Theorem gzsummhm2 14236

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 5866	. . 3 |- ( ph -> ( k e. ( M ... N ) |-> X ) : ( M ... N ) --> B )
10	1, 2, 3, 4, 5, 6, 7, 9	gzsummhm 14235	. 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 5877	. . 3 |- ( ph -> ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) = ( k e. ( M ... N ) |-> D ) )
15	14	oveq2d 6104	. 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 14201	. . . 4 |- ( ph -> G e. Mnd )
19	1, 2, 18, 5, 6, 9	gzsumcl 13863	. . 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 13831	. . . . . 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 5861	. . . . 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 5802	. 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 4193   o. ccom 4780   -->wf 5376   ` cfv 5380  (class class class)co 6088   ZZcz 9655   ...cfz 10428   Basecbs 13410   0gc0g 13669   gzsum_gz cgzsu 13670   Mndcmnd 13788   MndHom cmhm 13823  CMndccmn 14177

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-map 6928  df-en 7027  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-0g 13671  df-gzsum 13672  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-mhm 13825  df-cmn 14179

This theorem is used by: (None)