ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iseqf1olemnanb Unicode version

Theorem iseqf1olemnanb 10270
Description: Lemma for seq3f1o 10284. (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
) ) )
iseqf1olemnanb.a  |-  ( ph  ->  -.  A  e.  ( K ... ( `' J `  K ) ) )
iseqf1olemnanb.b  |-  ( ph  ->  -.  B  e.  ( K ... ( `' J `  K ) ) )
Assertion
Ref Expression
iseqf1olemnanb  |-  ( 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 iseqf1olemnanb
StepHypRef Expression
1 iseqf1olemnab.eq . . 3  |-  ( ph  ->  ( Q `  A
)  =  ( Q `
 B ) )
2 iseqf1olemqcl.k . . . . 5  |-  ( ph  ->  K  e.  ( M ... N ) )
3 iseqf1olemqcl.j . . . . 5  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
4 iseqf1olemqcl.a . . . . 5  |-  ( ph  ->  A  e.  ( M ... N ) )
5 iseqf1olemnab.q . . . . 5  |-  Q  =  ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )
62, 3, 4, 5iseqf1olemqval 10267 . . . 4  |-  ( ph  ->  ( Q `  A
)  =  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K , 
( J `  ( A  -  1 ) ) ) ,  ( J `  A ) ) )
7 iseqf1olemnanb.a . . . . 5  |-  ( ph  ->  -.  A  e.  ( K ... ( `' J `  K ) ) )
87iffalsed 3484 . . . 4  |-  ( ph  ->  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) ,  ( J `  A
) )  =  ( J `  A ) )
96, 8eqtrd 2172 . . 3  |-  ( ph  ->  ( Q `  A
)  =  ( J `
 A ) )
10 iseqf1olemnab.b . . . . 5  |-  ( ph  ->  B  e.  ( M ... N ) )
112, 3, 10, 5iseqf1olemqval 10267 . . . 4  |-  ( ph  ->  ( Q `  B
)  =  if ( B  e.  ( K ... ( `' J `  K ) ) ,  if ( B  =  K ,  K , 
( J `  ( B  -  1 ) ) ) ,  ( J `  B ) ) )
12 iseqf1olemnanb.b . . . . 5  |-  ( ph  ->  -.  B  e.  ( K ... ( `' J `  K ) ) )
1312iffalsed 3484 . . . 4  |-  ( ph  ->  if ( B  e.  ( K ... ( `' J `  K ) ) ,  if ( B  =  K ,  K ,  ( J `  ( B  -  1 ) ) ) ,  ( J `  B
) )  =  ( J `  B ) )
1411, 13eqtrd 2172 . . 3  |-  ( ph  ->  ( Q `  B
)  =  ( J `
 B ) )
151, 9, 143eqtr3d 2180 . 2  |-  ( ph  ->  ( J `  A
)  =  ( J `
 B ) )
16 f1of1 5366 . . . 4  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  J :
( M ... N
) -1-1-> ( M ... N ) )
173, 16syl 14 . . 3  |-  ( ph  ->  J : ( M ... N ) -1-1-> ( M ... N ) )
18 f1veqaeq 5670 . . 3  |-  ( ( J : ( M ... N ) -1-1-> ( M ... N )  /\  ( A  e.  ( M ... N
)  /\  B  e.  ( M ... N ) ) )  ->  (
( J `  A
)  =  ( J `
 B )  ->  A  =  B )
)
1917, 4, 10, 18syl12anc 1214 . 2  |-  ( ph  ->  ( ( J `  A )  =  ( J `  B )  ->  A  =  B ) )
2015, 19mpd 13 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1331    e. wcel 1480   ifcif 3474    |-> cmpt 3989   `'ccnv 4538   -1-1->wf1 5120   -1-1-onto->wf1o 5122   ` cfv 5123  (class class class)co 5774   1c1 7628    - cmin 7940   ...cfz 9797
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-ltadd 7743
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-inn 8728  df-n0 8985  df-z 9062  df-uz 9334  df-fz 9798
This theorem is referenced by:  iseqf1olemmo  10272
  Copyright terms: Public domain W3C validator