Theorem gsumfzmptfidmadd 13897

Description: The sum of two group sums expressed as mappings with finite domain. (Contributed by AV, 23-Jul-2019.) (Revised by Jim Kingdon, 31-Aug-2025.)

Hypotheses

Ref	Expression
gsummptfidmadd.b	|- B = ( Base ` G )
gsummptfidmadd.p	|- .+ = ( +g ` G )
gsummptfidmadd.g	|- ( ph -> G e. CMnd )
gsumfzmptfidmadd.m	|- ( ph -> M e. ZZ )
gsumfzmptfidmadd.n	|- ( ph -> N e. ZZ )
gsumfzmptfidmadd.c	|- ( ( ph /\ x e. ( M ... N ) ) -> C e. B )
gsumfzmptfidmadd.d	|- ( ( ph /\ x e. ( M ... N ) ) -> D e. B )
gsumfzmptfidmadd.f	|- F = ( x e. ( M ... N ) |-> C )
gsumfzmptfidmadd.h	|- H = ( x e. ( M ... N ) |-> D )

Assertion

Ref	Expression
gsumfzmptfidmadd	|- ( ph -> ( G gsumg ( x e. ( M ... N ) |-> ( C .+ D ) ) ) = ( ( G gsumg F ) .+ ( G gsumg H ) ) )

Distinct variable groups:   x, B   ph, x   x, .+   x, M   x, N

Allowed substitution hints:   C( x)   D( x)   F( x)   G( x)   H( x)

Proof of Theorem gsumfzmptfidmadd

Dummy variables k p q r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 |- ( ( ph /\ N < M ) -> N < M )
2	1	iftrued 3609	. . 3 |- ( ( ph /\ N < M ) -> if ( N < M , ( 0g ` G ) , ( seq M ( .+ , ( x e. ( M ... N ) |-> ( C .+ D ) ) ) ` N ) ) = ( 0g ` G ) )
3		gsummptfidmadd.b	. . . . 5 |- B = ( Base ` G )
4		eqid 2229	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
5		gsummptfidmadd.p	. . . . 5 |- .+ = ( +g ` G )
6		gsummptfidmadd.g	. . . . 5 |- ( ph -> G e. CMnd )
7		gsumfzmptfidmadd.m	. . . . 5 |- ( ph -> M e. ZZ )
8		gsumfzmptfidmadd.n	. . . . 5 |- ( ph -> N e. ZZ )
9	6	cmnmndd 13866	. . . . . . . 8 |- ( ph -> G e. Mnd )
10	9	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. ( M ... N ) ) -> G e. Mnd )
11		gsumfzmptfidmadd.c	. . . . . . 7 |- ( ( ph /\ x e. ( M ... N ) ) -> C e. B )
12		gsumfzmptfidmadd.d	. . . . . . 7 |- ( ( ph /\ x e. ( M ... N ) ) -> D e. B )
13	3, 5	mndcl 13477	. . . . . . 7 |- ( ( G e. Mnd /\ C e. B /\ D e. B ) -> ( C .+ D ) e. B )
14	10, 11, 12, 13	syl3anc 1271	. . . . . 6 |- ( ( ph /\ x e. ( M ... N ) ) -> ( C .+ D ) e. B )
15	14	fmpttd 5795	. . . . 5 |- ( ph -> ( x e. ( M ... N ) |-> ( C .+ D ) ) : ( M ... N ) --> B )
16	3, 4, 5, 6, 7, 8, 15	gsumfzval 13445	. . . 4 |- ( ph -> ( G gsumg ( x e. ( M ... N ) |-> ( C .+ D ) ) ) = if ( N < M , ( 0g ` G ) , ( seq M ( .+ , ( x e. ( M ... N ) |-> ( C .+ D ) ) ) ` N ) ) )
17	16	adantr 276	. . 3 |- ( ( ph /\ N < M ) -> ( G gsumg ( x e. ( M ... N ) |-> ( C .+ D ) ) ) = if ( N < M , ( 0g ` G ) , ( seq M ( .+ , ( x e. ( M ... N ) |-> ( C .+ D ) ) ) ` N ) ) )
18		gsumfzmptfidmadd.f	. . . . . . . . 9 |- F = ( x e. ( M ... N ) |-> C )
19	11, 18	fmptd 5794	. . . . . . . 8 |- ( ph -> F : ( M ... N ) --> B )
20	3, 4, 5, 6, 7, 8, 19	gsumfzval 13445	. . . . . . 7 |- ( ph -> ( G gsumg F ) = if ( N < M , ( 0g ` G ) , ( seq M ( .+ , F ) ` N ) ) )
21	20	adantr 276	. . . . . 6 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = if ( N < M , ( 0g ` G ) , ( seq M ( .+ , F ) ` N ) ) )
22	1	iftrued 3609	. . . . . 6 |- ( ( ph /\ N < M ) -> if ( N < M , ( 0g ` G ) , ( seq M ( .+ , F ) ` N ) ) = ( 0g ` G ) )
23	21, 22	eqtrd 2262	. . . . 5 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = ( 0g ` G ) )
24		gsumfzmptfidmadd.h	. . . . . . . . 9 |- H = ( x e. ( M ... N ) |-> D )
25	12, 24	fmptd 5794	. . . . . . . 8 |- ( ph -> H : ( M ... N ) --> B )
26	3, 4, 5, 6, 7, 8, 25	gsumfzval 13445	. . . . . . 7 |- ( ph -> ( G gsumg H ) = if ( N < M , ( 0g ` G ) , ( seq M ( .+ , H ) ` N ) ) )
27	26	adantr 276	. . . . . 6 |- ( ( ph /\ N < M ) -> ( G gsumg H ) = if ( N < M , ( 0g ` G ) , ( seq M ( .+ , H ) ` N ) ) )
28	1	iftrued 3609	. . . . . 6 |- ( ( ph /\ N < M ) -> if ( N < M , ( 0g ` G ) , ( seq M ( .+ , H ) ` N ) ) = ( 0g ` G ) )
29	27, 28	eqtrd 2262	. . . . 5 |- ( ( ph /\ N < M ) -> ( G gsumg H ) = ( 0g ` G ) )
30	23, 29	oveq12d 6028	. . . 4 |- ( ( ph /\ N < M ) -> ( ( G gsumg F ) .+ ( G gsumg H ) ) = ( ( 0g ` G ) .+ ( 0g ` G ) ) )
31	3, 4	mndidcl 13484	. . . . . 6 |- ( G e. Mnd -> ( 0g ` G ) e. B )
32	3, 5, 4	mndlid 13489	. . . . . 6 |- ( ( G e. Mnd /\ ( 0g ` G ) e. B ) -> ( ( 0g ` G ) .+ ( 0g ` G ) ) = ( 0g ` G ) )
33	9, 31, 32	syl2anc2 412	. . . . 5 |- ( ph -> ( ( 0g ` G ) .+ ( 0g ` G ) ) = ( 0g ` G ) )
34	33	adantr 276	. . . 4 |- ( ( ph /\ N < M ) -> ( ( 0g ` G ) .+ ( 0g ` G ) ) = ( 0g ` G ) )
35	30, 34	eqtrd 2262	. . 3 |- ( ( ph /\ N < M ) -> ( ( G gsumg F ) .+ ( G gsumg H ) ) = ( 0g ` G ) )
36	2, 17, 35	3eqtr4d 2272	. 2 |- ( ( ph /\ N < M ) -> ( G gsumg ( x e. ( M ... N ) |-> ( C .+ D ) ) ) = ( ( G gsumg F ) .+ ( G gsumg H ) ) )
37	9	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( p e. B /\ q e. B ) ) -> G e. Mnd )
38		simprl 529	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( p e. B /\ q e. B ) ) -> p e. B )
39		simprr 531	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( p e. B /\ q e. B ) ) -> q e. B )
40	3, 5	mndcl 13477	. . . . 5 |- ( ( G e. Mnd /\ p e. B /\ q e. B ) -> ( p .+ q ) e. B )
41	37, 38, 39, 40	syl3anc 1271	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( p e. B /\ q e. B ) ) -> ( p .+ q ) e. B )
42	6	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( p e. B /\ q e. B ) ) -> G e. CMnd )
43	3, 5	cmncom 13860	. . . . 5 |- ( ( G e. CMnd /\ p e. B /\ q e. B ) -> ( p .+ q ) = ( q .+ p ) )
44	42, 38, 39, 43	syl3anc 1271	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( p e. B /\ q e. B ) ) -> ( p .+ q ) = ( q .+ p ) )
45	9	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( p e. B /\ q e. B /\ r e. B ) ) -> G e. Mnd )
46	3, 5	mndass 13478	. . . . 5 |- ( ( G e. Mnd /\ ( p e. B /\ q e. B /\ r e. B ) ) -> ( ( p .+ q ) .+ r ) = ( p .+ ( q .+ r ) ) )
47	45, 46	sylancom 420	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( p e. B /\ q e. B /\ r e. B ) ) -> ( ( p .+ q ) .+ r ) = ( p .+ ( q .+ r ) ) )
48	7	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> M e. ZZ )
49	8	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> N e. ZZ )
50	48	zred 9585	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
51	49	zred 9585	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
52		simpr 110	. . . . . 6 |- ( ( ph /\ -. N < M ) -> -. N < M )
53	50, 51, 52	nltled 8283	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
54		eluz2 9744	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
55	48, 49, 53, 54	syl3anbrc 1205	. . . 4 |- ( ( ph /\ -. N < M ) -> N e. ( ZZ>= ` M ) )
56	19	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> F : ( M ... N ) --> B )
57	56	ffvelcdmda 5775	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( F ` k ) e. B )
58	25	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> H : ( M ... N ) --> B )
59	58	ffvelcdmda 5775	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( H ` k ) e. B )
60	7, 8	fzfigd 10670	. . . . . . . 8 |- ( ph -> ( M ... N ) e. Fin )
61	18	a1i 9	. . . . . . . 8 |- ( ph -> F = ( x e. ( M ... N ) |-> C ) )
62	24	a1i 9	. . . . . . . 8 |- ( ph -> H = ( x e. ( M ... N ) |-> D ) )
63	60, 11, 12, 61, 62	offval2 6243	. . . . . . 7 |- ( ph -> ( F oF .+ H ) = ( x e. ( M ... N ) |-> ( C .+ D ) ) )
64	63	fveq1d 5634	. . . . . 6 |- ( ph -> ( ( F oF .+ H ) ` k ) = ( ( x e. ( M ... N ) |-> ( C .+ D ) ) ` k ) )
65	64	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( ( F oF .+ H ) ` k ) = ( ( x e. ( M ... N ) |-> ( C .+ D ) ) ` k ) )
66	19	ffnd 5477	. . . . . . 7 |- ( ph -> F Fn ( M ... N ) )
67	25	ffnd 5477	. . . . . . 7 |- ( ph -> H Fn ( M ... N ) )
68		inidm 3413	. . . . . . 7 |- ( ( M ... N ) i^i ( M ... N ) ) = ( M ... N )
69		eqidd 2230	. . . . . . 7 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = ( F ` k ) )
70		eqidd 2230	. . . . . . 7 |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( H ` k ) )
71	9	adantr 276	. . . . . . . 8 |- ( ( ph /\ k e. ( M ... N ) ) -> G e. Mnd )
72	19	ffvelcdmda 5775	. . . . . . . 8 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. B )
73	25	ffvelcdmda 5775	. . . . . . . 8 |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) e. B )
74	3, 5	mndcl 13477	. . . . . . . 8 |- ( ( G e. Mnd /\ ( F ` k ) e. B /\ ( H ` k ) e. B ) -> ( ( F ` k ) .+ ( H ` k ) ) e. B )
75	71, 72, 73, 74	syl3anc 1271	. . . . . . 7 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( F ` k ) .+ ( H ` k ) ) e. B )
76	66, 67, 60, 60, 68, 69, 70, 75	ofvalg 6237	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( F oF .+ H ) ` k ) = ( ( F ` k ) .+ ( H ` k ) ) )
77	76	adantlr 477	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( ( F oF .+ H ) ` k ) = ( ( F ` k ) .+ ( H ` k ) ) )
78	65, 77	eqtr3d 2264	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( ( x e. ( M ... N ) |-> ( C .+ D ) ) ` k ) = ( ( F ` k ) .+ ( H ` k ) ) )
79		plusgslid 13166	. . . . . . . 8 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
80	79	slotex 13080	. . . . . . 7 |- ( G e. CMnd -> ( +g ` G ) e. _V )
81	6, 80	syl 14	. . . . . 6 |- ( ph -> ( +g ` G ) e. _V )
82	5, 81	eqeltrid 2316	. . . . 5 |- ( ph -> .+ e. _V )
83	82	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> .+ e. _V )
84	19, 60	fexd 5876	. . . . 5 |- ( ph -> F e. _V )
85	84	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
86	25, 60	fexd 5876	. . . . 5 |- ( ph -> H e. _V )
87	86	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> H e. _V )
88	15, 60	fexd 5876	. . . . 5 |- ( ph -> ( x e. ( M ... N ) |-> ( C .+ D ) ) e. _V )
89	88	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( x e. ( M ... N ) |-> ( C .+ D ) ) e. _V )
90	41, 44, 47, 55, 57, 59, 78, 83, 85, 87, 89	seqcaoprg 10735	. . 3 |- ( ( ph /\ -. N < M ) -> ( seq M ( .+ , ( x e. ( M ... N ) |-> ( C .+ D ) ) ) ` N ) = ( ( seq M ( .+ , F ) ` N ) .+ ( seq M ( .+ , H ) ` N ) ) )
91	16	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( G gsumg ( x e. ( M ... N ) |-> ( C .+ D ) ) ) = if ( N < M , ( 0g ` G ) , ( seq M ( .+ , ( x e. ( M ... N ) |-> ( C .+ D ) ) ) ` N ) ) )
92	52	iffalsed 3612	. . . 4 |- ( ( ph /\ -. N < M ) -> if ( N < M , ( 0g ` G ) , ( seq M ( .+ , ( x e. ( M ... N ) |-> ( C .+ D ) ) ) ` N ) ) = ( seq M ( .+ , ( x e. ( M ... N ) |-> ( C .+ D ) ) ) ` N ) )
93	91, 92	eqtrd 2262	. . 3 |- ( ( ph /\ -. N < M ) -> ( G gsumg ( x e. ( M ... N ) |-> ( C .+ D ) ) ) = ( seq M ( .+ , ( x e. ( M ... N ) |-> ( C .+ D ) ) ) ` N ) )
94	20	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = if ( N < M , ( 0g ` G ) , ( seq M ( .+ , F ) ` N ) ) )
95	52	iffalsed 3612	. . . . 5 |- ( ( ph /\ -. N < M ) -> if ( N < M , ( 0g ` G ) , ( seq M ( .+ , F ) ` N ) ) = ( seq M ( .+ , F ) ` N ) )
96	94, 95	eqtrd 2262	. . . 4 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = ( seq M ( .+ , F ) ` N ) )
97	26	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> ( G gsumg H ) = if ( N < M , ( 0g ` G ) , ( seq M ( .+ , H ) ` N ) ) )
98	52	iffalsed 3612	. . . . 5 |- ( ( ph /\ -. N < M ) -> if ( N < M , ( 0g ` G ) , ( seq M ( .+ , H ) ` N ) ) = ( seq M ( .+ , H ) ` N ) )
99	97, 98	eqtrd 2262	. . . 4 |- ( ( ph /\ -. N < M ) -> ( G gsumg H ) = ( seq M ( .+ , H ) ` N ) )
100	96, 99	oveq12d 6028	. . 3 |- ( ( ph /\ -. N < M ) -> ( ( G gsumg F ) .+ ( G gsumg H ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( seq M ( .+ , H ) ` N ) ) )
101	90, 93, 100	3eqtr4d 2272	. 2 |- ( ( ph /\ -. N < M ) -> ( G gsumg ( x e. ( M ... N ) |-> ( C .+ D ) ) ) = ( ( G gsumg F ) .+ ( G gsumg H ) ) )
102		zdclt 9540	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
103	8, 7, 102	syl2anc 411	. . 3 |- ( ph -> DECID N < M )
104		exmiddc 841	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
105	103, 104	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
106	36, 101, 105	mpjaodan 803	1 |- ( ph -> ( G gsumg ( x e. ( M ... N ) |-> ( C .+ D ) ) ) = ( ( G gsumg F ) .+ ( G gsumg H ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   _Vcvv 2799   ifcif 3602   class class class wbr 4083   |-> cmpt 4145   -->wf 5317   ` cfv 5321  (class class class)co 6010   oFcof 6225   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  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-rmo 2516  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-of 6227  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-1o 6573  df-er 6693  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-cmn 13844

This theorem is referenced by:  gsumfzmptfidmadd2  13898