Theorem gsumfzmhm 14052

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 2232	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
4	2, 3	mhm0 13673	. . . . 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 2232	. . . . . . 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 13596	. . . . . 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 3628	. . . . 5 |- ( ( ph /\ N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = .0. )
17	14, 16	eqtrd 2265	. . . 4 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = .0. )
18	17	fveq2d 5673	. . 3 |- ( ( ph /\ N < M ) -> ( K ` ( G gsumg F ) ) = ( K ` .0. ) )
19		eqid 2232	. . . . . 6 |- ( Base ` H ) = ( Base ` H )
20		eqid 2232	. . . . . 6 |- ( +g ` H ) = ( +g ` H )
21		gsummhm.h	. . . . . 6 |- ( ph -> H e. Mnd )
22	7, 19	mhmf 13670	. . . . . . . 8 |- ( K e. ( G MndHom H ) -> K : B --> ( Base ` H ) )
23	1, 22	syl 14	. . . . . . 7 |- ( ph -> K : B --> ( Base ` H ) )
24		fco 5526	. . . . . . 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 13596	. . . . 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 3628	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , ( 0g ` H ) , ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) ) = ( 0g ` H ) )
29	27, 28	eqtrd 2265	. . 3 |- ( ( ph /\ N < M ) -> ( H gsumg ( K o. F ) ) = ( 0g ` H ) )
30	6, 18, 29	3eqtr4rd 2276	. 2 |- ( ( ph /\ N < M ) -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )
31	9	cmnmndd 14017	. . . . . . 7 |- ( ph -> G e. Mnd )
32	31	adantr 276	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> G e. Mnd )
33		simprl 531	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> x e. B )
34		simprr 533	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> y e. B )
35	7, 8	mndcl 13628	. . . . . 6 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
36	32, 33, 34, 35	syl3anc 1274	. . . . 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 5811	. . . . 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 9699	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
43	41	zred 9699	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
44		simpr 110	. . . . . 6 |- ( ( ph /\ -. N < M ) -> -. N < M )
45	42, 43, 44	nltled 8393	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
46		eluz2 9858	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
47	40, 41, 45, 46	syl3anbrc 1208	. . . 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 531	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> x e. B )
50		simprr 533	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> y e. B )
51	7, 8, 20	mhmlin 13672	. . . . 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 1274	. . . 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 5747	. . . . . 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 2238	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> ( K ` ( F ` x ) ) = ( ( K o. F ) ` x ) )
58	10, 11	fzfigd 10792	. . . . . 6 |- ( ph -> ( M ... N ) e. Fin )
59	12, 58	fexd 5915	. . . . 5 |- ( ph -> F e. _V )
60	59	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
61		coexg 5306	. . . . . 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 13317	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
65	64	slotex 13231	. . . . . 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 13231	. . . . . 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 10891	. . 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 3631	. . . . 5 |- ( ( ph /\ -. N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
74	72, 73	eqtrd 2265	. . . 4 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
75	74	fveq2d 5673	. . 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 3631	. . . 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 2265	. . 3 |- ( ( ph /\ -. N < M ) -> ( H gsumg ( K o. F ) ) = ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) )
79	71, 75, 78	3eqtr4rd 2276	. 2 |- ( ( ph /\ -. N < M ) -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )
80		zdclt 9654	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
81	11, 10, 80	syl2anc 411	. . 3 |- ( ph -> DECID N < M )
82		exmiddc 844	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
83	81, 82	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
84	30, 79, 83	mpjaodan 806	1 |- ( ph -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2203   _Vcvv 2812   ifcif 3619   class class class wbr 4108   o. ccom 4752   -->wf 5347   ` cfv 5351  (class class class)co 6049   Fincfn 6974   < clt 8307   <_ cle 8308   ZZcz 9576   ZZ>=cuz 9852   ...cfz 10341   seqcseq 10808   Basecbs 13204   +g cplusg 13282   0gc0g 13461   gsum_g cgsu 13462   Mndcmnd 13621   MndHom cmhm 13662  CMndccmn 13993

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-1o 6646  df-er 6766  df-map 6883  df-en 6975  df-fin 6977  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-inn 9237  df-2 9295  df-n0 9496  df-z 9577  df-uz 9853  df-fz 10342  df-fzo 10476  df-seqfrec 10809  df-ndx 13207  df-slot 13208  df-base 13210  df-plusg 13295  df-0g 13463  df-igsum 13464  df-mgm 13561  df-sgrp 13607  df-mnd 13622  df-mhm 13664  df-cmn 13995

This theorem is referenced by:  gsumfzmhm2  14053