Theorem gsumfzmhm 13846

Description: Apply a monoid homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 6-Jun-2019.) (Revised by Jim Kingdon, 8-Sep-2025.)

Hypotheses

Ref	Expression
gsummhm.b	|- B = ( Base ` G )
gsummhm.z	|- .0. = ( 0g ` G )
gsummhm.g	|- ( ph -> G e. CMnd )
gsummhm.h	|- ( ph -> H e. Mnd )
gsummhm.m	|- ( ph -> M e. ZZ )
gsummhm.n	|- ( ph -> N e. ZZ )
gsummhm.k	|- ( ph -> K e. ( G MndHom H ) )
gsummhm.f	|- ( ph -> F : ( M ... N ) --> B )

Assertion

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

Proof of Theorem gsumfzmhm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsummhm.k	. . . . 5 |- ( ph -> K e. ( G MndHom H ) )
2		gsummhm.z	. . . . . 6 |- .0. = ( 0g ` G )
3		eqid 2209	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
4	2, 3	mhm0 13467	. . . . 5 |- ( K e. ( G MndHom H ) -> ( K ` .0. ) = ( 0g ` H ) )
5	1, 4	syl 14	. . . 4 |- ( ph -> ( K ` .0. ) = ( 0g ` H ) )
6	5	adantr 276	. . 3 |- ( ( ph /\ N < M ) -> ( K ` .0. ) = ( 0g ` H ) )
7		gsummhm.b	. . . . . . 7 |- B = ( Base ` G )
8		eqid 2209	. . . . . . 7 |- ( +g ` G ) = ( +g ` G )
9		gsummhm.g	. . . . . . 7 |- ( ph -> G e. CMnd )
10		gsummhm.m	. . . . . . 7 |- ( ph -> M e. ZZ )
11		gsummhm.n	. . . . . . 7 |- ( ph -> N e. ZZ )
12		gsummhm.f	. . . . . . 7 |- ( ph -> F : ( M ... N ) --> B )
13	7, 2, 8, 9, 10, 11, 12	gsumfzval 13390	. . . . . 6 |- ( ph -> ( G gsumg F ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
14	13	adantr 276	. . . . 5 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
15		simpr 110	. . . . . 6 |- ( ( ph /\ N < M ) -> N < M )
16	15	iftrued 3589	. . . . 5 |- ( ( ph /\ N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = .0. )
17	14, 16	eqtrd 2242	. . . 4 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = .0. )
18	17	fveq2d 5607	. . 3 |- ( ( ph /\ N < M ) -> ( K ` ( G gsumg F ) ) = ( K ` .0. ) )
19		eqid 2209	. . . . . 6 |- ( Base ` H ) = ( Base ` H )
20		eqid 2209	. . . . . 6 |- ( +g ` H ) = ( +g ` H )
21		gsummhm.h	. . . . . 6 |- ( ph -> H e. Mnd )
22	7, 19	mhmf 13464	. . . . . . . 8 |- ( K e. ( G MndHom H ) -> K : B --> ( Base ` H ) )
23	1, 22	syl 14	. . . . . . 7 |- ( ph -> K : B --> ( Base ` H ) )
24		fco 5465	. . . . . . 7 |- ( ( K : B --> ( Base ` H ) /\ F : ( M ... N ) --> B ) -> ( K o. F ) : ( M ... N ) --> ( Base ` H ) )
25	23, 12, 24	syl2anc 411	. . . . . 6 |- ( ph -> ( K o. F ) : ( M ... N ) --> ( Base ` H ) )
26	19, 3, 20, 21, 10, 11, 25	gsumfzval 13390	. . . . 5 |- ( ph -> ( H gsumg ( K o. F ) ) = if ( N < M , ( 0g ` H ) , ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) ) )
27	26	adantr 276	. . . 4 |- ( ( ph /\ N < M ) -> ( H gsumg ( K o. F ) ) = if ( N < M , ( 0g ` H ) , ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) ) )
28	15	iftrued 3589	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , ( 0g ` H ) , ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) ) = ( 0g ` H ) )
29	27, 28	eqtrd 2242	. . 3 |- ( ( ph /\ N < M ) -> ( H gsumg ( K o. F ) ) = ( 0g ` H ) )
30	6, 18, 29	3eqtr4rd 2253	. 2 |- ( ( ph /\ N < M ) -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )
31	9	cmnmndd 13811	. . . . . . 7 |- ( ph -> G e. Mnd )
32	31	adantr 276	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> G e. Mnd )
33		simprl 529	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> x e. B )
34		simprr 531	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> y e. B )
35	7, 8	mndcl 13422	. . . . . 6 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
36	32, 33, 34, 35	syl3anc 1252	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) e. B )
37	36	adantlr 477	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) e. B )
38	12	ffvelcdmda 5743	. . . . 5 |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) e. B )
39	38	adantlr 477	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> ( F ` x ) e. B )
40	10	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> M e. ZZ )
41	11	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> N e. ZZ )
42	40	zred 9537	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
43	41	zred 9537	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
44		simpr 110	. . . . . 6 |- ( ( ph /\ -. N < M ) -> -. N < M )
45	42, 43, 44	nltled 8235	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
46		eluz2 9696	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
47	40, 41, 45, 46	syl3anbrc 1186	. . . 4 |- ( ( ph /\ -. N < M ) -> N e. ( ZZ>= ` M ) )
48	1	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> K e. ( G MndHom H ) )
49		simprl 529	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> x e. B )
50		simprr 531	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> y e. B )
51	7, 8, 20	mhmlin 13466	. . . . 5 |- ( ( K e. ( G MndHom H ) /\ x e. B /\ y e. B ) -> ( K ` ( x ( +g ` G ) y ) ) = ( ( K ` x ) ( +g ` H ) ( K ` y ) ) )
52	48, 49, 50, 51	syl3anc 1252	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> ( K ` ( x ( +g ` G ) y ) ) = ( ( K ` x ) ( +g ` H ) ( K ` y ) ) )
53	12	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> F : ( M ... N ) --> B )
54		simpr 110	. . . . . 6 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> x e. ( M ... N ) )
55		fvco3 5678	. . . . . 6 |- ( ( F : ( M ... N ) --> B /\ x e. ( M ... N ) ) -> ( ( K o. F ) ` x ) = ( K ` ( F ` x ) ) )
56	53, 54, 55	syl2anc 411	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> ( ( K o. F ) ` x ) = ( K ` ( F ` x ) ) )
57	56	eqcomd 2215	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> ( K ` ( F ` x ) ) = ( ( K o. F ) ` x ) )
58	10, 11	fzfigd 10620	. . . . . 6 |- ( ph -> ( M ... N ) e. Fin )
59	12, 58	fexd 5842	. . . . 5 |- ( ph -> F e. _V )
60	59	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
61		coexg 5249	. . . . . 6 |- ( ( K e. ( G MndHom H ) /\ F e. _V ) -> ( K o. F ) e. _V )
62	1, 59, 61	syl2anc 411	. . . . 5 |- ( ph -> ( K o. F ) e. _V )
63	62	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( K o. F ) e. _V )
64		plusgslid 13111	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
65	64	slotex 13025	. . . . . 6 |- ( G e. CMnd -> ( +g ` G ) e. _V )
66	9, 65	syl 14	. . . . 5 |- ( ph -> ( +g ` G ) e. _V )
67	66	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( +g ` G ) e. _V )
68	64	slotex 13025	. . . . . 6 |- ( H e. Mnd -> ( +g ` H ) e. _V )
69	21, 68	syl 14	. . . . 5 |- ( ph -> ( +g ` H ) e. _V )
70	69	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( +g ` H ) e. _V )
71	37, 39, 47, 52, 57, 60, 63, 67, 70	seqhomog 10719	. . 3 |- ( ( ph /\ -. N < M ) -> ( K ` ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) )
72	13	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
73	44	iffalsed 3592	. . . . 5 |- ( ( ph /\ -. N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
74	72, 73	eqtrd 2242	. . . 4 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
75	74	fveq2d 5607	. . 3 |- ( ( ph /\ -. N < M ) -> ( K ` ( G gsumg F ) ) = ( K ` ( seq M ( ( +g ` G ) , F ) ` N ) ) )
76	26	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( H gsumg ( K o. F ) ) = if ( N < M , ( 0g ` H ) , ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) ) )
77	44	iffalsed 3592	. . . 4 |- ( ( ph /\ -. N < M ) -> if ( N < M , ( 0g ` H ) , ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) ) = ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) )
78	76, 77	eqtrd 2242	. . 3 |- ( ( ph /\ -. N < M ) -> ( H gsumg ( K o. F ) ) = ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) )
79	71, 75, 78	3eqtr4rd 2253	. 2 |- ( ( ph /\ -. N < M ) -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )
80		zdclt 9492	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
81	11, 10, 80	syl2anc 411	. . 3 |- ( ph -> DECID N < M )
82		exmiddc 840	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
83	81, 82	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
84	30, 79, 83	mpjaodan 802	1 |- ( ph -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 712  DECID wdc 838   = wceq 1375   e. wcel 2180   _Vcvv 2779   ifcif 3582   class class class wbr 4062   o. ccom 4700   -->wf 5290   ` cfv 5294  (class class class)co 5974   Fincfn 6857   < clt 8149   <_ cle 8150   ZZcz 9414   ZZ>=cuz 9690   ...cfz 10172   seqcseq 10636   Basecbs 12998   +g cplusg 13076   0gc0g 13255   gsum_g cgsu 13256   Mndcmnd 13415   MndHom cmhm 13456  CMndccmn 13787

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-addass 8069  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-frec 6507  df-1o 6532  df-er 6650  df-map 6767  df-en 6858  df-fin 6860  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-inn 9079  df-2 9137  df-n0 9338  df-z 9415  df-uz 9691  df-fz 10173  df-fzo 10307  df-seqfrec 10637  df-ndx 13001  df-slot 13002  df-base 13004  df-plusg 13089  df-0g 13257  df-igsum 13258  df-mgm 13355  df-sgrp 13401  df-mnd 13416  df-mhm 13458  df-cmn 13789

This theorem is referenced by:  gsumfzmhm2  13847