Theorem iseqf1olemfvp 10732

Description: Lemma for seq3f1o 10739. (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 3151	. . . 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 5572	. . . . . . 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 10220	. . . . . . . 8 |- ( K e. ( M ... N ) -> M e. ZZ )
8	6, 7	syl 14	. . . . . . 7 |- ( ph -> M e. ZZ )
9		elfzel2 10219	. . . . . . . 8 |- ( K e. ( M ... N ) -> N e. ZZ )
10	6, 9	syl 14	. . . . . . 7 |- ( ph -> N e. ZZ )
11	8, 10	fzfigd 10653	. . . . . 6 |- ( ph -> ( M ... N ) e. Fin )
12		fex 5868	. . . . . 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 2373	. . . . . 6 |- ( T e. _V -> F/_ f ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( T ` x ) ) , ( G ` M ) ) ) )
15		fveq1 5626	. . . . . . . . 9 |- ( f = T -> ( f ` x ) = ( T ` x ) )
16	15	fveq2d 5631	. . . . . . . 8 |- ( f = T -> ( G ` ( f ` x ) ) = ( G ` ( T ` x ) ) )
17	16	ifeq1d 3620	. . . . . . 7 |- ( f = T -> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) = if ( x <_ N , ( G ` ( T ` x ) ) , ( G ` M ) ) )
18	17	mpteq2dv 4175	. . . . . 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 3168	. . . . 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 2274	. . 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 4093	. . . 4 |- ( ( ph /\ x = A ) -> ( x <_ N <-> A <_ N ) )
24	22	fveq2d 5631	. . . . 5 |- ( ( ph /\ x = A ) -> ( T ` x ) = ( T ` A ) )
25	24	fveq2d 5631	. . . 4 |- ( ( ph /\ x = A ) -> ( G ` ( T ` x ) ) = ( G ` ( T ` A ) ) )
26	23, 25	ifbieq1d 3625	. . 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 10217	. . . 4 |- ( A e. ( M ... N ) -> A e. ( ZZ>= ` M ) )
29	27, 28	syl 14	. . 3 |- ( ph -> A e. ( ZZ>= ` M ) )
30		elfzle2 10224	. . . . . 6 |- ( A e. ( M ... N ) -> A <_ N )
31	27, 30	syl 14	. . . . 5 |- ( ph -> A <_ N )
32	31	iftrued 3609	. . . 4 |- ( ph -> if ( A <_ N , ( G ` ( T ` A ) ) , ( G ` M ) ) = ( G ` ( T ` A ) ) )
33		fveq2 5627	. . . . . 6 |- ( x = ( T ` A ) -> ( G ` x ) = ( G ` ( T ` A ) ) )
34	33	eleq1d 2298	. . . . 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 2603	. . . . 5 |- ( ph -> A. x e. ( ZZ>= ` M ) ( G ` x ) e. S )
37	5, 27	ffvelcdmd 5771	. . . . . 6 |- ( ph -> ( T ` A ) e. ( M ... N ) )
38		elfzuz 10217	. . . . . 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 2912	. . . 4 |- ( ph -> ( G ` ( T ` A ) ) e. S )
41	32, 40	eqeltrd 2306	. . 3 |- ( ph -> if ( A <_ N , ( G ` ( T ` A ) ) , ( G ` M ) ) e. S )
42	21, 26, 29, 41	fvmptd 5715	. 2 |- ( ph -> ( [_ T / f ]_ P ` A ) = if ( A <_ N , ( G ` ( T ` A ) ) , ( G ` M ) ) )
43	42, 32	eqtrd 2262	1 |- ( ph -> ( [_ T / f ]_ P ` A ) = ( G ` ( T ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   [_csb 3124   ifcif 3602   class class class wbr 4083   |-> cmpt 4145   -->wf 5314   -1-1-onto->wf1o 5317   ` cfv 5318  (class class class)co 6001   Fincfn 6887   <_ cle 8182   ZZcz 9446   ZZ>=cuz 9722   ...cfz 10204

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-er 6680  df-en 6888  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723  df-fz 10205

This theorem is referenced by:  seq3f1olemqsumkj  10733  seq3f1olemqsumk  10734