Theorem gsumfzmhm2 13724

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 5742	. . 3 |- ( ph -> ( k e. ( M ... N ) |-> X ) : ( M ... N ) --> B )
10	1, 2, 3, 4, 5, 6, 7, 9	gsumfzmhm 13723	. 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 2207	. . . 4 |- ( ph -> ( k e. ( M ... N ) |-> X ) = ( k e. ( M ... N ) |-> X ) )
12		eqidd 2207	. . . 4 |- ( ph -> ( x e. B |-> C ) = ( x e. B |-> C ) )
13		gsummhm2.1	. . . 4 |- ( x = X -> C = D )
14	8, 11, 12, 13	fmptco 5753	. . 3 |- ( ph -> ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) = ( k e. ( M ... N ) |-> D ) )
15	14	oveq2d 5967	. 2 |- ( ph -> ( H gsumg ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) ) = ( H gsumg ( k e. ( M ... N ) |-> D ) ) )
16		eqid 2206	. . 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 13688	. . . 4 |- ( ph -> G e. Mnd )
19	1, 2, 18, 5, 6, 9	gsumfzcl 13375	. . 3 |- ( ph -> ( G gsumg ( k e. ( M ... N ) |-> X ) ) e. B )
20	17	eleq1d 2275	. . . 4 |- ( x = ( G gsumg ( k e. ( M ... N ) |-> X ) ) -> ( C e. ( Base ` H ) <-> E e. ( Base ` H ) ) )
21		eqid 2206	. . . . . . 7 |- ( Base ` H ) = ( Base ` H )
22	1, 21	mhmf 13341	. . . . . 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 5737	. . . . 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 2883	. . 3 |- ( ph -> E e. ( Base ` H ) )
27	16, 17, 19, 26	fvmptd3 5680	. 2 |- ( ph -> ( ( x e. B |-> C ) ` ( G gsumg ( k e. ( M ... N ) |-> X ) ) ) = E )
28	10, 15, 27	3eqtr3d 2247	1 |- ( ph -> ( H gsumg ( k e. ( M ... N ) |-> D ) ) = E )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   A.wral 2485   |-> cmpt 4109   o. ccom 4683   -->wf 5272   ` cfv 5276  (class class class)co 5951   ZZcz 9379   ...cfz 10137   Basecbs 12876   0gc0g 13132   gsum_g cgsu 13133   Mndcmnd 13292   MndHom cmhm 13333  CMndccmn 13664

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-addcom 8032  ax-addass 8034  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-0id 8040  ax-rnegex 8041  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-frec 6484  df-1o 6509  df-er 6627  df-map 6744  df-en 6835  df-fin 6837  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-inn 9044  df-2 9102  df-n0 9303  df-z 9380  df-uz 9656  df-fz 10138  df-fzo 10272  df-seqfrec 10600  df-ndx 12879  df-slot 12880  df-base 12882  df-plusg 12966  df-0g 13134  df-igsum 13135  df-mgm 13232  df-sgrp 13278  df-mnd 13293  df-mhm 13335  df-cmn 13666

This theorem is referenced by:  lgseisenlem4  15594