Metamath Proof Explorer


Theorem gsumwun

Description: In a commutative ring, a group sum of a word W of characters taken from two submonoids E and F can be written as a simple sum. (Contributed by Thierry Arnoux, 6-Oct-2025)

Ref Expression
Hypotheses gsumwun.p + ˙ = + M
gsumwun.m φ M CMnd
gsumwun.e φ E SubMnd M
gsumwun.f φ F SubMnd M
gsumwun.w φ W Word E F
Assertion gsumwun φ e E f F M W = e + ˙ f

Proof

Step Hyp Ref Expression
1 gsumwun.p + ˙ = + M
2 gsumwun.m φ M CMnd
3 gsumwun.e φ E SubMnd M
4 gsumwun.f φ F SubMnd M
5 gsumwun.w φ W Word E F
6 oveq2 v = M v = M
7 6 eqeq1d v = M v = e + ˙ f M = e + ˙ f
8 7 2rexbidv v = e E f F M v = e + ˙ f e E f F M = e + ˙ f
9 8 imbi2d v = φ e E f F M v = e + ˙ f φ e E f F M = e + ˙ f
10 oveq2 v = w M v = M w
11 10 eqeq1d v = w M v = e + ˙ f M w = e + ˙ f
12 11 2rexbidv v = w e E f F M v = e + ˙ f e E f F M w = e + ˙ f
13 12 imbi2d v = w φ e E f F M v = e + ˙ f φ e E f F M w = e + ˙ f
14 oveq1 e = i e + ˙ f = i + ˙ f
15 14 eqeq2d e = i M v = e + ˙ f M v = i + ˙ f
16 oveq2 f = j i + ˙ f = i + ˙ j
17 16 eqeq2d f = j M v = i + ˙ f M v = i + ˙ j
18 15 17 cbvrex2vw e E f F M v = e + ˙ f i E j F M v = i + ˙ j
19 oveq2 v = w ++ ⟨“ x ”⟩ M v = M w ++ ⟨“ x ”⟩
20 19 eqeq1d v = w ++ ⟨“ x ”⟩ M v = i + ˙ j M w ++ ⟨“ x ”⟩ = i + ˙ j
21 20 2rexbidv v = w ++ ⟨“ x ”⟩ i E j F M v = i + ˙ j i E j F M w ++ ⟨“ x ”⟩ = i + ˙ j
22 18 21 bitrid v = w ++ ⟨“ x ”⟩ e E f F M v = e + ˙ f i E j F M w ++ ⟨“ x ”⟩ = i + ˙ j
23 22 imbi2d v = w ++ ⟨“ x ”⟩ φ e E f F M v = e + ˙ f φ i E j F M w ++ ⟨“ x ”⟩ = i + ˙ j
24 oveq2 v = W M v = M W
25 24 eqeq1d v = W M v = e + ˙ f M W = e + ˙ f
26 25 2rexbidv v = W e E f F M v = e + ˙ f e E f F M W = e + ˙ f
27 26 imbi2d v = W φ e E f F M v = e + ˙ f φ e E f F M W = e + ˙ f
28 oveq1 e = 0 M e + ˙ f = 0 M + ˙ f
29 28 eqeq2d e = 0 M M = e + ˙ f M = 0 M + ˙ f
30 oveq2 f = 0 M 0 M + ˙ f = 0 M + ˙ 0 M
31 30 eqeq2d f = 0 M M = 0 M + ˙ f M = 0 M + ˙ 0 M
32 eqid 0 M = 0 M
33 32 subm0cl E SubMnd M 0 M E
34 3 33 syl φ 0 M E
35 32 subm0cl F SubMnd M 0 M F
36 4 35 syl φ 0 M F
37 32 gsum0 M = 0 M
38 2 cmnmndd φ M Mnd
39 eqid Base M = Base M
40 39 32 mndidcl M Mnd 0 M Base M
41 39 1 32 mndlid M Mnd 0 M Base M 0 M + ˙ 0 M = 0 M
42 38 40 41 syl2anc2 φ 0 M + ˙ 0 M = 0 M
43 37 42 eqtr4id φ M = 0 M + ˙ 0 M
44 29 31 34 36 43 2rspcedvdw φ e E f F M = e + ˙ f
45 oveq1 i = e + ˙ x i + ˙ j = e + ˙ x + ˙ j
46 45 eqeq2d i = e + ˙ x M w ++ ⟨“ x ”⟩ = i + ˙ j M w ++ ⟨“ x ”⟩ = e + ˙ x + ˙ j
47 oveq2 j = f e + ˙ x + ˙ j = e + ˙ x + ˙ f
48 47 eqeq2d j = f M w ++ ⟨“ x ”⟩ = e + ˙ x + ˙ j M w ++ ⟨“ x ”⟩ = e + ˙ x + ˙ f
49 3 ad6antr φ w Word E F x E F e E f F M w = e + ˙ f x E E SubMnd M
50 simp-4r φ w Word E F x E F e E f F M w = e + ˙ f x E e E
51 simpr φ w Word E F x E F e E f F M w = e + ˙ f x E x E
52 1 49 50 51 submcld φ w Word E F x E F e E f F M w = e + ˙ f x E e + ˙ x E
53 simpllr φ w Word E F x E F e E f F M w = e + ˙ f x E f F
54 38 ad5antr φ w Word E F x E F e E f F M w = e + ˙ f M Mnd
55 39 submss E SubMnd M E Base M
56 3 55 syl φ E Base M
57 39 submss F SubMnd M F Base M
58 4 57 syl φ F Base M
59 56 58 unssd φ E F Base M
60 sswrd E F Base M Word E F Word Base M
61 59 60 syl φ Word E F Word Base M
62 61 sselda φ w Word E F w Word Base M
63 62 ad4antr φ w Word E F x E F e E f F M w = e + ˙ f w Word Base M
64 59 adantr φ w Word E F E F Base M
65 64 sselda φ w Word E F x E F x Base M
66 65 ad3antrrr φ w Word E F x E F e E f F M w = e + ˙ f x Base M
67 39 1 gsumccatsn M Mnd w Word Base M x Base M M w ++ ⟨“ x ”⟩ = M w + ˙ x
68 54 63 66 67 syl3anc φ w Word E F x E F e E f F M w = e + ˙ f M w ++ ⟨“ x ”⟩ = M w + ˙ x
69 simpr φ w Word E F x E F e E f F M w = e + ˙ f M w = e + ˙ f
70 69 oveq1d φ w Word E F x E F e E f F M w = e + ˙ f M w + ˙ x = e + ˙ f + ˙ x
71 56 ad2antrr φ w Word E F x E F E Base M
72 71 sselda φ w Word E F x E F e E e Base M
73 72 ad2antrr φ w Word E F x E F e E f F M w = e + ˙ f e Base M
74 58 ad3antrrr φ w Word E F x E F e E F Base M
75 74 sselda φ w Word E F x E F e E f F f Base M
76 75 adantr φ w Word E F x E F e E f F M w = e + ˙ f f Base M
77 2 ad5antr φ w Word E F x E F e E f F M w = e + ˙ f M CMnd
78 39 1 cmncom M CMnd f Base M x Base M f + ˙ x = x + ˙ f
79 77 76 66 78 syl3anc φ w Word E F x E F e E f F M w = e + ˙ f f + ˙ x = x + ˙ f
80 39 1 54 73 76 66 79 mnd32g φ w Word E F x E F e E f F M w = e + ˙ f e + ˙ f + ˙ x = e + ˙ x + ˙ f
81 68 70 80 3eqtrd φ w Word E F x E F e E f F M w = e + ˙ f M w ++ ⟨“ x ”⟩ = e + ˙ x + ˙ f
82 81 adantr φ w Word E F x E F e E f F M w = e + ˙ f x E M w ++ ⟨“ x ”⟩ = e + ˙ x + ˙ f
83 46 48 52 53 82 2rspcedvdw φ w Word E F x E F e E f F M w = e + ˙ f x E i E j F M w ++ ⟨“ x ”⟩ = i + ˙ j
84 oveq1 i = e i + ˙ j = e + ˙ j
85 84 eqeq2d i = e M w ++ ⟨“ x ”⟩ = i + ˙ j M w ++ ⟨“ x ”⟩ = e + ˙ j
86 oveq2 j = f + ˙ x e + ˙ j = e + ˙ f + ˙ x
87 86 eqeq2d j = f + ˙ x M w ++ ⟨“ x ”⟩ = e + ˙ j M w ++ ⟨“ x ”⟩ = e + ˙ f + ˙ x
88 simp-4r φ w Word E F x E F e E f F M w = e + ˙ f x F e E
89 4 ad6antr φ w Word E F x E F e E f F M w = e + ˙ f x F F SubMnd M
90 simpllr φ w Word E F x E F e E f F M w = e + ˙ f x F f F
91 simpr φ w Word E F x E F e E f F M w = e + ˙ f x F x F
92 1 89 90 91 submcld φ w Word E F x E F e E f F M w = e + ˙ f x F f + ˙ x F
93 39 1 54 73 76 66 mndassd φ w Word E F x E F e E f F M w = e + ˙ f e + ˙ f + ˙ x = e + ˙ f + ˙ x
94 68 70 93 3eqtrd φ w Word E F x E F e E f F M w = e + ˙ f M w ++ ⟨“ x ”⟩ = e + ˙ f + ˙ x
95 94 adantr φ w Word E F x E F e E f F M w = e + ˙ f x F M w ++ ⟨“ x ”⟩ = e + ˙ f + ˙ x
96 85 87 88 92 95 2rspcedvdw φ w Word E F x E F e E f F M w = e + ˙ f x F i E j F M w ++ ⟨“ x ”⟩ = i + ˙ j
97 elun x E F x E x F
98 97 biimpi x E F x E x F
99 98 ad4antlr φ w Word E F x E F e E f F M w = e + ˙ f x E x F
100 83 96 99 mpjaodan φ w Word E F x E F e E f F M w = e + ˙ f i E j F M w ++ ⟨“ x ”⟩ = i + ˙ j
101 100 r19.29ffa φ w Word E F x E F e E f F M w = e + ˙ f i E j F M w ++ ⟨“ x ”⟩ = i + ˙ j
102 101 ex φ w Word E F x E F e E f F M w = e + ˙ f i E j F M w ++ ⟨“ x ”⟩ = i + ˙ j
103 102 expl φ w Word E F x E F e E f F M w = e + ˙ f i E j F M w ++ ⟨“ x ”⟩ = i + ˙ j
104 103 com12 w Word E F x E F φ e E f F M w = e + ˙ f i E j F M w ++ ⟨“ x ”⟩ = i + ˙ j
105 104 a2d w Word E F x E F φ e E f F M w = e + ˙ f φ i E j F M w ++ ⟨“ x ”⟩ = i + ˙ j
106 9 13 23 27 44 105 wrdind W Word E F φ e E f F M W = e + ˙ f
107 5 106 mpcom φ e E f F M W = e + ˙ f