Theorem gzsummhm 14145

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

Assertion

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

Proof of Theorem gzsummhm

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

Step	Hyp	Ref	Expression
1		gzsummhm.k	. . . . 5 |- ( ph -> K e. ( G MndHom H ) )
2		gzsummhm.z	. . . . . 6 |- .0. = ( 0g ` G )
3		eqid 2238	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
4	2, 3	mhm0 13775	. . . . 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		gzsummhm.b	. . . . . . 7 |- B = ( Base ` G )
8		eqid 2238	. . . . . . 7 |- ( +g ` G ) = ( +g ` G )
9		gzsummhm.g	. . . . . . 7 |- ( ph -> G e. CMnd )
10		gzsummhm.m	. . . . . . 7 |- ( ph -> M e. ZZ )
11		gzsummhm.n	. . . . . . 7 |- ( ph -> N e. ZZ )
12		gzsummhm.f	. . . . . . 7 |- ( ph -> F : ( M ... N ) --> B )
13	7, 2, 8, 9, 10, 11, 12	gzsumfzval 13711	. . . . . 6 |- ( ph -> ( G gzsumgz F ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
14	13	adantr 276	. . . . 5 |- ( ( ph /\ N < M ) -> ( G gzsumgz F ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
15		simpr 110	. . . . . 6 |- ( ( ph /\ N < M ) -> N < M )
16	15	iftrued 3647	. . . . 5 |- ( ( ph /\ N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = .0. )
17	14, 16	eqtrd 2271	. . . 4 |- ( ( ph /\ N < M ) -> ( G gzsumgz F ) = .0. )
18	17	fveq2d 5699	. . 3 |- ( ( ph /\ N < M ) -> ( K ` ( G gzsumgz F ) ) = ( K ` .0. ) )
19		eqid 2238	. . . . . 6 |- ( Base ` H ) = ( Base ` H )
20		eqid 2238	. . . . . 6 |- ( +g ` H ) = ( +g ` H )
21		gzsummhm.h	. . . . . 6 |- ( ph -> H e. Mnd )
22	7, 19	mhmf 13772	. . . . . . . 8 |- ( K e. ( G MndHom H ) -> K : B --> ( Base ` H ) )
23	1, 22	syl 14	. . . . . . 7 |- ( ph -> K : B --> ( Base ` H ) )
24	23, 12	fcod 5553	. . . . . 6 |- ( ph -> ( K o. F ) : ( M ... N ) --> ( Base ` H ) )
25	19, 3, 20, 21, 10, 11, 24	gzsumfzval 13711	. . . . 5 |- ( ph -> ( H gzsumgz ( K o. F ) ) = if ( N < M , ( 0g ` H ) , ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) ) )
26	25	adantr 276	. . . 4 |- ( ( ph /\ N < M ) -> ( H gzsumgz ( K o. F ) ) = if ( N < M , ( 0g ` H ) , ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) ) )
27	15	iftrued 3647	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , ( 0g ` H ) , ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) ) = ( 0g ` H ) )
28	26, 27	eqtrd 2271	. . 3 |- ( ( ph /\ N < M ) -> ( H gzsumgz ( K o. F ) ) = ( 0g ` H ) )
29	6, 18, 28	3eqtr4rd 2282	. 2 |- ( ( ph /\ N < M ) -> ( H gzsumgz ( K o. F ) ) = ( K ` ( G gzsumgz F ) ) )
30	9	cmnmndd 14111	. . . . . . 7 |- ( ph -> G e. Mnd )
31	30	adantr 276	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> G e. Mnd )
32		simprl 535	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> x e. B )
33		simprr 537	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> y e. B )
34	7, 8	mndcl 13736	. . . . . 6 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
35	31, 32, 33, 34	syl3anc 1278	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) e. B )
36	35	adantlr 481	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) e. B )
37	12	ffvelcdmda 5843	. . . . 5 |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) e. B )
38	37	adantlr 481	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> ( F ` x ) e. B )
39	10	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> M e. ZZ )
40	11	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> N e. ZZ )
41	39	zred 9768	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
42	40	zred 9768	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
43		simpr 110	. . . . . 6 |- ( ( ph /\ -. N < M ) -> -. N < M )
44	41, 42, 43	nltled 8447	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
45		eluz2 9927	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
46	39, 40, 44, 45	syl3anbrc 1212	. . . 4 |- ( ( ph /\ -. N < M ) -> N e. ( ZZ>= ` M ) )
47	1	ad2antrr 492	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> K e. ( G MndHom H ) )
48		simprl 535	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> x e. B )
49		simprr 537	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> y e. B )
50	7, 8, 20	mhmlin 13774	. . . . 5 |- ( ( K e. ( G MndHom H ) /\ x e. B /\ y e. B ) -> ( K ` ( x ( +g ` G ) y ) ) = ( ( K ` x ) ( +g ` H ) ( K ` y ) ) )
51	47, 48, 49, 50	syl3anc 1278	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> ( K ` ( x ( +g ` G ) y ) ) = ( ( K ` x ) ( +g ` H ) ( K ` y ) ) )
52	12	ad2antrr 492	. . . . . 6 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> F : ( M ... N ) --> B )
53		simpr 110	. . . . . 6 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> x e. ( M ... N ) )
54		fvco3 5776	. . . . . 6 |- ( ( F : ( M ... N ) --> B /\ x e. ( M ... N ) ) -> ( ( K o. F ) ` x ) = ( K ` ( F ` x ) ) )
55	52, 53, 54	syl2anc 415	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> ( ( K o. F ) ` x ) = ( K ` ( F ` x ) ) )
56	55	eqcomd 2244	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> ( K ` ( F ` x ) ) = ( ( K o. F ) ` x ) )
57	10, 11	fzfigd 10868	. . . . . 6 |- ( ph -> ( M ... N ) e. Fin )
58	12, 57	fexd 5948	. . . . 5 |- ( ph -> F e. _V )
59	58	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
60		coexg 5332	. . . . . 6 |- ( ( K e. ( G MndHom H ) /\ F e. _V ) -> ( K o. F ) e. _V )
61	1, 58, 60	syl2anc 415	. . . . 5 |- ( ph -> ( K o. F ) e. _V )
62	61	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( K o. F ) e. _V )
63		plusgslid 13466	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
64	63	slotex 13379	. . . . . 6 |- ( G e. CMnd -> ( +g ` G ) e. _V )
65	9, 64	syl 14	. . . . 5 |- ( ph -> ( +g ` G ) e. _V )
66	65	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( +g ` G ) e. _V )
67	63	slotex 13379	. . . . . 6 |- ( H e. Mnd -> ( +g ` H ) e. _V )
68	21, 67	syl 14	. . . . 5 |- ( ph -> ( +g ` H ) e. _V )
69	68	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( +g ` H ) e. _V )
70	36, 38, 46, 51, 56, 59, 62, 66, 69	seqhomog 10967	. . 3 |- ( ( ph /\ -. N < M ) -> ( K ` ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) )
71	13	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> ( G gzsumgz F ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
72	43	iffalsed 3650	. . . . 5 |- ( ( ph /\ -. N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
73	71, 72	eqtrd 2271	. . . 4 |- ( ( ph /\ -. N < M ) -> ( G gzsumgz F ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
74	73	fveq2d 5699	. . 3 |- ( ( ph /\ -. N < M ) -> ( K ` ( G gzsumgz F ) ) = ( K ` ( seq M ( ( +g ` G ) , F ) ` N ) ) )
75	25	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( H gzsumgz ( K o. F ) ) = if ( N < M , ( 0g ` H ) , ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) ) )
76	43	iffalsed 3650	. . . 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 ) )
77	75, 76	eqtrd 2271	. . 3 |- ( ( ph /\ -. N < M ) -> ( H gzsumgz ( K o. F ) ) = ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) )
78	70, 74, 77	3eqtr4rd 2282	. 2 |- ( ( ph /\ -. N < M ) -> ( H gzsumgz ( K o. F ) ) = ( K ` ( G gzsumgz F ) ) )
79		zdclt 9722	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
80	11, 10, 79	syl2anc 415	. . 3 |- ( ph -> DECID N < M )
81		exmiddc 848	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
82	80, 81	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
83	29, 78, 82	mpjaodan 810	1 |- ( ph -> ( H gzsumgz ( K o. F ) ) = ( K ` ( G gzsumgz F ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   _Vcvv 2821   ifcif 3638   class class class wbr 4130   o. ccom 4778   -->wf 5373   ` cfv 5377  (class class class)co 6085   Fincfn 7022   < clt 8360   <_ cle 8361   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411   seqcseq 10884   Basecbs 13352   +g cplusg 13431   0gc0g 13610   gzsum_gz cgzsu 13611   Mndcmnd 13729   MndHom cmhm 13764  CMndccmn 14087

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-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

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-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-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  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-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

This theorem is used by:  gzsummhm2  14146  gsummhmfi  14164