Theorem gsumfzmhm 13723

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 2206	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
4	2, 3	mhm0 13344	. . . . 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 2206	. . . . . . 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 13267	. . . . . 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 3579	. . . . 5 |- ( ( ph /\ N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = .0. )
17	14, 16	eqtrd 2239	. . . 4 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = .0. )
18	17	fveq2d 5587	. . 3 |- ( ( ph /\ N < M ) -> ( K ` ( G gsumg F ) ) = ( K ` .0. ) )
19		eqid 2206	. . . . . 6 |- ( Base ` H ) = ( Base ` H )
20		eqid 2206	. . . . . 6 |- ( +g ` H ) = ( +g ` H )
21		gsummhm.h	. . . . . 6 |- ( ph -> H e. Mnd )
22	7, 19	mhmf 13341	. . . . . . . 8 |- ( K e. ( G MndHom H ) -> K : B --> ( Base ` H ) )
23	1, 22	syl 14	. . . . . . 7 |- ( ph -> K : B --> ( Base ` H ) )
24		fco 5447	. . . . . . 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 13267	. . . . 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 3579	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , ( 0g ` H ) , ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) ) = ( 0g ` H ) )
29	27, 28	eqtrd 2239	. . 3 |- ( ( ph /\ N < M ) -> ( H gsumg ( K o. F ) ) = ( 0g ` H ) )
30	6, 18, 29	3eqtr4rd 2250	. 2 |- ( ( ph /\ N < M ) -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )
31	9	cmnmndd 13688	. . . . . . 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 13299	. . . . . 6 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
36	32, 33, 34, 35	syl3anc 1250	. . . . 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 5722	. . . . 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 9502	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
43	41	zred 9502	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
44		simpr 110	. . . . . 6 |- ( ( ph /\ -. N < M ) -> -. N < M )
45	42, 43, 44	nltled 8200	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
46		eluz2 9661	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
47	40, 41, 45, 46	syl3anbrc 1184	. . . 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 13343	. . . . 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 1250	. . . 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 5657	. . . . . 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 2212	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> ( K ` ( F ` x ) ) = ( ( K o. F ) ` x ) )
58	10, 11	fzfigd 10583	. . . . . 6 |- ( ph -> ( M ... N ) e. Fin )
59	12, 58	fexd 5821	. . . . 5 |- ( ph -> F e. _V )
60	59	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
61		coexg 5232	. . . . . 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 12988	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
65	64	slotex 12903	. . . . . 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 12903	. . . . . 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 10682	. . 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 3582	. . . . 5 |- ( ( ph /\ -. N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
74	72, 73	eqtrd 2239	. . . 4 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
75	74	fveq2d 5587	. . 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 3582	. . . 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 2239	. . 3 |- ( ( ph /\ -. N < M ) -> ( H gsumg ( K o. F ) ) = ( seq M ( ( +g ` H ) , ( K o. F ) ) ` N ) )
79	71, 75, 78	3eqtr4rd 2250	. 2 |- ( ( ph /\ -. N < M ) -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )
80		zdclt 9457	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
81	11, 10, 80	syl2anc 411	. . 3 |- ( ph -> DECID N < M )
82		exmiddc 838	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
83	81, 82	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
84	30, 79, 83	mpjaodan 800	1 |- ( ph -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 710  DECID wdc 836   = wceq 1373   e. wcel 2177   _Vcvv 2773   ifcif 3572   class class class wbr 4047   o. ccom 4683   -->wf 5272   ` cfv 5276  (class class class)co 5951   Fincfn 6834   < clt 8114   <_ cle 8115   ZZcz 9379   ZZ>=cuz 9655   ...cfz 10137   seqcseq 10599   Basecbs 12876   +g cplusg 12953   0gc0g 13132   gsum_g cgsu 13133   Mndcmnd 13292   MndHom cmhm 13333  CMndccmn 13664

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-addcom 8032  ax-addass 8034  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-0id 8040  ax-rnegex 8041  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-frec 6484  df-1o 6509  df-er 6627  df-map 6744  df-en 6835  df-fin 6837  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-inn 9044  df-2 9102  df-n0 9303  df-z 9380  df-uz 9656  df-fz 10138  df-fzo 10272  df-seqfrec 10600  df-ndx 12879  df-slot 12880  df-base 12882  df-plusg 12966  df-0g 13134  df-igsum 13135  df-mgm 13232  df-sgrp 13278  df-mnd 13293  df-mhm 13335  df-cmn 13666

This theorem is referenced by:  gsumfzmhm2  13724