Theorem gsumfzmhm2 13933

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 5802	. . 3 |- ( ph -> ( k e. ( M ... N ) |-> X ) : ( M ... N ) --> B )
10	1, 2, 3, 4, 5, 6, 7, 9	gsumfzmhm 13932	. 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 2232	. . . 4 |- ( ph -> ( k e. ( M ... N ) |-> X ) = ( k e. ( M ... N ) |-> X ) )
12		eqidd 2232	. . . 4 |- ( ph -> ( x e. B |-> C ) = ( x e. B |-> C ) )
13		gsummhm2.1	. . . 4 |- ( x = X -> C = D )
14	8, 11, 12, 13	fmptco 5813	. . 3 |- ( ph -> ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) = ( k e. ( M ... N ) |-> D ) )
15	14	oveq2d 6034	. 2 |- ( ph -> ( H gsumg ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) ) = ( H gsumg ( k e. ( M ... N ) |-> D ) ) )
16		eqid 2231	. . 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 13897	. . . 4 |- ( ph -> G e. Mnd )
19	1, 2, 18, 5, 6, 9	gsumfzcl 13584	. . 3 |- ( ph -> ( G gsumg ( k e. ( M ... N ) |-> X ) ) e. B )
20	17	eleq1d 2300	. . . 4 |- ( x = ( G gsumg ( k e. ( M ... N ) |-> X ) ) -> ( C e. ( Base ` H ) <-> E e. ( Base ` H ) ) )
21		eqid 2231	. . . . . . 7 |- ( Base ` H ) = ( Base ` H )
22	1, 21	mhmf 13550	. . . . . 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 5797	. . . . 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 2915	. . 3 |- ( ph -> E e. ( Base ` H ) )
27	16, 17, 19, 26	fvmptd3 5740	. 2 |- ( ph -> ( ( x e. B |-> C ) ` ( G gsumg ( k e. ( M ... N ) |-> X ) ) ) = E )
28	10, 15, 27	3eqtr3d 2272	1 |- ( ph -> ( H gsumg ( k e. ( M ... N ) |-> D ) ) = E )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   A.wral 2510   |-> cmpt 4150   o. ccom 4729   -->wf 5322   ` cfv 5326  (class class class)co 6018   ZZcz 9479   ...cfz 10243   Basecbs 13084   0gc0g 13341   gsum_g cgsu 13342   Mndcmnd 13501   MndHom cmhm 13542  CMndccmn 13873

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-er 6702  df-map 6819  df-en 6910  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-fzo 10378  df-seqfrec 10711  df-ndx 13087  df-slot 13088  df-base 13090  df-plusg 13175  df-0g 13343  df-igsum 13344  df-mgm 13441  df-sgrp 13487  df-mnd 13502  df-mhm 13544  df-cmn 13875

This theorem is referenced by:  lgseisenlem4  15805