ( J `  B ) ) )
28 iseqf1olemab.b . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  ( K ... ( `' J `  K ) ) )
2928iftrued 3482 . . . . . . . . . . . . 13  |-  ( ph  ->  if ( B  e.  ( K ... ( `' J `  K ) ) ,  if ( B  =  K ,  K ,  ( J `  ( B  -  1 ) ) ) ,  ( J `  B
) )  =  if ( B  =  K ,  K ,  ( J `  ( B  -  1 ) ) ) )
3027, 29eqtrd 2173 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q `  B
)  =  if ( B  =  K ,  K ,  ( J `  ( B  -  1 ) ) ) )
3130ad2antrr 480 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  ( Q `  B
)  =  if ( B  =  K ,  K ,  ( J `  ( B  -  1 ) ) ) )
32 simplr 520 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  B  =  K )
3332iftrued 3482 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  if ( B  =  K ,  K , 
( J `  ( B  -  1 ) ) )  =  K )
3431, 33eqtrd 2173 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  ( Q `  B
)  =  K )
3525, 34eqtrd 2173 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  ( Q `  A
)  =  K )
36 simpr 109 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  -.  A  =  K )
3736iffalsed 3485 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  if ( A  =  K ,  K , 
( J `  ( A  -  1 ) ) )  =  ( J `  ( A  -  1 ) ) )
3823, 35, 373eqtr3d 2181 . . . . . . . 8  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  K  =  ( J `
 ( A  - 
1 ) ) )
3938fveq2d 5429 . . . . . . 7  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  ( `' J `  K )  =  ( `' J `  ( J `
 ( A  - 
1 ) ) ) )
4018ad2antrr 480 . . . . . . . 8  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
41 elfzel1 9832 . . . . . . . . . . . . 13  |-  ( K  e.  ( M ... N )  ->  M  e.  ZZ )
4217, 41syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  ZZ )
43 elfzel2 9831 . . . . . . . . . . . . 13  |-  ( K  e.  ( M ... N )  ->  N  e.  ZZ )
4417, 43syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  ZZ )
45 peano2zm 9112 . . . . . . . . . . . . 13  |-  ( A  e.  ZZ  ->  ( A  -  1 )  e.  ZZ )
466, 45syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  -  1 )  e.  ZZ )
4742, 44, 463jca 1162 . . . . . . . . . . 11  |-  ( ph  ->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( A  -  1
)  e.  ZZ ) )
4847adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  -.  A  =  K )  ->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( A  -  1 )  e.  ZZ ) )
4942zred 9193 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  RR )
5049adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  A  =  K )  ->  M  e.  RR )
51 elfzelz 9833 . . . . . . . . . . . . . . 15  |-  ( K  e.  ( M ... N )  ->  K  e.  ZZ )
5217, 51syl 14 . . . . . . . . . . . . . 14  |-  ( ph  ->  K  e.  ZZ )
5352zred 9193 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  RR )
5453adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  A  =  K )  ->  K  e.  RR )
5546zred 9193 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  -  1 )  e.  RR )
5655adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  RR )
57 elfzle1 9834 . . . . . . . . . . . . . 14  |-  ( K  e.  ( M ... N )  ->  M  <_  K )
5817, 57syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  M  <_  K )
5958adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  A  =  K )  ->  M  <_  K )
60 simpr 109 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  A  =  K )  ->  -.  A  =  K )
61 eqcom 2142 . . . . . . . . . . . . . . 15  |-  ( A  =  K  <->  K  =  A )
6260, 61sylnib 666 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  A  =  K )  ->  -.  K  =  A )
63 elfzle1 9834 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  ( K ... ( `' J `  K ) )  ->  K  <_  A )
6413, 63syl 14 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  K  <_  A )
65 zleloe 9121 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  A  e.  ZZ )  ->  ( K  <_  A  <->  ( K  <  A  \/  K  =  A )
) )
6652, 6, 65syl2anc 409 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( K  <_  A  <->  ( K  <  A  \/  K  =  A )
) )
6764, 66mpbid 146 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( K  <  A  \/  K  =  A
) )
6867adantr 274 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  A  =  K )  ->  ( K  <  A  \/  K  =  A ) )
6962, 68ecased 1328 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  A  =  K )  ->  K  <  A )
70 zltlem1 9131 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  ZZ  /\  A  e.  ZZ )  ->  ( K  <  A  <->  K  <_  ( A  - 
1 ) ) )
7152, 6, 70syl2anc 409 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( K  <  A  <->  K  <_  ( A  - 
1 ) ) )
7271adantr 274 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  A  =  K )  ->  ( K  <  A  <->  K  <_  ( A  -  1 ) ) )
7369, 72mpbid 146 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  A  =  K )  ->  K  <_  ( A  -  1 ) )
7450, 54, 56, 59, 73letrd 7906 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  A  =  K )  ->  M  <_  ( A  -  1 ) )
757adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  A  =  K )  ->  A  e.  RR )
7644zred 9193 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  RR )
7776adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  A  =  K )  ->  N  e.  RR )
7875lem1d 8711 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  A  =  K )  ->  ( A  -  1 )  <_  A )
79 elfzle2 9835 . . . . . . . . . . . . . 14  |-  ( A  e.  ( M ... N )  ->  A  <_  N )
804, 79syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  A  <_  N )
8180adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  A  =  K )  ->  A  <_  N )
8256, 75, 77, 78, 81letrd 7906 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  A  =  K )  ->  ( A  -  1 )  <_  N )
8374, 82jca 304 . . . . . . . . . 10  |-  ( (
ph  /\  -.  A  =  K )  ->  ( M  <_  ( A  - 
1 )  /\  ( A  -  1 )  <_  N ) )
84 elfz2 9824 . . . . . . . . . 10  |-  ( ( A  -  1 )  e.  ( M ... N )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( A  -  1 )  e.  ZZ )  /\  ( M  <_  ( A  - 
1 )  /\  ( A  -  1 )  <_  N ) ) )
8548, 83, 84sylanbrc 414 . . . . . . . . 9  |-  ( (
ph  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  ( M ... N ) )
8685adantlr 469 . . . . . . . 8  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  ( M ... N ) )
87 f1ocnvfv1 5682 . . . . . . . 8  |-  ( ( J : ( M ... N ) -1-1-onto-> ( M ... N )  /\  ( A  -  1
)  e.  ( M ... N ) )  ->  ( `' J `  ( J `  ( A  -  1 ) ) )  =  ( A  -  1 ) )
8840, 86, 87syl2anc 409 . . . . . . 7  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  ( `' J `  ( J `  ( A  -  1 ) ) )  =  ( A  -  1 ) )
8939, 88eqtrd 2173 . . . . . 6  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  ( `' J `  K )  =  ( A  -  1 ) )
9016, 89breqtrd 3958 . . . . 5  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  A  <_  ( A  -  1 ) )
9110, 12, 90lensymd 7904 . . . 4  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  -.  ( A  - 
1 )  <  A
)
929, 91pm2.21dd 610 . . 3  |-  ( ( ( ph  /\  B  =  K )  /\  -.  A  =  K )  ->  A  =  B )
93 zdceq 9146 . . . . . 6  |-  ( ( A  e.  ZZ  /\  K  e.  ZZ )  -> DECID  A  =  K )
946, 52, 93syl2anc 409 . . . . 5  |-  ( ph  -> DECID  A  =  K )
95 exmiddc 822 . . . . 5  |-  (DECID  A  =  K  ->  ( A  =  K  \/  -.  A  =  K )
)
9694, 95syl 14 . . . 4  |-  ( ph  ->  ( A  =  K  \/  -.  A  =  K ) )
9796adantr 274 . . 3  |-  ( (
ph  /\  B  =  K )  ->  ( A  =  K  \/  -.  A  =  K
) )
983, 92, 97mpjaodan 788 . 2  |-  ( (
ph  /\  B  =  K )  ->  A  =  B )
99 elfzelz 9833 . . . . . . . 8  |-  ( B  e.  ( M ... N )  ->  B  e.  ZZ )
10026, 99syl 14 . . . . . . 7  |-  ( ph  ->  B  e.  ZZ )
101100zred 9193 . . . . . 6  |-  ( ph  ->  B  e.  RR )
102101ltm1d 8710 . . . . 5  |-  ( ph  ->  ( B  -  1 )  <  B )
103102ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  ( B  -  1 )  < 
B )
104101ad2antrr 480 . . . . 5  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  B  e.  RR )
105 peano2rem 8049 . . . . . 6  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
106104, 105syl 14 . . . . 5  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  ( B  -  1 )  e.  RR )
107 elfzle2 9835 . . . . . . . 8  |-  ( B  e.  ( K ... ( `' J `  K ) )  ->  B  <_  ( `' J `  K ) )
10828, 107syl 14 . . . . . . 7  |-  ( ph  ->  B  <_  ( `' J `  K )
)
109108ad2antrr 480 . . . . . 6  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  B  <_  ( `' J `  K ) )
11030ad2antrr 480 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  ( Q `  B )  =  if ( B  =  K ,  K ,  ( J `  ( B  -  1 ) ) ) )
11124eqcomd 2146 . . . . . . . . . . 11  |-  ( ph  ->  ( Q `  B
)  =  ( Q `
 A ) )
112111ad2antrr 480 . . . . . . . . . 10  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  ( Q `  B )  =  ( Q `  A ) )
11322ad2antrr 480 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  ( Q `  A )  =  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) )
114 simpr 109 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  A  =  K )
115114iftrued 3482 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) )  =  K )
116113, 115eqtrd 2173 . . . . . . . . . 10  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  ( Q `  A )  =  K )
117112, 116eqtrd 2173 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  ( Q `  B )  =  K )
118 simplr 520 . . . . . . . . . 10  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  -.  B  =  K )
119118iffalsed 3485 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  if ( B  =  K ,  K ,  ( J `  ( B  -  1 ) ) )  =  ( J `  ( B  -  1 ) ) )
120110, 117, 1193eqtr3d 2181 . . . . . . . 8  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  K  =  ( J `  ( B  -  1 ) ) )
121120fveq2d 5429 . . . . . . 7  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  ( `' J `  K )  =  ( `' J `  ( J `  ( B  -  1 ) ) ) )
12218ad2antrr 480 . . . . . . . 8  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  J :
( M ... N
)
-1-1-onto-> ( M ... N ) )
123 peano2zm 9112 . . . . . . . . . . . . 13  |-  ( B  e.  ZZ  ->  ( B  -  1 )  e.  ZZ )
124100, 123syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  ( B  -  1 )  e.  ZZ )
12542, 44, 1243jca 1162 . . . . . . . . . . 11  |-  ( ph  ->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( B  -  1
)  e.  ZZ ) )
126125adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  K )  ->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( B  -  1 )  e.  ZZ ) )
12749adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  B  =  K )  ->  M  e.  RR )
12853adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  B  =  K )  ->  K  e.  RR )
129101, 105syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  ( B  -  1 )  e.  RR )
130129adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  B  =  K )  ->  ( B  -  1 )  e.  RR )
13158adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  B  =  K )  ->  M  <_  K )
132 simpr 109 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  B  =  K )  ->  -.  B  =  K )
133 eqcom 2142 . . . . . . . . . . . . . . 15  |-  ( B  =  K  <->  K  =  B )
134132, 133sylnib 666 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  B  =  K )  ->  -.  K  =  B )
135 elfzle1 9834 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  ( K ... ( `' J `  K ) )  ->  K  <_  B )
13628, 135syl 14 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  K  <_  B )
137136adantr 274 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  B  =  K )  ->  K  <_  B )
138 zleloe 9121 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  B  e.  ZZ )  ->  ( K  <_  B  <->  ( K  <  B  \/  K  =  B )
) )
13952, 100, 138syl2anc 409 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( K  <_  B  <->  ( K  <  B  \/  K  =  B )
) )
140139adantr 274 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  B  =  K )  ->  ( K  <_  B  <->  ( K  <  B  \/  K  =  B ) ) )
141137, 140mpbid 146 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  B  =  K )  ->  ( K  <  B  \/  K  =  B ) )
142134, 141ecased 1328 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  B  =  K )  ->  K  <  B )
143 zltlem1 9131 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  ZZ  /\  B  e.  ZZ )  ->  ( K  <  B  <->  K  <_  ( B  - 
1 ) ) )
14452, 100, 143syl2anc 409 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( K  <  B  <->  K  <_  ( B  - 
1 ) ) )
145144adantr 274 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  B  =  K )  ->  ( K  <  B  <->  K  <_  ( B  -  1 ) ) )
146142, 145mpbid 146 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  B  =  K )  ->  K  <_  ( B  -  1 ) )
147127, 128, 130, 131, 146letrd 7906 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  K )  ->  M  <_  ( B  -  1 ) )
148101lem1d 8711 . . . . . . . . . . . . 13  |-  ( ph  ->  ( B  -  1 )  <_  B )
149 elfzle2 9835 . . . . . . . . . . . . . 14  |-  ( B  e.  ( M ... N )  ->  B  <_  N )
15026, 149syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  B  <_  N )
151129, 101, 76, 148, 150letrd 7906 . . . . . . . . . . . 12  |-  ( ph  ->  ( B  -  1 )  <_  N )
152151adantr 274 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  K )  ->  ( B  -  1 )  <_  N )
153147, 152jca 304 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  K )  ->  ( M  <_  ( B  - 
1 )  /\  ( B  -  1 )  <_  N ) )
154 elfz2 9824 . . . . . . . . . 10  |-  ( ( B  -  1 )  e.  ( M ... N )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( B  -  1 )  e.  ZZ )  /\  ( M  <_  ( B  - 
1 )  /\  ( B  -  1 )  <_  N ) ) )
155126, 153, 154sylanbrc 414 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  K )  ->  ( B  -  1 )  e.  ( M ... N ) )
156155adantr 274 . . . . . . . 8  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  ( B  -  1 )  e.  ( M ... N
) )
157 f1ocnvfv1 5682 . . . . . . . 8  |-  ( ( J : ( M ... N ) -1-1-onto-> ( M ... N )  /\  ( B  -  1
)  e.  ( M ... N ) )  ->  ( `' J `  ( J `  ( B  -  1 ) ) )  =  ( B  -  1 ) )
158122, 156, 157syl2anc 409 . . . . . . 7  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  ( `' J `  ( J `  ( B  -  1 ) ) )  =  ( B  -  1 ) )
159121, 158eqtrd 2173 . . . . . 6  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  ( `' J `  K )  =  ( B  - 
1 ) )
160109, 159breqtrd 3958 . . . . 5  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  B  <_  ( B  -  1 ) )
161104, 106, 160lensymd 7904 . . . 4  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  -.  ( B  -  1 )  <  B )
162103, 161pm2.21dd 610 . . 3  |-  ( ( ( ph  /\  -.  B  =  K )  /\  A  =  K
)  ->  A  =  B )
1636zcnd 9194 . . . . 5  |-  ( ph  ->  A  e.  CC )
164163ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  A  e.  CC )
165100zcnd 9194 . . . . 5  |-  ( ph  ->  B  e.  CC )
166165ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  B  e.  CC )
167 1cnd 7802 . . . 4  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  1  e.  CC )
16824ad2antrr 480 . . . . . 6  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  ( Q `  A )  =  ( Q `  B ) )
16922ad2antrr 480 . . . . . . 7  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  ( Q `  A )  =  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) )
170 simpr 109 . . . . . . . 8  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  -.  A  =  K )
171170iffalsed 3485 . . . . . . 7  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) )  =  ( J `  ( A  -  1 ) ) )
172169, 171eqtrd 2173 . . . . . 6  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  ( Q `  A )  =  ( J `  ( A  -  1 ) ) )
17330ad2antrr 480 . . . . . . 7  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  ( Q `  B )  =  if ( B  =  K ,  K ,  ( J `  ( B  -  1 ) ) ) )
174 simplr 520 . . . . . . . 8  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  -.  B  =  K )
175174iffalsed 3485 . . . . . . 7  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  if ( B  =  K ,  K ,  ( J `  ( B  -  1 ) ) )  =  ( J `  ( B  -  1 ) ) )
176173, 175eqtrd 2173 . . . . . 6  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  ( Q `  B )  =  ( J `  ( B  -  1 ) ) )
177168, 172, 1763eqtr3d 2181 . . . . 5  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  ( J `  ( A  -  1 ) )  =  ( J `  ( B  -  1 ) ) )
178 f1of1 5370 . . . . . . . 8  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  J :
( M ... N
) -1-1-> ( M ... N ) )
17918, 178syl 14 . . . . . . 7  |-  ( ph  ->  J : ( M ... N ) -1-1-> ( M ... N ) )
180179ad2antrr 480 . . . . . 6  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  J :
( M ... N
) -1-1-> ( M ... N ) )
18185adantlr 469 . . . . . 6  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  ( M ... N
) )
182155adantr 274 . . . . . 6  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  ( B  -  1 )  e.  ( M ... N
) )
183 f1veqaeq 5674 . . . . . 6  |-  ( ( J : ( M ... N ) -1-1-> ( M ... N )  /\  ( ( A  -  1 )  e.  ( M ... N
)  /\  ( B  -  1 )  e.  ( M ... N
) ) )  -> 
( ( J `  ( A  -  1
) )  =  ( J `  ( B  -  1 ) )  ->  ( A  - 
1 )  =  ( B  -  1 ) ) )
184180, 181, 182, 183syl12anc 1215 . . . . 5  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  ( ( J `  ( A  -  1 ) )  =  ( J `  ( B  -  1
) )  ->  ( A  -  1 )  =  ( B  - 
1 ) ) )
185177, 184mpd 13 . . . 4  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  ( A  -  1 )  =  ( B  -  1 ) )
186164, 166, 167, 185subcan2d 8135 . . 3  |-  ( ( ( ph  /\  -.  B  =  K )  /\  -.  A  =  K )  ->  A  =  B )
18796adantr 274 . . 3  |-  ( (
ph  /\  -.  B  =  K )  ->  ( A  =  K  \/  -.  A  =  K
) )
188162, 186, 187mpjaodan 788 . 2  |-  ( (
ph  /\  -.  B  =  K )  ->  A  =  B )
189 zdceq 9146 . . . 4  |-  ( ( B  e.  ZZ  /\  K  e.  ZZ )  -> DECID  B  =  K )
190100, 52, 189syl2anc 409 . . 3  |-  ( ph  -> DECID  B  =  K )
191 exmiddc 822 . . 3  |-  (DECID  B  =  K  ->  ( B  =  K  \/  -.  B  =  K )
)
192190, 191syl 14 . 2  |-  ( ph  ->  ( B  =  K  \/  -.  B  =  K ) )
19398, 188, 192mpjaodan 788 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698  DECID wdc 820    /\ w3a 963    = wceq 1332    e. wcel 1481   ifcif 3475   class class class wbr 3933    |-> cmpt 3993   `'ccnv 4542   -1-1->wf1 5124   -1-1-onto->wf1o 5126   ` cfv 5127  (class class class)co 5778   CCcc 7638   RRcr 7639   1c1 7641    < clt 7820    <_ cle 7821    - cmin 7953   ZZcz 9074   ...cfz 9817
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-cnex 7731  ax-resscn 7732  ax-1cn 7733  ax-1re 7734  ax-icn 7735  ax-addcl 7736  ax-addrcl 7737  ax-mulcl 7738  ax-addcom 7740  ax-addass 7742  ax-distr 7744  ax-i2m1 7745  ax-0lt1 7746  ax-0id 7748  ax-rnegex 7749  ax-cnre 7751  ax-pre-ltirr 7752  ax-pre-ltwlin 7753  ax-pre-lttrn 7754  ax-pre-ltadd 7756
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-if 3476  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-br 3934  df-opab 3994  df-mpt 3995  df-id 4219  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-riota 5734  df-ov 5781  df-oprab 5782  df-mpo 5783  df-pnf 7822  df-mnf 7823  df-xr 7824  df-ltxr 7825  df-le 7826  df-sub 7955  df-neg 7956  df-inn 8741  df-n0 8998  df-z 9075  df-uz 9347  df-fz 9818
This theorem is referenced by:  iseqf1olemmo  10292
  Copyright terms: Public domain W3C validator