Metamath Proof Explorer


Theorem lincsum

Description: The sum of two linear combinations is a linear combination, see also the proof in Lang p. 129. (Contributed by AV, 4-Apr-2019) (Revised by AV, 28-Jul-2019)

Ref Expression
Hypotheses lincsum.p + ˙ = + M
lincsum.x X = A linC M V
lincsum.y Y = B linC M V
lincsum.s S = Scalar M
lincsum.r R = Base S
lincsum.b ˙ = + S
Assertion lincsum M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B X + ˙ Y = A ˙ f B linC M V

Proof

Step Hyp Ref Expression
1 lincsum.p + ˙ = + M
2 lincsum.x X = A linC M V
3 lincsum.y Y = B linC M V
4 lincsum.s S = Scalar M
5 lincsum.r R = Base S
6 lincsum.b ˙ = + S
7 eqid Base M = Base M
8 eqid 0 M = 0 M
9 lmodcmn M LMod M CMnd
10 9 adantr M LMod V 𝒫 Base M M CMnd
11 10 3ad2ant1 M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B M CMnd
12 simpr M LMod V 𝒫 Base M V 𝒫 Base M
13 12 3ad2ant1 M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B V 𝒫 Base M
14 simpl M LMod V 𝒫 Base M M LMod
15 14 3ad2ant1 M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B M LMod
16 15 adantr M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B x V M LMod
17 elmapi A R V A : V R
18 ffvelrn A : V R x V A x R
19 18 ex A : V R x V A x R
20 17 19 syl A R V x V A x R
21 20 adantr A R V B R V x V A x R
22 21 3ad2ant2 M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B x V A x R
23 22 imp M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B x V A x R
24 elelpwi x V V 𝒫 Base M x Base M
25 24 expcom V 𝒫 Base M x V x Base M
26 25 adantl M LMod V 𝒫 Base M x V x Base M
27 26 3ad2ant1 M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B x V x Base M
28 27 imp M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B x V x Base M
29 eqid M = M
30 7 4 29 5 lmodvscl M LMod A x R x Base M A x M x Base M
31 16 23 28 30 syl3anc M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B x V A x M x Base M
32 elmapi B R V B : V R
33 ffvelrn B : V R x V B x R
34 33 ex B : V R x V B x R
35 32 34 syl B R V x V B x R
36 35 adantl A R V B R V x V B x R
37 36 3ad2ant2 M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B x V B x R
38 37 imp M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B x V B x R
39 7 4 29 5 lmodvscl M LMod B x R x Base M B x M x Base M
40 16 38 28 39 syl3anc M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B x V B x M x Base M
41 eqidd M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B x V A x M x = x V A x M x
42 eqidd M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B x V B x M x = x V B x M x
43 id M LMod V 𝒫 Base M M LMod V 𝒫 Base M
44 simpl A R V B R V A R V
45 simpl finSupp 0 S A finSupp 0 S B finSupp 0 S A
46 4 5 scmfsupp M LMod V 𝒫 Base M A R V finSupp 0 S A finSupp 0 M x V A x M x
47 43 44 45 46 syl3an M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B finSupp 0 M x V A x M x
48 simpr A R V B R V B R V
49 simpr finSupp 0 S A finSupp 0 S B finSupp 0 S B
50 4 5 scmfsupp M LMod V 𝒫 Base M B R V finSupp 0 S B finSupp 0 M x V B x M x
51 43 48 49 50 syl3an M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B finSupp 0 M x V B x M x
52 7 8 1 11 13 31 40 41 42 47 51 gsummptfsadd M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B M x V A x M x + ˙ B x M x = M x V A x M x + ˙ M x V B x M x
53 12 adantr M LMod V 𝒫 Base M A R V B R V V 𝒫 Base M
54 elmapfn A R V A Fn V
55 54 ad2antrl M LMod V 𝒫 Base M A R V B R V A Fn V
56 elmapfn B R V B Fn V
57 56 ad2antll M LMod V 𝒫 Base M A R V B R V B Fn V
58 53 55 57 offvalfv M LMod V 𝒫 Base M A R V B R V A ˙ f B = y V A y ˙ B y
59 58 3adant3 M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B A ˙ f B = y V A y ˙ B y
60 4 lmodfgrp M LMod S Grp
61 60 grpmndd M LMod S Mnd
62 61 ad3antrrr M LMod V 𝒫 Base M A R V B R V y V S Mnd
63 ffvelrn A : V R y V A y R
64 63 ex A : V R y V A y R
65 17 64 syl A R V y V A y R
66 65 ad2antrl M LMod V 𝒫 Base M A R V B R V y V A y R
67 66 imp M LMod V 𝒫 Base M A R V B R V y V A y R
68 4 fveq2i Base S = Base Scalar M
69 5 68 eqtri R = Base Scalar M
70 67 69 eleqtrdi M LMod V 𝒫 Base M A R V B R V y V A y Base Scalar M
71 ffvelrn B : V R y V B y R
72 71 69 eleqtrdi B : V R y V B y Base Scalar M
73 72 ex B : V R y V B y Base Scalar M
74 32 73 syl B R V y V B y Base Scalar M
75 74 ad2antll M LMod V 𝒫 Base M A R V B R V y V B y Base Scalar M
76 75 imp M LMod V 𝒫 Base M A R V B R V y V B y Base Scalar M
77 4 eqcomi Scalar M = S
78 77 fveq2i Base Scalar M = Base S
79 78 6 mndcl S Mnd A y Base Scalar M B y Base Scalar M A y ˙ B y Base Scalar M
80 62 70 76 79 syl3anc M LMod V 𝒫 Base M A R V B R V y V A y ˙ B y Base Scalar M
81 80 fmpttd M LMod V 𝒫 Base M A R V B R V y V A y ˙ B y : V Base Scalar M
82 fvex Base Scalar M V
83 elmapg Base Scalar M V V 𝒫 Base M y V A y ˙ B y Base Scalar M V y V A y ˙ B y : V Base Scalar M
84 82 53 83 sylancr M LMod V 𝒫 Base M A R V B R V y V A y ˙ B y Base Scalar M V y V A y ˙ B y : V Base Scalar M
85 81 84 mpbird M LMod V 𝒫 Base M A R V B R V y V A y ˙ B y Base Scalar M V
86 85 3adant3 M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B y V A y ˙ B y Base Scalar M V
87 59 86 eqeltrd M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B A ˙ f B Base Scalar M V
88 lincval M LMod A ˙ f B Base Scalar M V V 𝒫 Base M A ˙ f B linC M V = M x V A ˙ f B x M x
89 15 87 13 88 syl3anc M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B A ˙ f B linC M V = M x V A ˙ f B x M x
90 54 56 anim12i A R V B R V A Fn V B Fn V
91 90 adantl M LMod V 𝒫 Base M A R V B R V A Fn V B Fn V
92 91 adantr M LMod V 𝒫 Base M A R V B R V x V A Fn V B Fn V
93 53 anim1i M LMod V 𝒫 Base M A R V B R V x V V 𝒫 Base M x V
94 fnfvof A Fn V B Fn V V 𝒫 Base M x V A ˙ f B x = A x ˙ B x
95 92 93 94 syl2anc M LMod V 𝒫 Base M A R V B R V x V A ˙ f B x = A x ˙ B x
96 6 a1i M LMod V 𝒫 Base M A R V B R V x V ˙ = + S
97 96 oveqd M LMod V 𝒫 Base M A R V B R V x V A x ˙ B x = A x + S B x
98 95 97 eqtrd M LMod V 𝒫 Base M A R V B R V x V A ˙ f B x = A x + S B x
99 98 oveq1d M LMod V 𝒫 Base M A R V B R V x V A ˙ f B x M x = A x + S B x M x
100 14 adantr M LMod V 𝒫 Base M A R V B R V M LMod
101 100 adantr M LMod V 𝒫 Base M A R V B R V x V M LMod
102 20 ad2antrl M LMod V 𝒫 Base M A R V B R V x V A x R
103 102 imp M LMod V 𝒫 Base M A R V B R V x V A x R
104 35 ad2antll M LMod V 𝒫 Base M A R V B R V x V B x R
105 104 imp M LMod V 𝒫 Base M A R V B R V x V B x R
106 26 adantr M LMod V 𝒫 Base M A R V B R V x V x Base M
107 106 imp M LMod V 𝒫 Base M A R V B R V x V x Base M
108 eqid Scalar M = Scalar M
109 4 fveq2i + S = + Scalar M
110 7 1 108 29 69 109 lmodvsdir M LMod A x R B x R x Base M A x + S B x M x = A x M x + ˙ B x M x
111 101 103 105 107 110 syl13anc M LMod V 𝒫 Base M A R V B R V x V A x + S B x M x = A x M x + ˙ B x M x
112 99 111 eqtrd M LMod V 𝒫 Base M A R V B R V x V A ˙ f B x M x = A x M x + ˙ B x M x
113 112 mpteq2dva M LMod V 𝒫 Base M A R V B R V x V A ˙ f B x M x = x V A x M x + ˙ B x M x
114 113 oveq2d M LMod V 𝒫 Base M A R V B R V M x V A ˙ f B x M x = M x V A x M x + ˙ B x M x
115 114 3adant3 M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B M x V A ˙ f B x M x = M x V A x M x + ˙ B x M x
116 89 115 eqtrd M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B A ˙ f B linC M V = M x V A x M x + ˙ B x M x
117 2 3 oveq12i X + ˙ Y = A linC M V + ˙ B linC M V
118 69 oveq1i R V = Base Scalar M V
119 118 eleq2i A R V A Base Scalar M V
120 119 biimpi A R V A Base Scalar M V
121 120 ad2antrl M LMod V 𝒫 Base M A R V B R V A Base Scalar M V
122 lincval M LMod A Base Scalar M V V 𝒫 Base M A linC M V = M x V A x M x
123 100 121 53 122 syl3anc M LMod V 𝒫 Base M A R V B R V A linC M V = M x V A x M x
124 118 eleq2i B R V B Base Scalar M V
125 124 biimpi B R V B Base Scalar M V
126 125 ad2antll M LMod V 𝒫 Base M A R V B R V B Base Scalar M V
127 lincval M LMod B Base Scalar M V V 𝒫 Base M B linC M V = M x V B x M x
128 100 126 53 127 syl3anc M LMod V 𝒫 Base M A R V B R V B linC M V = M x V B x M x
129 123 128 oveq12d M LMod V 𝒫 Base M A R V B R V A linC M V + ˙ B linC M V = M x V A x M x + ˙ M x V B x M x
130 129 3adant3 M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B A linC M V + ˙ B linC M V = M x V A x M x + ˙ M x V B x M x
131 117 130 syl5eq M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B X + ˙ Y = M x V A x M x + ˙ M x V B x M x
132 52 116 131 3eqtr4rd M LMod V 𝒫 Base M A R V B R V finSupp 0 S A finSupp 0 S B X + ˙ Y = A ˙ f B linC M V