Theorem gsummhmfi 14141

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 11209	. . . 4 |- ( A e. Fin -> ( 1 ... (♯ ` A ) ) ~~ A )
3	1, 2	syl 14	. . 3 |- ( ph -> ( 1 ... (♯ ` A ) ) ~~ A )
4		bren 7020	. . 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 14088	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> H e. Mnd )
13		1zzd 9650	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> 1 e. ZZ )
14		hashcl 11198	. . . . . . 7 |- ( A e. Fin -> (♯ ` A ) e. NN0 )
15	1, 14	syl 14	. . . . . 6 |- ( ph -> (♯ ` A ) e. NN0 )
16	15	nn0zd 9745	. . . . 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 5634	. . . . . 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 5548	. . . 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 14122	. . 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 13749	. . . . . . . 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 5548	. . . . . 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 14129	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( H gsumg ( K o. F ) ) = ( H gzsumgz ( ( K o. F ) o. f ) ) )
34		coass 5301	. . . . 5 |- ( ( K o. F ) o. f ) = ( K o. ( F o. f ) )
35	34	oveq2i 6086	. . . 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 14129	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( G gsumg F ) = ( G gzsumgz ( F o. f ) ) )
38	37	fveq2d 5694	. . 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   class class class wbr 4125   o. ccom 4773   -->wf 5368   -1-1-onto->wf1o 5371   ` cfv 5372  (class class class)co 6075   ~~ cen 7010   Fincfn 7012   1c1 8170   NN0cn0 9542   ZZcz 9623   ...cfz 10390  ♯chash 11192   Basecbs 13330   0gc0g 13587   gzsum_gz cgzsu 13588   MndHom cmhm 13741  CMndccmn 14064   gsum_g cgsu 14127

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-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem 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 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-dom 7014  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-reap 8893  df-ap 8900  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-ihash 11193  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  df-gsumfi 14128

This theorem is referenced by:  gsummhm2fi  14142