Theorem gzsummhm 14122

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 13752	. . . . 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 13688	. . . . . 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 3644	. . . . 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 5694	. . 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 13749	. . . . . . . 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 5548	. . . . . 6 |- ( ph -> ( K o. F ) : ( M ... N ) --> ( Base ` H ) )
25	19, 3, 20, 21, 10, 11, 24	gzsumfzval 13688	. . . . 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 3644	. . . 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 14088	. . . . . . 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 13713	. . . . . 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 5834	. . . . 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 9747	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
42	40	zred 9747	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
43		simpr 110	. . . . . 6 |- ( ( ph /\ -. N < M ) -> -. N < M )
44	41, 42, 43	nltled 8437	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
45		eluz2 9906	. . . . 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 13751	. . . . 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 5770	. . . . . 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 10846	. . . . . 6 |- ( ph -> ( M ... N ) e. Fin )
58	12, 57	fexd 5938	. . . . 5 |- ( ph -> F e. _V )
59	58	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
60		coexg 5327	. . . . . 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 13443	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
64	63	slotex 13357	. . . . . 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 13357	. . . . . 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 10945	. . 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 3647	. . . . 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 5694	. . 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 3647	. . . 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 9701	. . . 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 ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   _Vcvv 2821   ifcif 3635   class class class wbr 4125   o. ccom 4773   -->wf 5368   ` cfv 5372  (class class class)co 6075   Fincfn 7012   < clt 8350   <_ cle 8351   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   seqcseq 10862   Basecbs 13330   +g cplusg 13408   0gc0g 13587   gzsum_gz cgzsu 13588   Mndcmnd 13706   MndHom cmhm 13741  CMndccmn 14064

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-map 6914  df-en 7013  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-gzsum 13590  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-mhm 13743  df-cmn 14066

This theorem is referenced by:  gzsummhm2  14123  gsummhmfi  14141