Theorem gsumfzmptfidmadd 14140

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 3633	. . 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 2234	. . . . 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 14109	. . . . . . . 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 13720	. . . . . . 7 |- ( ( G e. Mnd /\ C e. B /\ D e. B ) -> ( C .+ D ) e. B )
14	10, 11, 12, 13	syl3anc 1274	. . . . . 6 |- ( ( ph /\ x e. ( M ... N ) ) -> ( C .+ D ) e. B )
15	14	fmpttd 5837	. . . . 5 |- ( ph -> ( x e. ( M ... N ) |-> ( C .+ D ) ) : ( M ... N ) --> B )
16	3, 4, 5, 6, 7, 8, 15	gsumfzval 13688	. . . 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 5836	. . . . . . . 8 |- ( ph -> F : ( M ... N ) --> B )
20	3, 4, 5, 6, 7, 8, 19	gsumfzval 13688	. . . . . . 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 3633	. . . . . 6 |- ( ( ph /\ N < M ) -> if ( N < M , ( 0g ` G ) , ( seq M ( .+ , F ) ` N ) ) = ( 0g ` G ) )
23	21, 22	eqtrd 2267	. . . . 5 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = ( 0g ` G ) )
24		gsumfzmptfidmadd.h	. . . . . . . . 9 |- H = ( x e. ( M ... N ) |-> D )
25	12, 24	fmptd 5836	. . . . . . . 8 |- ( ph -> H : ( M ... N ) --> B )
26	3, 4, 5, 6, 7, 8, 25	gsumfzval 13688	. . . . . . 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 3633	. . . . . 6 |- ( ( ph /\ N < M ) -> if ( N < M , ( 0g ` G ) , ( seq M ( .+ , H ) ` N ) ) = ( 0g ` G ) )
29	27, 28	eqtrd 2267	. . . . 5 |- ( ( ph /\ N < M ) -> ( G gsumg H ) = ( 0g ` G ) )
30	23, 29	oveq12d 6076	. . . 4 |- ( ( ph /\ N < M ) -> ( ( G gsumg F ) .+ ( G gsumg H ) ) = ( ( 0g ` G ) .+ ( 0g ` G ) ) )
31	3, 4	mndidcl 13727	. . . . . 6 |- ( G e. Mnd -> ( 0g ` G ) e. B )
32	3, 5, 4	mndlid 13732	. . . . . 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 2267	. . 3 |- ( ( ph /\ N < M ) -> ( ( G gsumg F ) .+ ( G gsumg H ) ) = ( 0g ` G ) )
36	2, 17, 35	3eqtr4d 2277	. 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 531	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( p e. B /\ q e. B ) ) -> p e. B )
39		simprr 533	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( p e. B /\ q e. B ) ) -> q e. B )
40	3, 5	mndcl 13720	. . . . 5 |- ( ( G e. Mnd /\ p e. B /\ q e. B ) -> ( p .+ q ) e. B )
41	37, 38, 39, 40	syl3anc 1274	. . . 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 14103	. . . . 5 |- ( ( G e. CMnd /\ p e. B /\ q e. B ) -> ( p .+ q ) = ( q .+ p ) )
44	42, 38, 39, 43	syl3anc 1274	. . . 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 13721	. . . . 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 9718	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
51	49	zred 9718	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
52		simpr 110	. . . . . 6 |- ( ( ph /\ -. N < M ) -> -. N < M )
53	50, 51, 52	nltled 8410	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
54		eluz2 9877	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
55	48, 49, 53, 54	syl3anbrc 1208	. . . 4 |- ( ( ph /\ -. N < M ) -> N e. ( ZZ>= ` M ) )
56	19	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> F : ( M ... N ) --> B )
57	56	ffvelcdmda 5817	. . . 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 5817	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( H ` k ) e. B )
60	7, 8	fzfigd 10817	. . . . . . . 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 6291	. . . . . . 7 |- ( ph -> ( F oF .+ H ) = ( x e. ( M ... N ) |-> ( C .+ D ) ) )
64	63	fveq1d 5677	. . . . . 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 5514	. . . . . . 7 |- ( ph -> F Fn ( M ... N ) )
67	25	ffnd 5514	. . . . . . 7 |- ( ph -> H Fn ( M ... N ) )
68		inidm 3434	. . . . . . 7 |- ( ( M ... N ) i^i ( M ... N ) ) = ( M ... N )
69		eqidd 2235	. . . . . . 7 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = ( F ` k ) )
70		eqidd 2235	. . . . . . 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 5817	. . . . . . . 8 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. B )
73	25	ffvelcdmda 5817	. . . . . . . 8 |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) e. B )
74	3, 5	mndcl 13720	. . . . . . . 8 |- ( ( G e. Mnd /\ ( F ` k ) e. B /\ ( H ` k ) e. B ) -> ( ( F ` k ) .+ ( H ` k ) ) e. B )
75	71, 72, 73, 74	syl3anc 1274	. . . . . . 7 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( F ` k ) .+ ( H ` k ) ) e. B )
76	66, 67, 60, 60, 68, 69, 70, 75	ofvalg 6285	. . . . . 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 2269	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( ( x e. ( M ... N ) |-> ( C .+ D ) ) ` k ) = ( ( F ` k ) .+ ( H ` k ) ) )
79		plusgslid 13409	. . . . . . . 8 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
80	79	slotex 13323	. . . . . . 7 |- ( G e. CMnd -> ( +g ` G ) e. _V )
81	6, 80	syl 14	. . . . . 6 |- ( ph -> ( +g ` G ) e. _V )
82	5, 81	eqeltrid 2321	. . . . 5 |- ( ph -> .+ e. _V )
83	82	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> .+ e. _V )
84	19, 60	fexd 5921	. . . . 5 |- ( ph -> F e. _V )
85	84	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
86	25, 60	fexd 5921	. . . . 5 |- ( ph -> H e. _V )
87	86	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> H e. _V )
88	15, 60	fexd 5921	. . . . 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 10882	. . 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 3636	. . . 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 2267	. . 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 3636	. . . . 5 |- ( ( ph /\ -. N < M ) -> if ( N < M , ( 0g ` G ) , ( seq M ( .+ , F ) ` N ) ) = ( seq M ( .+ , F ) ` N ) )
96	94, 95	eqtrd 2267	. . . 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 3636	. . . . 5 |- ( ( ph /\ -. N < M ) -> if ( N < M , ( 0g ` G ) , ( seq M ( .+ , H ) ` N ) ) = ( seq M ( .+ , H ) ` N ) )
99	97, 98	eqtrd 2267	. . . 4 |- ( ( ph /\ -. N < M ) -> ( G gsumg H ) = ( seq M ( .+ , H ) ` N ) )
100	96, 99	oveq12d 6076	. . 3 |- ( ( ph /\ -. N < M ) -> ( ( G gsumg F ) .+ ( G gsumg H ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( seq M ( .+ , H ) ` N ) ) )
101	90, 93, 100	3eqtr4d 2277	. 2 |- ( ( ph /\ -. N < M ) -> ( G gsumg ( x e. ( M ... N ) |-> ( C .+ D ) ) ) = ( ( G gsumg F ) .+ ( G gsumg H ) ) )
102		zdclt 9672	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
103	8, 7, 102	syl2anc 411	. . 3 |- ( ph -> DECID N < M )
104		exmiddc 844	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
105	103, 104	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
106	36, 101, 105	mpjaodan 806	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 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   _Vcvv 2815   ifcif 3624   class class class wbr 4114   |-> cmpt 4176   -->wf 5353   ` cfv 5357  (class class class)co 6058   oFcof 6273   Fincfn 6988   < clt 8324   <_ cle 8325   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361   seqcseq 10833   Basecbs 13296   +g cplusg 13374   0gc0g 13553   gsum_g cgsu 13554   Mndcmnd 13713  CMndccmn 14085

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-of 6275  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-igsum 13556  df-mgm 13653  df-sgrp 13699  df-mnd 13714  df-cmn 14087

This theorem is referenced by:  gsumfzmptfidmadd2  14141