Theorem iseqf1olemnanb 10612

Description: Lemma for seq3f1o 10626. (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 10609	. . . 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 3572	. . . 4 |- ( ph -> if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) = ( J ` A ) )
9	6, 8	eqtrd 2229	. . 3 |- ( ph -> ( Q ` A ) = ( J ` A ) )
10		iseqf1olemnab.b	. . . . 5 |- ( ph -> B e. ( M ... N ) )
11	2, 3, 10, 5	iseqf1olemqval 10609	. . . 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 3572	. . . 4 |- ( ph -> if ( B e. ( K ... ( `' J ` K ) ) , if ( B = K , K , ( J ` ( B - 1 ) ) ) , ( J ` B ) ) = ( J ` B ) )
14	11, 13	eqtrd 2229	. . 3 |- ( ph -> ( Q ` B ) = ( J ` B ) )
15	1, 9, 14	3eqtr3d 2237	. 2 |- ( ph -> ( J ` A ) = ( J ` B ) )
16		f1of1 5506	. . . 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 5819	. . 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 1247	. 2 |- ( ph -> ( ( J ` A ) = ( J ` B ) -> A = B ) )
20	15, 19	mpd 13	1 |- ( ph -> A = B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   e. wcel 2167   ifcif 3562   |-> cmpt 4095   `'ccnv 4663   -1-1->wf1 5256   -1-1-onto->wf1o 5258   ` cfv 5259  (class class class)co 5925   1c1 7897   - cmin 8214   ...cfz 10100

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 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012

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 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-fz 10101

This theorem is referenced by:  iseqf1olemmo  10614