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Theorem iseqf1olemmo 10942
Description: Lemma for seq3f1o 10954. 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 492 . . . 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 492 . . . 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 492 . . . 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 492 . . . 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 492 . . . 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 533 . . . 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 10939 . . 3 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ B e. ( K ... ( `' J ` K ) ) ) -> A = B )
15 simplr 533 . . . . 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 10938 . . . . 5 |- ( ph -> -. ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) )
1918ad2antrr 492 . . . 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 629 . . 3 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) -> A = B )
21 elfzelz 10428 . . . . . . 7 |- ( B e. ( M ... N ) -> B e. ZZ )
227, 21syl 14 . . . . . 6 |- ( ph -> B e. ZZ )
23 elfzelz 10428 . . . . . . 7 |- ( K e. ( M ... N ) -> K e. ZZ )
241, 23syl 14 . . . . . 6 |- ( ph -> K e. ZZ )
25 f1ocnv 5652 . . . . . . . . 9 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
26 f1of 5639 . . . . . . . . 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 5844 . . . . . . 7 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
29 elfzelz 10428 . . . . . . 7 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ZZ )
3028, 29syl 14 . . . . . 6 |- ( ph -> ( `' J ` K ) e. ZZ )
31 fzdcel 10444 . . . . . 6 |- ( ( B e. ZZ /\ K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> DECID B e. ( K ... ( `' J ` K ) ) )
3222, 24, 30, 31syl3anc 1278 . . . . 5 |- ( ph -> DECID B e. ( K ... ( `' J ` K ) ) )
33 exmiddc 848 . . . . 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 810 . 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 533 . . . . 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 2244 . . . . . 6 |- ( ph -> ( Q ` B ) = ( Q ` A ) )
411, 3, 7, 5, 40, 11iseqf1olemnab 10938 . . . . 5 |- ( ph -> -. ( B e. ( K ... ( `' J ` K ) ) /\ -. A e. ( K ... ( `' J ` K ) ) ) )
4241ad2antrr 492 . . . 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 629 . . 3 |- ( ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) /\ B e. ( K ... ( `' J ` K ) ) ) -> A = B )
441ad2antrr 492 . . . 4 |- ( ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) -> K e. ( M ... N ) )
453ad2antrr 492 . . . 4 |- ( ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
465ad2antrr 492 . . . 4 |- ( ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) -> A e. ( M ... N ) )
477ad2antrr 492 . . . 4 |- ( ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) -> B e. ( M ... N ) )
489ad2antrr 492 . . . 4 |- ( ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) -> ( Q ` A ) = ( Q ` B ) )
49 simplr 533 . . . 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 10940 . . 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 810 . 2 |- ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) -> A = B )
54 elfzelz 10428 . . . . 5 |- ( A e. ( M ... N ) -> A e. ZZ )
555, 54syl 14 . . . 4 |- ( ph -> A e. ZZ )
56 fzdcel 10444 . . . 4 |- ( ( A e. ZZ /\ K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> DECID A e. ( K ... ( `' J ` K ) ) )
5755, 24, 30, 56syl3anc 1278 . . 3 |- ( ph -> DECID A e. ( K ... ( `' J ` K ) ) )
58 exmiddc 848 . . 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 810 1 |- ( ph -> A = B )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   ifcif 3638   |-> cmpt 4192   `'ccnv 4773   -->wf 5373   -1-1-onto->wf1o 5376   ` cfv 5377  (class class class)co 6085   1c1 8180   - cmin 8497   ZZcz 9644   ...cfz 10411
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295
This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412
This theorem is used by:  iseqf1olemqf1o  10943
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