Theorem gsumfzmhm2 14103

Description: Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.) (Revised by AV, 6-Jun-2019.) (Revised by Jim Kingdon, 9-Sep-2025.)

Hypotheses

Ref	Expression
gsummhm2.b	|- B = ( Base ` G )
gsummhm2.z	|- .0. = ( 0g ` G )
gsummhm2.g	|- ( ph -> G e. CMnd )
gsummhm2.h	|- ( ph -> H e. Mnd )
gsumfzmhm2.m	|- ( ph -> M e. ZZ )
gsumfzmhm2.n	|- ( ph -> N e. ZZ )
gsummhm2.k	|- ( ph -> ( x e. B |-> C ) e. ( G MndHom H ) )
gsumfzmhm2.f	|- ( ( ph /\ k e. ( M ... N ) ) -> X e. B )
gsummhm2.1	|- ( x = X -> C = D )
gsumfzmhm2.2	|- ( x = ( G gsumg ( k e. ( M ... N ) |-> X ) ) -> C = E )

Assertion

Ref	Expression
gsumfzmhm2	|- ( ph -> ( H gsumg ( k e. ( M ... N ) |-> D ) ) = E )

Distinct variable groups:   x, k, N   k, M, x   B, k, x   C, k   x, D   x, E   ph, k   x, G   x, H   x, X

Allowed substitution hints:   ph( x)   C( x)   D( k)   E( k)   G( k)   H( k)   X( k)   .0. ( x, k)

Proof of Theorem gsumfzmhm2

Step	Hyp	Ref	Expression
1		gsummhm2.b	. . 3 |- B = ( Base ` G )
2		gsummhm2.z	. . 3 |- .0. = ( 0g ` G )
3		gsummhm2.g	. . 3 |- ( ph -> G e. CMnd )
4		gsummhm2.h	. . 3 |- ( ph -> H e. Mnd )
5		gsumfzmhm2.m	. . 3 |- ( ph -> M e. ZZ )
6		gsumfzmhm2.n	. . 3 |- ( ph -> N e. ZZ )
7		gsummhm2.k	. . 3 |- ( ph -> ( x e. B |-> C ) e. ( G MndHom H ) )
8		gsumfzmhm2.f	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> X e. B )
9	8	fmpttd 5839	. . 3 |- ( ph -> ( k e. ( M ... N ) |-> X ) : ( M ... N ) --> B )
10	1, 2, 3, 4, 5, 6, 7, 9	gsumfzmhm 14102	. 2 |- ( ph -> ( H gsumg ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) ) = ( ( x e. B |-> C ) ` ( G gsumg ( k e. ( M ... N ) |-> X ) ) ) )
11		eqidd 2235	. . . 4 |- ( ph -> ( k e. ( M ... N ) |-> X ) = ( k e. ( M ... N ) |-> X ) )
12		eqidd 2235	. . . 4 |- ( ph -> ( x e. B |-> C ) = ( x e. B |-> C ) )
13		gsummhm2.1	. . . 4 |- ( x = X -> C = D )
14	8, 11, 12, 13	fmptco 5850	. . 3 |- ( ph -> ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) = ( k e. ( M ... N ) |-> D ) )
15	14	oveq2d 6076	. 2 |- ( ph -> ( H gsumg ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) ) = ( H gsumg ( k e. ( M ... N ) |-> D ) ) )
16		eqid 2234	. . 3 |- ( x e. B |-> C ) = ( x e. B |-> C )
17		gsumfzmhm2.2	. . 3 |- ( x = ( G gsumg ( k e. ( M ... N ) |-> X ) ) -> C = E )
18	3	cmnmndd 14067	. . . 4 |- ( ph -> G e. Mnd )
19	1, 2, 18, 5, 6, 9	gsumfzcl 13760	. . 3 |- ( ph -> ( G gsumg ( k e. ( M ... N ) |-> X ) ) e. B )
20	17	eleq1d 2303	. . . 4 |- ( x = ( G gsumg ( k e. ( M ... N ) |-> X ) ) -> ( C e. ( Base ` H ) <-> E e. ( Base ` H ) ) )
21		eqid 2234	. . . . . . 7 |- ( Base ` H ) = ( Base ` H )
22	1, 21	mhmf 13726	. . . . . 6 |- ( ( x e. B |-> C ) e. ( G MndHom H ) -> ( x e. B |-> C ) : B --> ( Base ` H ) )
23	7, 22	syl 14	. . . . 5 |- ( ph -> ( x e. B |-> C ) : B --> ( Base ` H ) )
24	16	fmpt 5834	. . . . 5 |- ( A. x e. B C e. ( Base ` H ) <-> ( x e. B |-> C ) : B --> ( Base ` H ) )
25	23, 24	sylibr 134	. . . 4 |- ( ph -> A. x e. B C e. ( Base ` H ) )
26	20, 25, 19	rspcdva 2928	. . 3 |- ( ph -> E e. ( Base ` H ) )
27	16, 17, 19, 26	fvmptd3 5778	. 2 |- ( ph -> ( ( x e. B |-> C ) ` ( G gsumg ( k e. ( M ... N ) |-> X ) ) ) = E )
28	10, 15, 27	3eqtr3d 2275	1 |- ( ph -> ( H gsumg ( k e. ( M ... N ) |-> D ) ) = E )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   A.wral 2522   |-> cmpt 4177   o. ccom 4760   -->wf 5355   ` cfv 5359  (class class class)co 6060   ZZcz 9599   ...cfz 10366   Basecbs 13302   0gc0g 13559   gsum_g cgsu 13560   Mndcmnd 13683   MndHom cmhm 13718  CMndccmn 14043

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-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4666  ax-iinf 4717  ax-cnex 8236  ax-resscn 8237  ax-1cn 8238  ax-1re 8239  ax-icn 8240  ax-addcl 8241  ax-addrcl 8242  ax-mulcl 8243  ax-addcom 8245  ax-addass 8247  ax-distr 8249  ax-i2m1 8250  ax-0lt1 8251  ax-0id 8253  ax-rnegex 8254  ax-cnre 8256  ax-pre-ltirr 8257  ax-pre-ltwlin 8258  ax-pre-lttrn 8259  ax-pre-apti 8260  ax-pre-ltadd 8261

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-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-iord 4493  df-on 4495  df-ilim 4496  df-suc 4498  df-iom 4720  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-ima 4769  df-iota 5319  df-fun 5361  df-fn 5362  df-f 5363  df-f1 5364  df-fo 5365  df-f1o 5366  df-fv 5367  df-riota 6013  df-ov 6063  df-oprab 6064  df-mpo 6065  df-1st 6349  df-2nd 6350  df-recs 6551  df-frec 6637  df-1o 6662  df-er 6782  df-map 6899  df-en 6991  df-fin 6993  df-pnf 8328  df-mnf 8329  df-xr 8330  df-ltxr 8331  df-le 8332  df-sub 8465  df-neg 8466  df-inn 9260  df-2 9318  df-n0 9519  df-z 9600  df-uz 9877  df-fz 10367  df-fzo 10504  df-seqfrec 10839  df-ndx 13305  df-slot 13306  df-base 13308  df-plusg 13393  df-0g 13561  df-igsum 13562  df-mgm 13625  df-sgrp 13671  df-mnd 13684  df-mhm 13720  df-cmn 14045

This theorem is referenced by:  lgseisenlem4  16078