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Theorem iseqf1olemmo 10667
Description: Lemma for seq3f1o 10679. 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 10664 . . 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 10663 . . . . 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 10162 . . . . . . 7 |- ( B e. ( M ... N ) -> B e. ZZ )
227, 21syl 14 . . . . . 6 |- ( ph -> B e. ZZ )
23 elfzelz 10162 . . . . . . 7 |- ( K e. ( M ... N ) -> K e. ZZ )
241, 23syl 14 . . . . . 6 |- ( ph -> K e. ZZ )
25 f1ocnv 5546 . . . . . . . . 9 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
26 f1of 5533 . . . . . . . . 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 5728 . . . . . . 7 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
29 elfzelz 10162 . . . . . . 7 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ZZ )
3028, 29syl 14 . . . . . 6 |- ( ph -> ( `' J ` K ) e. ZZ )
31 fzdcel 10177 . . . . . 6 |- ( ( B e. ZZ /\ K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> DECID B e. ( K ... ( `' J ` K ) ) )
3222, 24, 30, 31syl3anc 1250 . . . . 5 |- ( ph -> DECID B e. ( K ... ( `' J ` K ) ) )
33 exmiddc 838 . . . . 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 800 . 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 2212 . . . . . 6 |- ( ph -> ( Q ` B ) = ( Q ` A ) )
411, 3, 7, 5, 40, 11iseqf1olemnab 10663 . . . . 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 10665 . . 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 800 . 2 |- ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) -> A = B )
54 elfzelz 10162 . . . . 5 |- ( A e. ( M ... N ) -> A e. ZZ )
555, 54syl 14 . . . 4 |- ( ph -> A e. ZZ )
56 fzdcel 10177 . . . 4 |- ( ( A e. ZZ /\ K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> DECID A e. ( K ... ( `' J ` K ) ) )
5755, 24, 30, 56syl3anc 1250 . . 3 |- ( ph -> DECID A e. ( K ... ( `' J ` K ) ) )
58 exmiddc 838 . . 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 800 1 |- ( ph -> A = B )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 710  DECID wdc 836   = wceq 1373   e. wcel 2177   ifcif 3575   |-> cmpt 4112   `'ccnv 4681   -->wf 5275   -1-1-onto->wf1o 5278   ` cfv 5279  (class class class)co 5956   1c1 7941   - cmin 8258   ZZcz 9387   ...cfz 10145
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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4169  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-cnex 8031  ax-resscn 8032  ax-1cn 8033  ax-1re 8034  ax-icn 8035  ax-addcl 8036  ax-addrcl 8037  ax-mulcl 8038  ax-addcom 8040  ax-addass 8042  ax-distr 8044  ax-i2m1 8045  ax-0lt1 8046  ax-0id 8048  ax-rnegex 8049  ax-cnre 8051  ax-pre-ltirr 8052  ax-pre-ltwlin 8053  ax-pre-lttrn 8054  ax-pre-ltadd 8056
This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-if 3576  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-br 4051  df-opab 4113  df-mpt 4114  df-id 4347  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-riota 5911  df-ov 5959  df-oprab 5960  df-mpo 5961  df-pnf 8124  df-mnf 8125  df-xr 8126  df-ltxr 8127  df-le 8128  df-sub 8260  df-neg 8261  df-inn 9052  df-n0 9311  df-z 9388  df-uz 9664  df-fz 10146
This theorem is referenced by:  iseqf1olemqf1o  10668
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