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Theorem iseqf1olemmo 10597
Description: Lemma for seq3f1o 10609. 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 10594 . . 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 10593 . . . . 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 621 . . 3 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) -> A = B )
21 elfzelz 10100 . . . . . . 7 |- ( B e. ( M ... N ) -> B e. ZZ )
227, 21syl 14 . . . . . 6 |- ( ph -> B e. ZZ )
23 elfzelz 10100 . . . . . . 7 |- ( K e. ( M ... N ) -> K e. ZZ )
241, 23syl 14 . . . . . 6 |- ( ph -> K e. ZZ )
25 f1ocnv 5517 . . . . . . . . 9 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
26 f1of 5504 . . . . . . . . 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 5698 . . . . . . 7 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
29 elfzelz 10100 . . . . . . 7 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ZZ )
3028, 29syl 14 . . . . . 6 |- ( ph -> ( `' J ` K ) e. ZZ )
31 fzdcel 10115 . . . . . 6 |- ( ( B e. ZZ /\ K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> DECID B e. ( K ... ( `' J ` K ) ) )
3222, 24, 30, 31syl3anc 1249 . . . . 5 |- ( ph -> DECID B e. ( K ... ( `' J ` K ) ) )
33 exmiddc 837 . . . . 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 799 . 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 2202 . . . . . 6 |- ( ph -> ( Q ` B ) = ( Q ` A ) )
411, 3, 7, 5, 40, 11iseqf1olemnab 10593 . . . . 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 621 . . 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 10595 . . 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 799 . 2 |- ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) -> A = B )
54 elfzelz 10100 . . . . 5 |- ( A e. ( M ... N ) -> A e. ZZ )
555, 54syl 14 . . . 4 |- ( ph -> A e. ZZ )
56 fzdcel 10115 . . . 4 |- ( ( A e. ZZ /\ K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> DECID A e. ( K ... ( `' J ` K ) ) )
5755, 24, 30, 56syl3anc 1249 . . 3 |- ( ph -> DECID A e. ( K ... ( `' J ` K ) ) )
58 exmiddc 837 . . 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 799 1 |- ( ph -> A = B )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2167   ifcif 3561   |-> cmpt 4094   `'ccnv 4662   -->wf 5254   -1-1-onto->wf1o 5257   ` cfv 5258  (class class class)co 5922   1c1 7880   - cmin 8197   ZZcz 9326   ...cfz 10083
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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084
This theorem is referenced by:  iseqf1olemqf1o  10598
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