Theorem gsumfzmhm 13901

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 2229	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
4	2, 3	mhm0 13522	. . . . 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 2229	. . . . . . 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 13445	. . . . . 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 3609	. . . . 5 |- ( ( ph /\ N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = .0. )
17	14, 16	eqtrd 2262	. . . 4 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = .0. )
18	17	fveq2d 5636	. . 3 |- ( ( ph /\ N < M ) -> ( K ` ( G gsumg F ) ) = ( K ` .0. ) )
19		eqid 2229	. . . . . 6 |- ( Base ` H ) = ( Base ` H )
20		eqid 2229	. . . . . 6 |- ( +g ` H ) = ( +g ` H )
21		gsummhm.h	. . . . . 6 |- ( ph -> H e. Mnd )
22	7, 19	mhmf 13519	. . . . . . . 8 |- ( K e. ( G MndHom H ) -> K : B --> ( Base ` H ) )
23	1, 22	syl 14	. . . . . . 7 |- ( ph -> K : B --> ( Base ` H ) )
24		fco 5494	. . . . . . 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 13445	. . . . 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 3609	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , ( 0g ` H ) , ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) ) = ( 0g ` H ) )
29	27, 28	eqtrd 2262	. . 3 |- ( ( ph /\ N < M ) -> ( H gsumg ( K o. F ) ) = ( 0g ` H ) )
30	6, 18, 29	3eqtr4rd 2273	. 2 |- ( ( ph /\ N < M ) -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )
31	9	cmnmndd 13866	. . . . . . 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 13477	. . . . . 6 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
36	32, 33, 34, 35	syl3anc 1271	. . . . 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 5775	. . . . 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 9585	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
43	41	zred 9585	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
44		simpr 110	. . . . . 6 |- ( ( ph /\ -. N < M ) -> -. N < M )
45	42, 43, 44	nltled 8283	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
46		eluz2 9744	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
47	40, 41, 45, 46	syl3anbrc 1205	. . . 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 13521	. . . . 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 1271	. . . 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 5710	. . . . . 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 2235	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> ( K ` ( F ` x ) ) = ( ( K o. F ) ` x ) )
58	10, 11	fzfigd 10670	. . . . . 6 |- ( ph -> ( M ... N ) e. Fin )
59	12, 58	fexd 5876	. . . . 5 |- ( ph -> F e. _V )
60	59	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
61		coexg 5276	. . . . . 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 13166	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
65	64	slotex 13080	. . . . . 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 13080	. . . . . 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 10769	. . 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 3612	. . . . 5 |- ( ( ph /\ -. N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
74	72, 73	eqtrd 2262	. . . 4 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
75	74	fveq2d 5636	. . 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 3612	. . . 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 2262	. . 3 |- ( ( ph /\ -. N < M ) -> ( H gsumg ( K o. F ) ) = ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) )
79	71, 75, 78	3eqtr4rd 2273	. 2 |- ( ( ph /\ -. N < M ) -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )
80		zdclt 9540	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
81	11, 10, 80	syl2anc 411	. . 3 |- ( ph -> DECID N < M )
82		exmiddc 841	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
83	81, 82	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
84	30, 79, 83	mpjaodan 803	1 |- ( ph -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   _Vcvv 2799   ifcif 3602   class class class wbr 4083   o. ccom 4724   -->wf 5317   ` cfv 5321  (class class class)co 6010   Fincfn 6900   < clt 8197   <_ cle 8198   ZZcz 9462   ZZ>=cuz 9738   ...cfz 10221   seqcseq 10686   Basecbs 13053   +g cplusg 13131   0gc0g 13310   gsum_g cgsu 13311   Mndcmnd 13470   MndHom cmhm 13511  CMndccmn 13842

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-1o 6573  df-er 6693  df-map 6810  df-en 6901  df-fin 6903  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-2 9185  df-n0 9386  df-z 9463  df-uz 9739  df-fz 10222  df-fzo 10356  df-seqfrec 10687  df-ndx 13056  df-slot 13057  df-base 13059  df-plusg 13144  df-0g 13312  df-igsum 13313  df-mgm 13410  df-sgrp 13456  df-mnd 13471  df-mhm 13513  df-cmn 13844

This theorem is referenced by:  gsumfzmhm2  13902