Theorem iseqf1olemnanb 10918

Description: Lemma for seq3f1o 10932. (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 10915	. . . 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 3647	. . . 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 10915	. . . 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 3647	. . . 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 5633	. . . 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 5965	. . 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 )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1402   e. wcel 2209   ifcif 3635   |-> cmpt 4187   `'ccnv 4768   -1-1->wf1 5369   -1-1-onto->wf1o 5371   ` cfv 5372  (class class class)co 6075   1c1 8170   - cmin 8487   ...cfz 10390

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 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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391

This theorem is referenced by:  iseqf1olemmo  10920