Theorem iseqf1olemnanb 10425

Description: Lemma for seq3f1o 10439. (Contributed by Jim Kingdon, 27-Aug-2022.)

Hypotheses

Ref	Expression
iseqf1olemqcl.k	|- ( ph -> K e. ( M ... N ) )
iseqf1olemqcl.j	|- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
iseqf1olemqcl.a	|- ( ph -> A e. ( M ... N ) )
iseqf1olemnab.b	|- ( ph -> B e. ( M ... N ) )
iseqf1olemnab.eq	|- ( ph -> ( Q ` A ) = ( Q ` B ) )
iseqf1olemnab.q	|- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )
iseqf1olemnanb.a	|- ( ph -> -. A e. ( K ... ( `' J ` K ) ) )
iseqf1olemnanb.b	|- ( ph -> -. B e. ( K ... ( `' J ` K ) ) )

Assertion

Ref	Expression
iseqf1olemnanb	|- ( 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 iseqf1olemnanb

Step	Hyp	Ref	Expression
1		iseqf1olemnab.eq	. . 3 |- ( ph -> ( Q ` A ) = ( Q ` B ) )
2		iseqf1olemqcl.k	. . . . 5 |- ( ph -> K e. ( M ... N ) )
3		iseqf1olemqcl.j	. . . . 5 |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
4		iseqf1olemqcl.a	. . . . 5 |- ( ph -> A e. ( M ... N ) )
5		iseqf1olemnab.q	. . . . 5 |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )
6	2, 3, 4, 5	iseqf1olemqval 10422	. . . 4 |- ( ph -> ( Q ` A ) = if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) )
7		iseqf1olemnanb.a	. . . . 5 |- ( ph -> -. A e. ( K ... ( `' J ` K ) ) )
8	7	iffalsed 3530	. . . 4 |- ( ph -> if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) = ( J ` A ) )
9	6, 8	eqtrd 2198	. . 3 |- ( ph -> ( Q ` A ) = ( J ` A ) )
10		iseqf1olemnab.b	. . . . 5 |- ( ph -> B e. ( M ... N ) )
11	2, 3, 10, 5	iseqf1olemqval 10422	. . . 4 |- ( ph -> ( Q ` B ) = if ( B e. ( K ... ( `' J ` K ) ) , if ( B = K , K , ( J ` ( B - 1 ) ) ) , ( J ` B ) ) )
12		iseqf1olemnanb.b	. . . . 5 |- ( ph -> -. B e. ( K ... ( `' J ` K ) ) )
13	12	iffalsed 3530	. . . 4 |- ( ph -> if ( B e. ( K ... ( `' J ` K ) ) , if ( B = K , K , ( J ` ( B - 1 ) ) ) , ( J ` B ) ) = ( J ` B ) )
14	11, 13	eqtrd 2198	. . 3 |- ( ph -> ( Q ` B ) = ( J ` B ) )
15	1, 9, 14	3eqtr3d 2206	. 2 |- ( ph -> ( J ` A ) = ( J ` B ) )
16		f1of1 5431	. . . 4 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> J : ( M ... N ) -1-1-> ( M ... N ) )
17	3, 16	syl 14	. . 3 |- ( ph -> J : ( M ... N ) -1-1-> ( M ... N ) )
18		f1veqaeq 5737	. . 3 |- ( ( J : ( M ... N ) -1-1-> ( M ... N ) /\ ( A e. ( M ... N ) /\ B e. ( M ... N ) ) ) -> ( ( J ` A ) = ( J ` B ) -> A = B ) )
19	17, 4, 10, 18	syl12anc 1226	. 2 |- ( ph -> ( ( J ` A ) = ( J ` B ) -> A = B ) )
20	15, 19	mpd 13	1 |- ( ph -> A = B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1343   e. wcel 2136   ifcif 3520   |-> cmpt 4043   `'ccnv 4603   -1-1->wf1 5185   -1-1-onto->wf1o 5187   ` cfv 5188  (class class class)co 5842   1c1 7754   - cmin 8069   ...cfz 9944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-fz 9945

This theorem is referenced by:  iseqf1olemmo  10427