Metamath Proof Explorer


Theorem dprdfadd

Description: Take the sum of group sums over two families of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016) (Revised by AV, 14-Jul-2019)

Ref Expression
Hypotheses eldprdi.0 0 ˙ = 0 G
eldprdi.w W = h i I S i | finSupp 0 ˙ h
eldprdi.1 φ G dom DProd S
eldprdi.2 φ dom S = I
eldprdi.3 φ F W
dprdfadd.4 φ H W
dprdfadd.b + ˙ = + G
Assertion dprdfadd φ F + ˙ f H W G F + ˙ f H = G F + ˙ G H

Proof

Step Hyp Ref Expression
1 eldprdi.0 0 ˙ = 0 G
2 eldprdi.w W = h i I S i | finSupp 0 ˙ h
3 eldprdi.1 φ G dom DProd S
4 eldprdi.2 φ dom S = I
5 eldprdi.3 φ F W
6 dprdfadd.4 φ H W
7 dprdfadd.b + ˙ = + G
8 3 4 dprddomcld φ I V
9 2 3 4 5 dprdfcl φ x I F x S x
10 2 3 4 6 dprdfcl φ x I H x S x
11 eqid Base G = Base G
12 2 3 4 5 11 dprdff φ F : I Base G
13 12 feqmptd φ F = x I F x
14 2 3 4 6 11 dprdff φ H : I Base G
15 14 feqmptd φ H = x I H x
16 8 9 10 13 15 offval2 φ F + ˙ f H = x I F x + ˙ H x
17 3 4 dprdf2 φ S : I SubGrp G
18 17 ffvelrnda φ x I S x SubGrp G
19 7 subgcl S x SubGrp G F x S x H x S x F x + ˙ H x S x
20 18 9 10 19 syl3anc φ x I F x + ˙ H x S x
21 2 3 4 5 dprdffsupp φ finSupp 0 ˙ F
22 2 3 4 6 dprdffsupp φ finSupp 0 ˙ H
23 21 22 fsuppunfi φ supp 0 ˙ F supp 0 ˙ H Fin
24 ssun1 F supp 0 ˙ supp 0 ˙ F supp 0 ˙ H
25 24 a1i φ F supp 0 ˙ supp 0 ˙ F supp 0 ˙ H
26 1 fvexi 0 ˙ V
27 26 a1i φ 0 ˙ V
28 12 25 8 27 suppssr φ x I supp 0 ˙ F supp 0 ˙ H F x = 0 ˙
29 ssun2 H supp 0 ˙ supp 0 ˙ F supp 0 ˙ H
30 29 a1i φ H supp 0 ˙ supp 0 ˙ F supp 0 ˙ H
31 14 30 8 27 suppssr φ x I supp 0 ˙ F supp 0 ˙ H H x = 0 ˙
32 28 31 oveq12d φ x I supp 0 ˙ F supp 0 ˙ H F x + ˙ H x = 0 ˙ + ˙ 0 ˙
33 dprdgrp G dom DProd S G Grp
34 3 33 syl φ G Grp
35 11 1 grpidcl G Grp 0 ˙ Base G
36 11 7 1 grplid G Grp 0 ˙ Base G 0 ˙ + ˙ 0 ˙ = 0 ˙
37 34 35 36 syl2anc2 φ 0 ˙ + ˙ 0 ˙ = 0 ˙
38 37 adantr φ x I supp 0 ˙ F supp 0 ˙ H 0 ˙ + ˙ 0 ˙ = 0 ˙
39 32 38 eqtrd φ x I supp 0 ˙ F supp 0 ˙ H F x + ˙ H x = 0 ˙
40 39 8 suppss2 φ x I F x + ˙ H x supp 0 ˙ supp 0 ˙ F supp 0 ˙ H
41 23 40 ssfid φ x I F x + ˙ H x supp 0 ˙ Fin
42 funmpt Fun x I F x + ˙ H x
43 42 a1i φ Fun x I F x + ˙ H x
44 8 mptexd φ x I F x + ˙ H x V
45 funisfsupp Fun x I F x + ˙ H x x I F x + ˙ H x V 0 ˙ V finSupp 0 ˙ x I F x + ˙ H x x I F x + ˙ H x supp 0 ˙ Fin
46 43 44 27 45 syl3anc φ finSupp 0 ˙ x I F x + ˙ H x x I F x + ˙ H x supp 0 ˙ Fin
47 41 46 mpbird φ finSupp 0 ˙ x I F x + ˙ H x
48 2 3 4 20 47 dprdwd φ x I F x + ˙ H x W
49 16 48 eqeltrd φ F + ˙ f H W
50 eqid Cntz G = Cntz G
51 34 grpmndd φ G Mnd
52 eqid F H supp 0 ˙ = F H supp 0 ˙
53 2 3 4 5 50 dprdfcntz φ ran F Cntz G ran F
54 2 3 4 6 50 dprdfcntz φ ran H Cntz G ran H
55 2 3 4 49 50 dprdfcntz φ ran F + ˙ f H Cntz G ran F + ˙ f H
56 51 adantr φ x I k I x G Mnd
57 vex x V
58 57 a1i φ x I k I x x V
59 eldifi k I x k I
60 59 adantl x I k I x k I
61 ffvelrn F : I Base G k I F k Base G
62 12 60 61 syl2an φ x I k I x F k Base G
63 62 snssd φ x I k I x F k Base G
64 11 50 cntzsubm G Mnd F k Base G Cntz G F k SubMnd G
65 56 63 64 syl2anc φ x I k I x Cntz G F k SubMnd G
66 14 adantr φ x I k I x H : I Base G
67 66 ffnd φ x I k I x H Fn I
68 simprl φ x I k I x x I
69 fnssres H Fn I x I H x Fn x
70 67 68 69 syl2anc φ x I k I x H x Fn x
71 fvres y x H x y = H y
72 71 adantl φ x I k I x y x H x y = H y
73 3 ad2antrr φ x I k I x y x G dom DProd S
74 4 ad2antrr φ x I k I x y x dom S = I
75 73 74 dprdf2 φ x I k I x y x S : I SubGrp G
76 60 ad2antlr φ x I k I x y x k I
77 75 76 ffvelrnd φ x I k I x y x S k SubGrp G
78 11 subgss S k SubGrp G S k Base G
79 77 78 syl φ x I k I x y x S k Base G
80 5 ad2antrr φ x I k I x y x F W
81 2 73 74 80 dprdfcl φ x I k I x y x k I F k S k
82 76 81 mpdan φ x I k I x y x F k S k
83 82 snssd φ x I k I x y x F k S k
84 11 50 cntz2ss S k Base G F k S k Cntz G S k Cntz G F k
85 79 83 84 syl2anc φ x I k I x y x Cntz G S k Cntz G F k
86 68 sselda φ x I k I x y x y I
87 simpr φ x I k I x y x y x
88 simplrr φ x I k I x y x k I x
89 88 eldifbd φ x I k I x y x ¬ k x
90 nelne2 y x ¬ k x y k
91 87 89 90 syl2anc φ x I k I x y x y k
92 73 74 86 76 91 50 dprdcntz φ x I k I x y x S y Cntz G S k
93 6 ad2antrr φ x I k I x y x H W
94 2 73 74 93 dprdfcl φ x I k I x y x y I H y S y
95 86 94 mpdan φ x I k I x y x H y S y
96 92 95 sseldd φ x I k I x y x H y Cntz G S k
97 85 96 sseldd φ x I k I x y x H y Cntz G F k
98 72 97 eqeltrd φ x I k I x y x H x y Cntz G F k
99 98 ralrimiva φ x I k I x y x H x y Cntz G F k
100 ffnfv H x : x Cntz G F k H x Fn x y x H x y Cntz G F k
101 70 99 100 sylanbrc φ x I k I x H x : x Cntz G F k
102 resss H x H
103 102 rnssi ran H x ran H
104 50 cntzidss ran H Cntz G ran H ran H x ran H ran H x Cntz G ran H x
105 54 103 104 sylancl φ ran H x Cntz G ran H x
106 105 adantr φ x I k I x ran H x Cntz G ran H x
107 22 27 fsuppres φ finSupp 0 ˙ H x
108 107 adantr φ x I k I x finSupp 0 ˙ H x
109 1 50 56 58 65 101 106 108 gsumzsubmcl φ x I k I x G H x Cntz G F k
110 109 snssd φ x I k I x G H x Cntz G F k
111 66 68 fssresd φ x I k I x H x : x Base G
112 11 1 50 56 58 111 106 108 gsumzcl φ x I k I x G H x Base G
113 112 snssd φ x I k I x G H x Base G
114 11 50 cntzrec G H x Base G F k Base G G H x Cntz G F k F k Cntz G G H x
115 113 63 114 syl2anc φ x I k I x G H x Cntz G F k F k Cntz G G H x
116 110 115 mpbid φ x I k I x F k Cntz G G H x
117 fvex F k V
118 117 snss F k Cntz G G H x F k Cntz G G H x
119 116 118 sylibr φ x I k I x F k Cntz G G H x
120 11 1 7 50 51 8 21 22 52 12 14 53 54 55 119 gsumzaddlem φ G F + ˙ f H = G F + ˙ G H
121 49 120 jca φ F + ˙ f H W G F + ˙ f H = G F + ˙ G H