Theorem iseqf1olemfvp 10876

Description: Lemma for seq3f1o 10883. (Contributed by Jim Kingdon, 30-Aug-2022.)

Hypotheses

Ref	Expression
iseqf1olemfvp.k	|- ( ph -> K e. ( M ... N ) )
iseqf1olemfvp.t	|- ( ph -> T : ( M ... N ) -1-1-onto-> ( M ... N ) )
iseqf1olemfvp.a	|- ( ph -> A e. ( M ... N ) )
iseqf1olemfvp.g	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )
iseqf1olemfvp.p	|- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )

Assertion

Ref	Expression
iseqf1olemfvp	|- ( ph -> ( [_ T / f ]_ P ` A ) = ( G ` ( T ` A ) ) )

Distinct variable groups:   x, A   f, G, x   x, K   f, M, x   f, N, x   x, S   T, f, x   ph, x

Allowed substitution hints:   ph( f)   A( f)   P( x, f)   S( f)   K( f)

Proof of Theorem iseqf1olemfvp

Step	Hyp	Ref	Expression
1		iseqf1olemfvp.p	. . . . 5 |- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )
2	1	csbeq2i 3167	. . . 4 |- [_ T / f ]_ P = [_ T / f ]_ ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )
3		iseqf1olemfvp.t	. . . . . . 7 |- ( ph -> T : ( M ... N ) -1-1-onto-> ( M ... N ) )
4		f1of 5616	. . . . . . 7 |- ( T : ( M ... N ) -1-1-onto-> ( M ... N ) -> T : ( M ... N ) --> ( M ... N ) )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> T : ( M ... N ) --> ( M ... N ) )
6		iseqf1olemfvp.k	. . . . . . . 8 |- ( ph -> K e. ( M ... N ) )
7		elfzel1 10361	. . . . . . . 8 |- ( K e. ( M ... N ) -> M e. ZZ )
8	6, 7	syl 14	. . . . . . 7 |- ( ph -> M e. ZZ )
9		elfzel2 10360	. . . . . . . 8 |- ( K e. ( M ... N ) -> N e. ZZ )
10	6, 9	syl 14	. . . . . . 7 |- ( ph -> N e. ZZ )
11	8, 10	fzfigd 10797	. . . . . 6 |- ( ph -> ( M ... N ) e. Fin )
12		fex 5917	. . . . . 6 |- ( ( T : ( M ... N ) --> ( M ... N ) /\ ( M ... N ) e. Fin ) -> T e. _V )
13	5, 11, 12	syl2anc 411	. . . . 5 |- ( ph -> T e. _V )
14		nfcvd 2387	. . . . . 6 |- ( T e. _V -> F/_ f ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( T ` x ) ) , ( G ` M ) ) ) )
15		fveq1 5671	. . . . . . . . 9 |- ( f = T -> ( f ` x ) = ( T ` x ) )
16	15	fveq2d 5676	. . . . . . . 8 |- ( f = T -> ( G ` ( f ` x ) ) = ( G ` ( T ` x ) ) )
17	16	ifeq1d 3642	. . . . . . 7 |- ( f = T -> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) = if ( x <_ N , ( G ` ( T ` x ) ) , ( G ` M ) ) )
18	17	mpteq2dv 4203	. . . . . 6 |- ( f = T -> ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) ) = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( T ` x ) ) , ( G ` M ) ) ) )
19	14, 18	csbiegf 3184	. . . . 5 |- ( T e. _V -> [_ T / f ]_ ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) ) = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( T ` x ) ) , ( G ` M ) ) ) )
20	13, 19	syl 14	. . . 4 |- ( ph -> [_ T / f ]_ ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) ) = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( T ` x ) ) , ( G ` M ) ) ) )
21	2, 20	eqtrid 2279	. . 3 |- ( ph -> [_ T / f ]_ P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( T ` x ) ) , ( G ` M ) ) ) )
22		simpr 110	. . . . 5 |- ( ( ph /\ x = A ) -> x = A )
23	22	breq1d 4121	. . . 4 |- ( ( ph /\ x = A ) -> ( x <_ N <-> A <_ N ) )
24	22	fveq2d 5676	. . . . 5 |- ( ( ph /\ x = A ) -> ( T ` x ) = ( T ` A ) )
25	24	fveq2d 5676	. . . 4 |- ( ( ph /\ x = A ) -> ( G ` ( T ` x ) ) = ( G ` ( T ` A ) ) )
26	23, 25	ifbieq1d 3647	. . 3 |- ( ( ph /\ x = A ) -> if ( x <_ N , ( G ` ( T ` x ) ) , ( G ` M ) ) = if ( A <_ N , ( G ` ( T ` A ) ) , ( G ` M ) ) )
27		iseqf1olemfvp.a	. . . 4 |- ( ph -> A e. ( M ... N ) )
28		elfzuz 10358	. . . 4 |- ( A e. ( M ... N ) -> A e. ( ZZ>= ` M ) )
29	27, 28	syl 14	. . 3 |- ( ph -> A e. ( ZZ>= ` M ) )
30		elfzle2 10365	. . . . . 6 |- ( A e. ( M ... N ) -> A <_ N )
31	27, 30	syl 14	. . . . 5 |- ( ph -> A <_ N )
32	31	iftrued 3631	. . . 4 |- ( ph -> if ( A <_ N , ( G ` ( T ` A ) ) , ( G ` M ) ) = ( G ` ( T ` A ) ) )
33		fveq2 5672	. . . . . 6 |- ( x = ( T ` A ) -> ( G ` x ) = ( G ` ( T ` A ) ) )
34	33	eleq1d 2303	. . . . 5 |- ( x = ( T ` A ) -> ( ( G ` x ) e. S <-> ( G ` ( T ` A ) ) e. S ) )
35		iseqf1olemfvp.g	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )
36	35	ralrimiva 2617	. . . . 5 |- ( ph -> A. x e. ( ZZ>= ` M ) ( G ` x ) e. S )
37	5, 27	ffvelcdmd 5815	. . . . . 6 |- ( ph -> ( T ` A ) e. ( M ... N ) )
38		elfzuz 10358	. . . . . 6 |- ( ( T ` A ) e. ( M ... N ) -> ( T ` A ) e. ( ZZ>= ` M ) )
39	37, 38	syl 14	. . . . 5 |- ( ph -> ( T ` A ) e. ( ZZ>= ` M ) )
40	34, 36, 39	rspcdva 2928	. . . 4 |- ( ph -> ( G ` ( T ` A ) ) e. S )
41	32, 40	eqeltrd 2311	. . 3 |- ( ph -> if ( A <_ N , ( G ` ( T ` A ) ) , ( G ` M ) ) e. S )
42	21, 26, 29, 41	fvmptd 5760	. 2 |- ( ph -> ( [_ T / f ]_ P ` A ) = if ( A <_ N , ( G ` ( T ` A ) ) , ( G ` M ) ) )
43	42, 32	eqtrd 2267	1 |- ( ph -> ( [_ T / f ]_ P ` A ) = ( G ` ( T ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   [_csb 3140   ifcif 3622   class class class wbr 4111   |-> cmpt 4173   -->wf 5350   -1-1-onto->wf1o 5353   ` cfv 5354  (class class class)co 6052   Fincfn 6977   <_ cle 8311   ZZcz 9579   ZZ>=cuz 9856   ...cfz 10345

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-1o 6649  df-er 6769  df-en 6978  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857  df-fz 10346

This theorem is referenced by:  seq3f1olemqsumkj  10877  seq3f1olemqsumk  10878