Step | Hyp | Ref
| Expression |
1 | | eqid 2622 |
. . . . 5
|
2 | | eqid 2622 |
. . . . 5
Scalar Scalar |
3 | | lincscmcl.r |
. . . . 5
Scalar |
4 | 1, 2, 3 | lcoval 42201 |
. . . 4
LinCo
finSupp Scalar linC |
5 | 4 | adantr 481 |
. . 3
LinCo finSupp Scalar
linC |
6 | | simpl 473 |
. . . . . . 7
|
7 | 6 | ad2antrr 762 |
. . . . . 6
finSupp Scalar
linC
|
8 | | simpr 477 |
. . . . . . 7
|
9 | 8 | adantr 481 |
. . . . . 6
finSupp Scalar
linC
|
10 | | simprl 794 |
. . . . . 6
finSupp Scalar
linC
|
11 | | lincscmcl.s |
. . . . . . 7
|
12 | 1, 2, 11, 3 | lmodvscl 18880 |
. . . . . 6
|
13 | 7, 9, 10, 12 | syl3anc 1326 |
. . . . 5
finSupp Scalar
linC
|
14 | 2 | lmodring 18871 |
. . . . . . . . . . . . . . . . 17
Scalar |
15 | 14 | ad2antrr 762 |
. . . . . . . . . . . . . . . 16
Scalar |
16 | 15 | adantl 482 |
. . . . . . . . . . . . . . 15
finSupp Scalar
linC
Scalar |
17 | 16 | adantr 481 |
. . . . . . . . . . . . . 14
finSupp Scalar
linC
Scalar |
18 | 8 | adantl 482 |
. . . . . . . . . . . . . . 15
finSupp Scalar
linC
|
19 | 18 | adantr 481 |
. . . . . . . . . . . . . 14
finSupp Scalar
linC
|
20 | | elmapi 7879 |
. . . . . . . . . . . . . . . . . 18
|
21 | | ffvelrn 6357 |
. . . . . . . . . . . . . . . . . . 19
|
22 | 21 | ex 450 |
. . . . . . . . . . . . . . . . . 18
|
23 | 20, 22 | syl 17 |
. . . . . . . . . . . . . . . . 17
|
24 | 23 | adantr 481 |
. . . . . . . . . . . . . . . 16
finSupp Scalar linC
|
25 | 24 | ad2antrr 762 |
. . . . . . . . . . . . . . 15
finSupp Scalar
linC
|
26 | 25 | imp 445 |
. . . . . . . . . . . . . 14
finSupp Scalar
linC
|
27 | | eqid 2622 |
. . . . . . . . . . . . . . 15
Scalar Scalar |
28 | 3, 27 | ringcl 18561 |
. . . . . . . . . . . . . 14
Scalar
Scalar |
29 | 17, 19, 26, 28 | syl3anc 1326 |
. . . . . . . . . . . . 13
finSupp Scalar
linC
Scalar |
30 | | eqid 2622 |
. . . . . . . . . . . . 13
Scalar Scalar |
31 | 29, 30 | fmptd 6385 |
. . . . . . . . . . . 12
finSupp Scalar
linC
Scalar |
32 | | fvex 6201 |
. . . . . . . . . . . . . 14
Scalar |
33 | 3, 32 | eqeltri 2697 |
. . . . . . . . . . . . 13
|
34 | | simpr 477 |
. . . . . . . . . . . . . . 15
|
35 | 34 | adantr 481 |
. . . . . . . . . . . . . 14
|
36 | 35 | adantl 482 |
. . . . . . . . . . . . 13
finSupp Scalar
linC
|
37 | | elmapg 7870 |
. . . . . . . . . . . . 13
Scalar Scalar |
38 | 33, 36, 37 | sylancr 695 |
. . . . . . . . . . . 12
finSupp Scalar
linC
Scalar Scalar |
39 | 31, 38 | mpbird 247 |
. . . . . . . . . . 11
finSupp Scalar
linC
Scalar |
40 | 15, 35, 8 | 3jca 1242 |
. . . . . . . . . . . . 13
Scalar
|
41 | 40 | adantl 482 |
. . . . . . . . . . . 12
finSupp Scalar
linC
Scalar
|
42 | | simpl 473 |
. . . . . . . . . . . . 13
finSupp Scalar linC
|
43 | 42 | ad2antrr 762 |
. . . . . . . . . . . 12
finSupp Scalar
linC
|
44 | | simprl 794 |
. . . . . . . . . . . . 13
finSupp Scalar linC
finSupp Scalar |
45 | 44 | ad2antrr 762 |
. . . . . . . . . . . 12
finSupp Scalar
linC
finSupp Scalar |
46 | 3 | rmfsupp 42155 |
. . . . . . . . . . . 12
Scalar
finSupp Scalar Scalar finSupp Scalar |
47 | 41, 43, 45, 46 | syl3anc 1326 |
. . . . . . . . . . 11
finSupp Scalar
linC
Scalar finSupp Scalar |
48 | | oveq2 6658 |
. . . . . . . . . . . . . . 15
linC linC |
49 | 48 | adantl 482 |
. . . . . . . . . . . . . 14
finSupp Scalar linC linC |
50 | 49 | adantl 482 |
. . . . . . . . . . . . 13
finSupp Scalar linC
linC |
51 | 50 | ad2antrr 762 |
. . . . . . . . . . . 12
finSupp Scalar
linC
linC |
52 | | simprl 794 |
. . . . . . . . . . . . 13
finSupp Scalar
linC
|
53 | 42 | adantr 481 |
. . . . . . . . . . . . . 14
finSupp Scalar linC |
54 | 53, 8 | anim12i 590 |
. . . . . . . . . . . . 13
finSupp Scalar
linC
|
55 | | eqid 2622 |
. . . . . . . . . . . . . 14
linC linC |
56 | 11, 27, 55, 3, 30 | lincscm 42219 |
. . . . . . . . . . . . 13
finSupp Scalar linC Scalar linC |
57 | 52, 54, 45, 56 | syl3anc 1326 |
. . . . . . . . . . . 12
finSupp Scalar
linC
linC Scalar linC |
58 | 51, 57 | eqtrd 2656 |
. . . . . . . . . . 11
finSupp Scalar
linC
Scalar linC |
59 | | breq1 4656 |
. . . . . . . . . . . . 13
Scalar finSupp Scalar
Scalar finSupp Scalar |
60 | | oveq1 6657 |
. . . . . . . . . . . . . 14
Scalar linC Scalar linC |
61 | 60 | eqeq2d 2632 |
. . . . . . . . . . . . 13
Scalar linC
Scalar linC |
62 | 59, 61 | anbi12d 747 |
. . . . . . . . . . . 12
Scalar finSupp
Scalar
linC Scalar finSupp Scalar
Scalar linC |
63 | 62 | rspcev 3309 |
. . . . . . . . . . 11
Scalar Scalar finSupp Scalar
Scalar linC finSupp Scalar linC |
64 | 39, 47, 58, 63 | syl12anc 1324 |
. . . . . . . . . 10
finSupp Scalar
linC
finSupp Scalar linC |
65 | 64 | ex 450 |
. . . . . . . . 9
finSupp Scalar linC
finSupp Scalar linC |
66 | 65 | ex 450 |
. . . . . . . 8
finSupp Scalar linC
finSupp Scalar linC |
67 | 66 | rexlimiva 3028 |
. . . . . . 7
finSupp Scalar linC
finSupp Scalar linC |
68 | 67 | impcom 446 |
. . . . . 6
finSupp Scalar
linC finSupp Scalar linC |
69 | 68 | impcom 446 |
. . . . 5
finSupp Scalar
linC
finSupp Scalar linC |
70 | 1, 2, 3 | lcoval 42201 |
. . . . . 6
LinCo
finSupp Scalar linC |
71 | 70 | ad2antrr 762 |
. . . . 5
finSupp Scalar
linC
LinCo finSupp Scalar
linC |
72 | 13, 69, 71 | mpbir2and 957 |
. . . 4
finSupp Scalar
linC
LinCo |
73 | 72 | ex 450 |
. . 3
finSupp Scalar
linC LinCo |
74 | 5, 73 | sylbid 230 |
. 2
LinCo LinCo |
75 | 74 | 3impia 1261 |
1
LinCo LinCo |