Metamath Proof Explorer


Theorem zlmodzxzequa

Description: Example of an equation within the ZZ-module ZZ X. ZZ (see example in Roman p. 112 for a linearly dependent set). (Contributed by AV, 22-May-2019) (Revised by AV, 10-Jun-2019)

Ref Expression
Hypotheses zlmodzxzequa.z Z = ring freeLMod 0 1
zlmodzxzequa.o 0 ˙ = 0 0 1 0
zlmodzxzequa.t ˙ = Z
zlmodzxzequa.m - ˙ = - Z
zlmodzxzequa.a A = 0 3 1 6
zlmodzxzequa.b B = 0 2 1 4
Assertion zlmodzxzequa 2 ˙ A - ˙ 3 ˙ B = 0 ˙

Proof

Step Hyp Ref Expression
1 zlmodzxzequa.z Z = ring freeLMod 0 1
2 zlmodzxzequa.o 0 ˙ = 0 0 1 0
3 zlmodzxzequa.t ˙ = Z
4 zlmodzxzequa.m - ˙ = - Z
5 zlmodzxzequa.a A = 0 3 1 6
6 zlmodzxzequa.b B = 0 2 1 4
7 3cn 3
8 7 2timesi 2 3 = 3 + 3
9 3p3e6 3 + 3 = 6
10 8 9 eqtri 2 3 = 6
11 3t2e6 3 2 = 6
12 10 11 oveq12i 2 3 3 2 = 6 6
13 6cn 6
14 13 subidi 6 6 = 0
15 12 14 eqtri 2 3 3 2 = 0
16 15 opeq2i 0 2 3 3 2 = 0 0
17 2t6m3t4e0 2 6 3 4 = 0
18 17 opeq2i 1 2 6 3 4 = 1 0
19 16 18 preq12i 0 2 3 3 2 1 2 6 3 4 = 0 0 1 0
20 5 oveq2i 2 ˙ A = 2 ˙ 0 3 1 6
21 2z 2
22 3z 3
23 6nn 6
24 23 nnzi 6
25 1 3 zlmodzxzscm 2 3 6 2 ˙ 0 3 1 6 = 0 2 3 1 2 6
26 21 22 24 25 mp3an 2 ˙ 0 3 1 6 = 0 2 3 1 2 6
27 20 26 eqtri 2 ˙ A = 0 2 3 1 2 6
28 6 oveq2i 3 ˙ B = 3 ˙ 0 2 1 4
29 4z 4
30 1 3 zlmodzxzscm 3 2 4 3 ˙ 0 2 1 4 = 0 3 2 1 3 4
31 22 21 29 30 mp3an 3 ˙ 0 2 1 4 = 0 3 2 1 3 4
32 28 31 eqtri 3 ˙ B = 0 3 2 1 3 4
33 27 32 oveq12i 2 ˙ A - ˙ 3 ˙ B = 0 2 3 1 2 6 - ˙ 0 3 2 1 3 4
34 zmulcl 2 3 2 3
35 21 22 34 mp2an 2 3
36 zmulcl 3 2 3 2
37 22 21 36 mp2an 3 2
38 zmulcl 2 6 2 6
39 21 24 38 mp2an 2 6
40 zmulcl 3 4 3 4
41 22 29 40 mp2an 3 4
42 1 4 zlmodzxzsub 2 3 3 2 2 6 3 4 0 2 3 1 2 6 - ˙ 0 3 2 1 3 4 = 0 2 3 3 2 1 2 6 3 4
43 35 37 39 41 42 mp4an 0 2 3 1 2 6 - ˙ 0 3 2 1 3 4 = 0 2 3 3 2 1 2 6 3 4
44 33 43 eqtri 2 ˙ A - ˙ 3 ˙ B = 0 2 3 3 2 1 2 6 3 4
45 19 44 2 3eqtr4i 2 ˙ A - ˙ 3 ˙ B = 0 ˙