Theorem iseqf1olemqval 10862

Description: Lemma for seq3f1o 10879. Value of the function Q. (Contributed by Jim Kingdon, 28-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 ) )
iseqf1olemqval.q	|- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )

Assertion

Ref	Expression
iseqf1olemqval	|- ( ph -> ( Q ` A ) = if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) )

Distinct variable groups:   u, A   u, J   u, K   u, M   u, N

Allowed substitution hints:   ph( u)   Q( u)

Proof of Theorem iseqf1olemqval

Step	Hyp	Ref	Expression
1		iseqf1olemqcl.a	. 2 |- ( ph -> A e. ( M ... N ) )
2		iseqf1olemqcl.k	. . 3 |- ( ph -> K e. ( M ... N ) )
3		iseqf1olemqcl.j	. . 3 |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
4	2, 3, 1	iseqf1olemqcl 10861	. 2 |- ( ph -> if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) e. ( M ... N ) )
5		eleq1 2295	. . . 4 |- ( u = A -> ( u e. ( K ... ( `' J ` K ) ) <-> A e. ( K ... ( `' J ` K ) ) ) )
6		eqeq1 2239	. . . . 5 |- ( u = A -> ( u = K <-> A = K ) )
7		oveq1 6057	. . . . . 6 |- ( u = A -> ( u - 1 ) = ( A - 1 ) )
8	7	fveq2d 5674	. . . . 5 |- ( u = A -> ( J ` ( u - 1 ) ) = ( J ` ( A - 1 ) ) )
9	6, 8	ifbieq2d 3647	. . . 4 |- ( u = A -> if ( u = K , K , ( J ` ( u - 1 ) ) ) = if ( A = K , K , ( J ` ( A - 1 ) ) ) )
10		fveq2 5670	. . . 4 |- ( u = A -> ( J ` u ) = ( J ` A ) )
11	5, 9, 10	ifbieq12d 3649	. . 3 |- ( u = A -> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) = if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) )
12		iseqf1olemqval.q	. . 3 |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )
13	11, 12	fvmptg 5753	. 2 |- ( ( A e. ( M ... N ) /\ if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) e. ( M ... N ) ) -> ( Q ` A ) = if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) )
14	1, 4, 13	syl2anc 411	1 |- ( ph -> ( Q ` A ) = if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   ifcif 3620   |-> cmpt 4171   `'ccnv 4748   -1-1-onto->wf1o 5351   ` cfv 5352  (class class class)co 6050   1c1 8128   - cmin 8444   ...cfz 10342

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343

This theorem is referenced by:  iseqf1olemnab  10863  iseqf1olemab  10864  iseqf1olemnanb  10865  iseqf1olemqk  10869  seq3f1olemqsumkj  10873  seq3f1olemqsumk  10874