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Theorem iseqf1olemmo 10478
Description: Lemma for seq3f1o 10490. 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 488 . . . 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 488 . . . 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 488 . . . 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 488 . . . 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 488 . . . 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 528 . . . 4  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  A  e.  ( K ... ( `' J `  K ) ) )
13 simpr 110 . . . 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 10475 . . 3  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  A  =  B )
15 simplr 528 . . . . 5  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  A  e.  ( K ... ( `' J `  K ) ) )
16 simpr 110 . . . . 5  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  -.  B  e.  ( K ... ( `' J `  K ) ) )
1715, 16jca 306 . . . 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 10474 . . . . 5  |-  ( ph  ->  -.  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) ) )
1918ad2antrr 488 . . . 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 620 . . 3  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  A  =  B )
21 elfzelz 10011 . . . . . . 7  |-  ( B  e.  ( M ... N )  ->  B  e.  ZZ )
227, 21syl 14 . . . . . 6  |-  ( ph  ->  B  e.  ZZ )
23 elfzelz 10011 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  K  e.  ZZ )
241, 23syl 14 . . . . . 6  |-  ( ph  ->  K  e.  ZZ )
25 f1ocnv 5470 . . . . . . . . 9  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  `' J : ( M ... N ) -1-1-onto-> ( M ... N
) )
26 f1of 5457 . . . . . . . . 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, 1ffvelcdmd 5648 . . . . . . 7  |-  ( ph  ->  ( `' J `  K )  e.  ( M ... N ) )
29 elfzelz 10011 . . . . . . 7  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  ( `' J `  K )  e.  ZZ )
3028, 29syl 14 . . . . . 6  |-  ( ph  ->  ( `' J `  K )  e.  ZZ )
31 fzdcel 10026 . . . . . 6  |-  ( ( B  e.  ZZ  /\  K  e.  ZZ  /\  ( `' J `  K )  e.  ZZ )  -> DECID  B  e.  ( K ... ( `' J `  K ) ) )
3222, 24, 30, 31syl3anc 1238 . . . . 5  |-  ( ph  -> DECID  B  e.  ( K ... ( `' J `  K ) ) )
33 exmiddc 836 . . . . 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 276 . . 3  |-  ( (
ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  ->  ( B  e.  ( K ... ( `' J `  K ) )  \/ 
-.  B  e.  ( K ... ( `' J `  K ) ) ) )
3614, 20, 35mpjaodan 798 . 2  |-  ( (
ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  ->  A  =  B )
37 simpr 110 . . . . 5  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  B  e.  ( K ... ( `' J `  K ) ) )
38 simplr 528 . . . . 5  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  -.  A  e.  ( K ... ( `' J `  K ) ) )
3937, 38jca 306 . . . 4  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  ( B  e.  ( K ... ( `' J `  K ) )  /\  -.  A  e.  ( K ... ( `' J `  K ) ) ) )
409eqcomd 2183 . . . . . 6  |-  ( ph  ->  ( Q `  B
)  =  ( Q `
 A ) )
411, 3, 7, 5, 40, 11iseqf1olemnab 10474 . . . . 5  |-  ( ph  ->  -.  ( B  e.  ( K ... ( `' J `  K ) )  /\  -.  A  e.  ( K ... ( `' J `  K ) ) ) )
4241ad2antrr 488 . . . 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 620 . . 3  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  B  e.  ( K ... ( `' J `  K ) ) )  ->  A  =  B )
441ad2antrr 488 . . . 4  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  K  e.  ( M ... N
) )
453ad2antrr 488 . . . 4  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  J : ( M ... N ) -1-1-onto-> ( M ... N
) )
465ad2antrr 488 . . . 4  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  A  e.  ( M ... N
) )
477ad2antrr 488 . . . 4  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  B  e.  ( M ... N
) )
489ad2antrr 488 . . . 4  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  ( Q `  A )  =  ( Q `  B ) )
49 simplr 528 . . . 4  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  -.  A  e.  ( K ... ( `' J `  K ) ) )
50 simpr 110 . . . 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 10476 . . 3  |-  ( ( ( ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  B  e.  ( K ... ( `' J `  K ) ) )  ->  A  =  B )
5234adantr 276 . . 3  |-  ( (
ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  ->  ( B  e.  ( K ... ( `' J `  K ) )  \/ 
-.  B  e.  ( K ... ( `' J `  K ) ) ) )
5343, 51, 52mpjaodan 798 . 2  |-  ( (
ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  ->  A  =  B )
54 elfzelz 10011 . . . . 5  |-  ( A  e.  ( M ... N )  ->  A  e.  ZZ )
555, 54syl 14 . . . 4  |-  ( ph  ->  A  e.  ZZ )
56 fzdcel 10026 . . . 4  |-  ( ( A  e.  ZZ  /\  K  e.  ZZ  /\  ( `' J `  K )  e.  ZZ )  -> DECID  A  e.  ( K ... ( `' J `  K ) ) )
5755, 24, 30, 56syl3anc 1238 . . 3  |-  ( ph  -> DECID  A  e.  ( K ... ( `' J `  K ) ) )
58 exmiddc 836 . . 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 798 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 708  DECID wdc 834    = wceq 1353    e. wcel 2148   ifcif 3534    |-> cmpt 4061   `'ccnv 4622   -->wf 5208   -1-1-onto->wf1o 5211   ` cfv 5212  (class class class)co 5869   1c1 7803    - cmin 8118   ZZcz 9242   ...cfz 9995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996
This theorem is referenced by:  iseqf1olemqf1o  10479
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