Theorem gzsummhm2 14146

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
gzsummhm2.b	|- B = ( Base ` G )
gzsummhm2.z	|- .0. = ( 0g ` G )
gzsummhm2.g	|- ( ph -> G e. CMnd )
gzsummhm2.h	|- ( ph -> H e. Mnd )
gzsummhm2.m	|- ( ph -> M e. ZZ )
gzsummhm2.n	|- ( ph -> N e. ZZ )
gzsummhm2.k	|- ( ph -> ( x e. B |-> C ) e. ( G MndHom H ) )
gzsummhm2.f	|- ( ( ph /\ k e. ( M ... N ) ) -> X e. B )
gzsummhm2.1	|- ( x = X -> C = D )
gzsummhm2.2	|- ( x = ( G gzsumgz ( k e. ( M ... N ) |-> X ) ) -> C = E )

Assertion

Ref	Expression
gzsummhm2	|- ( ph -> ( H gzsumgz ( 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 gzsummhm2

Step	Hyp	Ref	Expression
1		gzsummhm2.b	. . 3 |- B = ( Base ` G )
2		gzsummhm2.z	. . 3 |- .0. = ( 0g ` G )
3		gzsummhm2.g	. . 3 |- ( ph -> G e. CMnd )
4		gzsummhm2.h	. . 3 |- ( ph -> H e. Mnd )
5		gzsummhm2.m	. . 3 |- ( ph -> M e. ZZ )
6		gzsummhm2.n	. . 3 |- ( ph -> N e. ZZ )
7		gzsummhm2.k	. . 3 |- ( ph -> ( x e. B |-> C ) e. ( G MndHom H ) )
8		gzsummhm2.f	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> X e. B )
9	8	fmpttd 5863	. . 3 |- ( ph -> ( k e. ( M ... N ) |-> X ) : ( M ... N ) --> B )
10	1, 2, 3, 4, 5, 6, 7, 9	gzsummhm 14145	. 2 |- ( ph -> ( H gzsumgz ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) ) = ( ( x e. B |-> C ) ` ( G gzsumgz ( k e. ( M ... N ) |-> X ) ) ) )
11		eqidd 2239	. . . 4 |- ( ph -> ( k e. ( M ... N ) |-> X ) = ( k e. ( M ... N ) |-> X ) )
12		eqidd 2239	. . . 4 |- ( ph -> ( x e. B |-> C ) = ( x e. B |-> C ) )
13		gzsummhm2.1	. . . 4 |- ( x = X -> C = D )
14	8, 11, 12, 13	fmptco 5874	. . 3 |- ( ph -> ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) = ( k e. ( M ... N ) |-> D ) )
15	14	oveq2d 6101	. 2 |- ( ph -> ( H gzsumgz ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) ) = ( H gzsumgz ( k e. ( M ... N ) |-> D ) ) )
16		eqid 2238	. . 3 |- ( x e. B |-> C ) = ( x e. B |-> C )
17		gzsummhm2.2	. . 3 |- ( x = ( G gzsumgz ( k e. ( M ... N ) |-> X ) ) -> C = E )
18	3	cmnmndd 14111	. . . 4 |- ( ph -> G e. Mnd )
19	1, 2, 18, 5, 6, 9	gzsumcl 13804	. . 3 |- ( ph -> ( G gzsumgz ( k e. ( M ... N ) |-> X ) ) e. B )
20	17	eleq1d 2307	. . . 4 |- ( x = ( G gzsumgz ( k e. ( M ... N ) |-> X ) ) -> ( C e. ( Base ` H ) <-> E e. ( Base ` H ) ) )
21		eqid 2238	. . . . . . 7 |- ( Base ` H ) = ( Base ` H )
22	1, 21	mhmf 13772	. . . . . 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 5858	. . . . 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 2934	. . 3 |- ( ph -> E e. ( Base ` H ) )
27	16, 17, 19, 26	fvmptd3 5799	. 2 |- ( ph -> ( ( x e. B |-> C ) ` ( G gzsumgz ( k e. ( M ... N ) |-> X ) ) ) = E )
28	10, 15, 27	3eqtr3d 2279	1 |- ( ph -> ( H gzsumgz ( k e. ( M ... N ) |-> D ) ) = E )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   |-> cmpt 4192   o. ccom 4778   -->wf 5373   ` cfv 5377  (class class class)co 6085   ZZcz 9644   ...cfz 10411   Basecbs 13352   0gc0g 13610   gzsum_gz cgzsu 13611   Mndcmnd 13729   MndHom cmhm 13764  CMndccmn 14087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-map 6924  df-en 7023  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-0g 13612  df-gzsum 13613  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-mhm 13766  df-cmn 14089

This theorem is used by: (None)