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Theorem iseqf1olemmo 10106
Description: Lemma for seq3f1o 10118. 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 475 . . . 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 475 . . . 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 475 . . . 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 475 . . . 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 475 . . . 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 500 . . . 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 10103 . . 3 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ B e. ( K ... ( `' J ` K ) ) ) -> A = B )
15 simplr 500 . . . . 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 302 . . . 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 10102 . . . . 5 |- ( ph -> -. ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) )
1918ad2antrr 475 . . . 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 590 . . 3 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) -> A = B )
21 elfzelz 9647 . . . . . . 7 |- ( B e. ( M ... N ) -> B e. ZZ )
227, 21syl 14 . . . . . 6 |- ( ph -> B e. ZZ )
23 elfzelz 9647 . . . . . . 7 |- ( K e. ( M ... N ) -> K e. ZZ )
241, 23syl 14 . . . . . 6 |- ( ph -> K e. ZZ )
25 f1ocnv 5314 . . . . . . . . 9 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
26 f1of 5301 . . . . . . . . 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 5488 . . . . . . 7 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
29 elfzelz 9647 . . . . . . 7 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ZZ )
3028, 29syl 14 . . . . . 6 |- ( ph -> ( `' J ` K ) e. ZZ )
31 fzdcel 9661 . . . . . 6 |- ( ( B e. ZZ /\ K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> DECID B e. ( K ... ( `' J ` K ) ) )
3222, 24, 30, 31syl3anc 1184 . . . . 5 |- ( ph -> DECID B e. ( K ... ( `' J ` K ) ) )
33 exmiddc 788 . . . . 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 272 . . 3 |- ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) -> ( B e. ( K ... ( `' J ` K ) ) \/ -. B e. ( K ... ( `' J ` K ) ) ) )
3614, 20, 35mpjaodan 753 . 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 500 . . . . 5 |- ( ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) /\ B e. ( K ... ( `' J ` K ) ) ) -> -. A e. ( K ... ( `' J ` K ) ) )
3937, 38jca 302 . . . 4 |- ( ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) /\ B e. ( K ... ( `' J ` K ) ) ) -> ( B e. ( K ... ( `' J ` K ) ) /\ -. A e. ( K ... ( `' J ` K ) ) ) )
409eqcomd 2105 . . . . . 6 |- ( ph -> ( Q ` B ) = ( Q ` A ) )
411, 3, 7, 5, 40, 11iseqf1olemnab 10102 . . . . 5 |- ( ph -> -. ( B e. ( K ... ( `' J ` K ) ) /\ -. A e. ( K ... ( `' J ` K ) ) ) )
4241ad2antrr 475 . . . 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 590 . . 3 |- ( ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) /\ B e. ( K ... ( `' J ` K ) ) ) -> A = B )
441ad2antrr 475 . . . 4 |- ( ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) -> K e. ( M ... N ) )
453ad2antrr 475 . . . 4 |- ( ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
465ad2antrr 475 . . . 4 |- ( ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) -> A e. ( M ... N ) )
477ad2antrr 475 . . . 4 |- ( ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) -> B e. ( M ... N ) )
489ad2antrr 475 . . . 4 |- ( ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) -> ( Q ` A ) = ( Q ` B ) )
49 simplr 500 . . . 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 10104 . . 3 |- ( ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) -> A = B )
5234adantr 272 . . 3 |- ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) -> ( B e. ( K ... ( `' J ` K ) ) \/ -. B e. ( K ... ( `' J ` K ) ) ) )
5343, 51, 52mpjaodan 753 . 2 |- ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) -> A = B )
54 elfzelz 9647 . . . . 5 |- ( A e. ( M ... N ) -> A e. ZZ )
555, 54syl 14 . . . 4 |- ( ph -> A e. ZZ )
56 fzdcel 9661 . . . 4 |- ( ( A e. ZZ /\ K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> DECID A e. ( K ... ( `' J ` K ) ) )
5755, 24, 30, 56syl3anc 1184 . . 3 |- ( ph -> DECID A e. ( K ... ( `' J ` K ) ) )
58 exmiddc 788 . . 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 753 1 |- ( ph -> A = B )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 670  DECID wdc 786   = wceq 1299   e. wcel 1448   ifcif 3421   |-> cmpt 3929   `'ccnv 4476   -->wf 5055   -1-1-onto->wf1o 5058   ` cfv 5059  (class class class)co 5706   1c1 7501   - cmin 7804   ZZcz 8906   ...cfz 9631
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 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-ltadd 7611
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177  df-fz 9632
This theorem is referenced by:  iseqf1olemqf1o  10107
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