Theorem iseqf1olemnanb 10940

Description: Lemma for seq3f1o 10954. (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 10937	. . . 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 3650	. . . 4 |- ( ph -> if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) = ( J ` A ) )
9	6, 8	eqtrd 2271	. . 3 |- ( ph -> ( Q ` A ) = ( J ` A ) )
10		iseqf1olemnab.b	. . . . 5 |- ( ph -> B e. ( M ... N ) )
11	2, 3, 10, 5	iseqf1olemqval 10937	. . . 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 3650	. . . 4 |- ( ph -> if ( B e. ( K ... ( `' J ` K ) ) , if ( B = K , K , ( J ` ( B - 1 ) ) ) , ( J ` B ) ) = ( J ` B ) )
14	11, 13	eqtrd 2271	. . 3 |- ( ph -> ( Q ` B ) = ( J ` B ) )
15	1, 9, 14	3eqtr3d 2279	. 2 |- ( ph -> ( J ` A ) = ( J ` B ) )
16		f1of1 5638	. . . 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 5975	. . 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 1276	. 2 |- ( ph -> ( ( J ` A ) = ( J ` B ) -> A = B ) )
20	15, 19	mpd 13	1 |- ( ph -> A = B )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   = wceq 1402   e. wcel 2209   ifcif 3638   |-> cmpt 4192   `'ccnv 4773   -1-1->wf1 5374   -1-1-onto->wf1o 5376   ` cfv 5377  (class class class)co 6085   1c1 8180   - cmin 8497   ...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:  iseqf1olemmo  10942