Theorem gsummhmfi 14164

Description: Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 6-Jun-2019.)

Hypotheses

Ref	Expression
gsummhm.b	|- B = ( Base ` G )
gsummhm.z	|- .0. = ( 0g ` G )
gsummhm.g	|- ( ph -> G e. CMnd )
gsummhmfi.h	|- ( ph -> H e. CMnd )
gsummhmfi.a	|- ( ph -> A e. Fin )
gsummhm.k	|- ( ph -> K e. ( G MndHom H ) )
gsummhm.f	|- ( ph -> F : A --> B )

Assertion

Ref	Expression
gsummhmfi	|- ( ph -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )

Proof of Theorem gsummhmfi

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsummhmfi.a	. . . 4 |- ( ph -> A e. Fin )
2		isfinite4im 11231	. . . 4 |- ( A e. Fin -> ( 1 ... (♯ ` A ) ) ~~ A )
3	1, 2	syl 14	. . 3 |- ( ph -> ( 1 ... (♯ ` A ) ) ~~ A )
4		bren 7030	. . 3 |- ( ( 1 ... (♯ ` A ) ) ~~ A <-> E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
5	3, 4	sylib 122	. 2 |- ( ph -> E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
6		gsummhm.b	. . . 4 |- B = ( Base ` G )
7		gsummhm.z	. . . 4 |- .0. = ( 0g ` G )
8		gsummhm.g	. . . . 5 |- ( ph -> G e. CMnd )
9	8	adantr 276	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> G e. CMnd )
10		gsummhmfi.h	. . . . . 6 |- ( ph -> H e. CMnd )
11	10	adantr 276	. . . . 5 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> H e. CMnd )
12	11	cmnmndd 14111	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> H e. Mnd )
13		1zzd 9671	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> 1 e. ZZ )
14		hashcl 11220	. . . . . . 7 |- ( A e. Fin -> (♯ ` A ) e. NN0 )
15	1, 14	syl 14	. . . . . 6 |- ( ph -> (♯ ` A ) e. NN0 )
16	15	nn0zd 9766	. . . . 5 |- ( ph -> (♯ ` A ) e. ZZ )
17	16	adantr 276	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> (♯ ` A ) e. ZZ )
18		gsummhm.k	. . . . 5 |- ( ph -> K e. ( G MndHom H ) )
19	18	adantr 276	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> K e. ( G MndHom H ) )
20		gsummhm.f	. . . . . 6 |- ( ph -> F : A --> B )
21	20	adantr 276	. . . . 5 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> F : A --> B )
22		f1of 5639	. . . . . 6 |- ( f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A -> f : ( 1 ... (♯ ` A ) ) --> A )
23	22	adantl 277	. . . . 5 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> f : ( 1 ... (♯ ` A ) ) --> A )
24	21, 23	fcod 5553	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( F o. f ) : ( 1 ... (♯ ` A ) ) --> B )
25	6, 7, 9, 12, 13, 17, 19, 24	gzsummhm 14145	. . 3 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( H gzsumgz ( K o. ( F o. f ) ) ) = ( K ` ( G gzsumgz ( F o. f ) ) ) )
26		eqid 2238	. . . . 5 |- ( Base ` H ) = ( Base ` H )
27	6, 26	mhmf 13772	. . . . . . . 8 |- ( K e. ( G MndHom H ) -> K : B --> ( Base ` H ) )
28	18, 27	syl 14	. . . . . . 7 |- ( ph -> K : B --> ( Base ` H ) )
29	28, 20	fcod 5553	. . . . . 6 |- ( ph -> ( K o. F ) : A --> ( Base ` H ) )
30	29	adantr 276	. . . . 5 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( K o. F ) : A --> ( Base ` H ) )
31	1	adantr 276	. . . . 5 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> A e. Fin )
32		simpr 110	. . . . 5 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
33	26, 11, 30, 31, 32	gsumvalfi 14152	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( H gsumg ( K o. F ) ) = ( H gzsumgz ( ( K o. F ) o. f ) ) )
34		coass 5306	. . . . 5 |- ( ( K o. F ) o. f ) = ( K o. ( F o. f ) )
35	34	oveq2i 6096	. . . 4 |- ( H gzsumgz ( ( K o. F ) o. f ) ) = ( H gzsumgz ( K o. ( F o. f ) ) )
36	33, 35	eqtrdi 2287	. . 3 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( H gsumg ( K o. F ) ) = ( H gzsumgz ( K o. ( F o. f ) ) ) )
37	6, 9, 21, 31, 32	gsumvalfi 14152	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( G gsumg F ) = ( G gzsumgz ( F o. f ) ) )
38	37	fveq2d 5699	. . 3 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( K ` ( G gsumg F ) ) = ( K ` ( G gzsumgz ( F o. f ) ) ) )
39	25, 36, 38	3eqtr4d 2281	. 2 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )
40	5, 39	exlimddv 1954	1 |- ( ph -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   class class class wbr 4130   o. ccom 4778   -->wf 5373   -1-1-onto->wf1o 5376   ` cfv 5377  (class class class)co 6085   ~~ cen 7020   Fincfn 7022   1c1 8180   NN0cn0 9563   ZZcz 9644   ...cfz 10411  ♯chash 11214   Basecbs 13352   0gc0g 13610   gzsum_gz cgzsu 13611   MndHom cmhm 13764  CMndccmn 14087   gsum_g cgsu 14150

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-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof depends on definitions:  df-bi 117  df-stab 843  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-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-dom 7024  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-reap 8903  df-ap 8910  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-ihash 11215  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  df-gsumfi 14151

This theorem is used by:  gsummhm2fi  14165