Theorem gzsummhm2 14123

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 5854	. . 3 |- ( ph -> ( k e. ( M ... N ) |-> X ) : ( M ... N ) --> B )
10	1, 2, 3, 4, 5, 6, 7, 9	gzsummhm 14122	. 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 5865	. . 3 |- ( ph -> ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) = ( k e. ( M ... N ) |-> D ) )
15	14	oveq2d 6091	. 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 14088	. . . 4 |- ( ph -> G e. Mnd )
19	1, 2, 18, 5, 6, 9	gzsumcl 13781	. . 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 13749	. . . . . 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 5849	. . . . 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 5793	. 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 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   |-> cmpt 4187   o. ccom 4773   -->wf 5368   ` cfv 5372  (class class class)co 6075   ZZcz 9623   ...cfz 10390   Basecbs 13330   0gc0g 13587   gzsum_gz cgzsu 13588   Mndcmnd 13706   MndHom cmhm 13741  CMndccmn 14064

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-map 6914  df-en 7013  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-gzsum 13590  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-mhm 13743  df-cmn 14066

This theorem is referenced by: (None)