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Theorem iseqf1olemnanb 9973
Description: Lemma for seq3f1o 9987. (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 9970 . . . 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 3407 . . . 4  |-  ( ph  ->  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) ,  ( J `  A
) )  =  ( J `  A ) )
96, 8eqtrd 2121 . . 3  |-  ( ph  ->  ( Q `  A
)  =  ( J `
 A ) )
10 iseqf1olemnab.b . . . . 5  |-  ( ph  ->  B  e.  ( M ... N ) )
112, 3, 10, 5iseqf1olemqval 9970 . . . 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 3407 . . . 4  |-  ( ph  ->  if ( B  e.  ( K ... ( `' J `  K ) ) ,  if ( B  =  K ,  K ,  ( J `  ( B  -  1 ) ) ) ,  ( J `  B
) )  =  ( J `  B ) )
1411, 13eqtrd 2121 . . 3  |-  ( ph  ->  ( Q `  B
)  =  ( J `
 B ) )
151, 9, 143eqtr3d 2129 . 2  |-  ( ph  ->  ( J `  A
)  =  ( J `
 B ) )
16 f1of1 5265 . . . 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 5562 . . 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 1173 . 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 1290    e. wcel 1439   ifcif 3397    |-> cmpt 3905   `'ccnv 4450   -1-1->wf1 5025   -1-1-onto->wf1o 5027   ` cfv 5028  (class class class)co 5666   1c1 7405    - cmin 7707   ...cfz 9478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-addcom 7499  ax-addass 7501  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-0id 7507  ax-rnegex 7508  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-ltadd 7515
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-inn 8477  df-n0 8728  df-z 8805  df-uz 9074  df-fz 9479
This theorem is referenced by:  iseqf1olemmo  9975
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