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Theorem iseqf1olemmo 10294
Description: Lemma for seq3f1o 10306. Showing that  Q is one-to-one. (Contributed by Jim Kingdon, 27-Aug-2022.)
Hypotheses
Ref Expression
iseqf1olemqf.k  |-  ( ph  ->  K  e.  ( M ... N ) )
iseqf1olemqf.j  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
iseqf1olemqf.q  |-  Q  =  ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )
iseqf1olemmo.a  |-  ( ph  ->  A  e.  ( M ... N ) )
iseqf1olemmo.b  |-  ( ph  ->  B  e.  ( M ... N ) )
iseqf1olemmo.eq  |-  ( ph  ->  ( Q `  A
)  =  ( Q `
 B ) )
Assertion
Ref Expression
iseqf1olemmo  |-  ( 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 iseqf1olemmo
StepHypRef Expression
1 iseqf1olemqf.k . . . . 5  |-  ( ph  ->  K  e.  ( M ... N ) )
21ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  K  e.  ( M ... N
) )
3 iseqf1olemqf.j . . . . 5  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
43ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  J : ( M ... N ) -1-1-onto-> ( M ... N
) )
5 iseqf1olemmo.a . . . . 5  |-  ( ph  ->  A  e.  ( M ... N ) )
65ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  A  e.  ( M ... N
) )
7 iseqf1olemmo.b . . . . 5  |-  ( ph  ->  B  e.  ( M ... N ) )
87ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  B  e.  ( M ... N
) )
9 iseqf1olemmo.eq . . . . 5  |-  ( ph  ->  ( Q `  A
)  =  ( Q `
 B ) )
109ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  ( Q `  A )  =  ( Q `  B ) )
11 iseqf1olemqf.q . . . 4  |-  Q  =  ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )
12 simplr 520 . . . 4  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  A  e.  ( K ... ( `' J `  K ) ) )
13 simpr 109 . . . 4  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  B  e.  ( K ... ( `' J `  K ) ) )
142, 4, 6, 8, 10, 11, 12, 13iseqf1olemab 10291 . . 3  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  A  =  B )
15 simplr 520 . . . . 5  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  A  e.  ( K ... ( `' J `  K ) ) )
16 simpr 109 . . . . 5  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  -.  B  e.  ( K ... ( `' J `  K ) ) )
1715, 16jca 304 . . . 4  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )
181, 3, 5, 7, 9, 11iseqf1olemnab 10290 . . . . 5  |-  ( ph  ->  -.  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )
1918ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  -.  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )
2017, 19pm2.21dd 610 . . 3  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  A  =  B )
21 elfzelz 9835 . . . . . . 7  |-  ( B  e.  ( M ... N )  ->  B  e.  ZZ )
227, 21syl 14 . . . . . 6  |-  ( ph  ->  B  e.  ZZ )
23 elfzelz 9835 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  K  e.  ZZ )
241, 23syl 14 . . . . . 6  |-  ( ph  ->  K  e.  ZZ )
25 f1ocnv 5386 . . . . . . . . 9  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  `' J : ( M ... N ) -1-1-onto-> ( M ... N
) )
26 f1of 5373 . . . . . . . . 9  |-  ( `' J : ( M ... N ) -1-1-onto-> ( M ... N )  ->  `' J : ( M ... N ) --> ( M ... N ) )
273, 25, 263syl 17 . . . . . . . 8  |-  ( ph  ->  `' J : ( M ... N ) --> ( M ... N ) )
2827, 1ffvelrnd 5562 . . . . . . 7  |-  ( ph  ->  ( `' J `  K )  e.  ( M ... N ) )
29 elfzelz 9835 . . . . . . 7  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  ( `' J `  K )  e.  ZZ )
3028, 29syl 14 . . . . . 6  |-  ( ph  ->  ( `' J `  K )  e.  ZZ )
31 fzdcel 9849 . . . . . 6  |-  ( ( B  e.  ZZ  /\  K  e.  ZZ  /\  ( `' J `  K )  e.  ZZ )  -> DECID  B  e.  ( K ... ( `' J `  K ) ) )
3222, 24, 30, 31syl3anc 1217 . . . . 5  |-  ( ph  -> DECID  B  e.  ( K ... ( `' J `  K ) ) )
33 exmiddc 822 . . . . 5  |-  (DECID  B  e.  ( K ... ( `' J `  K ) )  ->  ( B  e.  ( K ... ( `' J `  K ) )  \/  -.  B  e.  ( K ... ( `' J `  K ) ) ) )
3432, 33syl 14 . . . 4  |-  ( ph  ->  ( B  e.  ( K ... ( `' J `  K ) )  \/  -.  B  e.  ( K ... ( `' J `  K ) ) ) )
3534adantr 274 . . 3  |-  ( (
ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  ->  ( B  e.  ( K ... ( `' J `  K ) )  \/ 
-.  B  e.  ( K ... ( `' J `  K ) ) ) )
3614, 20, 35mpjaodan 788 . 2  |-  ( (
ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  ->  A  =  B )
37 simpr 109 . . . . 5  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  B  e.  ( K ... ( `' J `  K ) ) )
38 simplr 520 . . . . 5  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  -.  A  e.  ( K ... ( `' J `  K ) ) )
3937, 38jca 304 . . . 4  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  ( B  e.  ( K ... ( `' J `  K ) )  /\  -.  A  e.  ( K ... ( `' J `  K ) ) ) )
409eqcomd 2146 . . . . . 6  |-  ( ph  ->  ( Q `  B
)  =  ( Q `
 A ) )
411, 3, 7, 5, 40, 11iseqf1olemnab 10290 . . . . 5  |-  ( ph  ->  -.  ( B  e.  ( K ... ( `' J `  K ) )  /\  -.  A  e.  ( K ... ( `' J `  K ) ) ) )
4241ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  -.  ( B  e.  ( K ... ( `' J `  K ) )  /\  -.  A  e.  ( K ... ( `' J `  K ) ) ) )
4339, 42pm2.21dd 610 . . 3  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  A  =  B )
441ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  K  e.  ( M ... N
) )
453ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  J : ( M ... N ) -1-1-onto-> ( M ... N
) )
465ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  A  e.  ( M ... N
) )
477ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  B  e.  ( M ... N
) )
489ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  ( Q `  A )  =  ( Q `  B ) )
49 simplr 520 . . . 4  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  -.  A  e.  ( K ... ( `' J `  K ) ) )
50 simpr 109 . . . 4  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  -.  B  e.  ( K ... ( `' J `  K ) ) )
5144, 45, 46, 47, 48, 11, 49, 50iseqf1olemnanb 10292 . . 3  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  A  =  B )
5234adantr 274 . . 3  |-  ( (
ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  ->  ( B  e.  ( K ... ( `' J `  K ) )  \/ 
-.  B  e.  ( K ... ( `' J `  K ) ) ) )
5343, 51, 52mpjaodan 788 . 2  |-  ( (
ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  ->  A  =  B )
54 elfzelz 9835 . . . . 5  |-  ( A  e.  ( M ... N )  ->  A  e.  ZZ )
555, 54syl 14 . . . 4  |-  ( ph  ->  A  e.  ZZ )
56 fzdcel 9849 . . . 4  |-  ( ( A  e.  ZZ  /\  K  e.  ZZ  /\  ( `' J `  K )  e.  ZZ )  -> DECID  A  e.  ( K ... ( `' J `  K ) ) )
5755, 24, 30, 56syl3anc 1217 . . 3  |-  ( ph  -> DECID  A  e.  ( K ... ( `' J `  K ) ) )
58 exmiddc 822 . . 3  |-  (DECID  A  e.  ( K ... ( `' J `  K ) )  ->  ( A  e.  ( K ... ( `' J `  K ) )  \/  -.  A  e.  ( K ... ( `' J `  K ) ) ) )
5957, 58syl 14 . 2  |-  ( ph  ->  ( A  e.  ( K ... ( `' J `  K ) )  \/  -.  A  e.  ( K ... ( `' J `  K ) ) ) )
6036, 53, 59mpjaodan 788 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 698  DECID wdc 820    = wceq 1332    e. wcel 1481   ifcif 3477    |-> cmpt 3995   `'ccnv 4544   -->wf 5125   -1-1-onto->wf1o 5128   ` cfv 5129  (class class class)co 5780   1c1 7643    - cmin 7955   ZZcz 9076   ...cfz 9819
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 4052  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458  ax-cnex 7733  ax-resscn 7734  ax-1cn 7735  ax-1re 7736  ax-icn 7737  ax-addcl 7738  ax-addrcl 7739  ax-mulcl 7740  ax-addcom 7742  ax-addass 7744  ax-distr 7746  ax-i2m1 7747  ax-0lt1 7748  ax-0id 7750  ax-rnegex 7751  ax-cnre 7753  ax-pre-ltirr 7754  ax-pre-ltwlin 7755  ax-pre-lttrn 7756  ax-pre-ltadd 7758
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 2691  df-sbc 2913  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-if 3478  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-int 3778  df-br 3936  df-opab 3996  df-mpt 3997  df-id 4221  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-fv 5137  df-riota 5736  df-ov 5783  df-oprab 5784  df-mpo 5785  df-pnf 7824  df-mnf 7825  df-xr 7826  df-ltxr 7827  df-le 7828  df-sub 7957  df-neg 7958  df-inn 8743  df-n0 9000  df-z 9077  df-uz 9349  df-fz 9820
This theorem is referenced by:  iseqf1olemqf1o  10295
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