Step | Hyp | Ref
| Expression |
1 | | eqidd 2824 |
. 2
⊢ (𝜑 → (Base‘𝑌) = (Base‘𝑌)) |
2 | | eqidd 2824 |
. 2
⊢ (𝜑 → (+g‘𝑌) = (+g‘𝑌)) |
3 | | prdslmodd.y |
. . 3
⊢ 𝑌 = (𝑆Xs𝑅) |
4 | | prdslmodd.s |
. . 3
⊢ (𝜑 → 𝑆 ∈ Ring) |
5 | | prdslmodd.rm |
. . . 4
⊢ (𝜑 → 𝑅:𝐼⟶LMod) |
6 | | prdslmodd.i |
. . . 4
⊢ (𝜑 → 𝐼 ∈ 𝑉) |
7 | | fex 6991 |
. . . 4
⊢ ((𝑅:𝐼⟶LMod ∧ 𝐼 ∈ 𝑉) → 𝑅 ∈ V) |
8 | 5, 6, 7 | syl2anc 586 |
. . 3
⊢ (𝜑 → 𝑅 ∈ V) |
9 | 3, 4, 8 | prdssca 16731 |
. 2
⊢ (𝜑 → 𝑆 = (Scalar‘𝑌)) |
10 | | eqidd 2824 |
. 2
⊢ (𝜑 → (
·𝑠 ‘𝑌) = ( ·𝑠
‘𝑌)) |
11 | | eqidd 2824 |
. 2
⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆)) |
12 | | eqidd 2824 |
. 2
⊢ (𝜑 → (+g‘𝑆) = (+g‘𝑆)) |
13 | | eqidd 2824 |
. 2
⊢ (𝜑 → (.r‘𝑆) = (.r‘𝑆)) |
14 | | eqidd 2824 |
. 2
⊢ (𝜑 → (1r‘𝑆) = (1r‘𝑆)) |
15 | | lmodgrp 19643 |
. . . . 5
⊢ (𝑎 ∈ LMod → 𝑎 ∈ Grp) |
16 | 15 | ssriv 3973 |
. . . 4
⊢ LMod
⊆ Grp |
17 | | fss 6529 |
. . . 4
⊢ ((𝑅:𝐼⟶LMod ∧ LMod ⊆ Grp) →
𝑅:𝐼⟶Grp) |
18 | 5, 16, 17 | sylancl 588 |
. . 3
⊢ (𝜑 → 𝑅:𝐼⟶Grp) |
19 | 3, 6, 4, 18 | prdsgrpd 18211 |
. 2
⊢ (𝜑 → 𝑌 ∈ Grp) |
20 | | eqid 2823 |
. . . 4
⊢
(Base‘𝑌) =
(Base‘𝑌) |
21 | | eqid 2823 |
. . . 4
⊢ (
·𝑠 ‘𝑌) = ( ·𝑠
‘𝑌) |
22 | | eqid 2823 |
. . . 4
⊢
(Base‘𝑆) =
(Base‘𝑆) |
23 | 4 | adantr 483 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring) |
24 | 6 | elexd 3516 |
. . . . 5
⊢ (𝜑 → 𝐼 ∈ V) |
25 | 24 | adantr 483 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝐼 ∈ V) |
26 | 5 | adantr 483 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod) |
27 | | simprl 769 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆)) |
28 | | simprr 771 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌)) |
29 | | prdslmodd.rs |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (Scalar‘(𝑅‘𝑦)) = 𝑆) |
30 | 29 | adantlr 713 |
. . . 4
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (Scalar‘(𝑅‘𝑦)) = 𝑆) |
31 | 3, 20, 21, 22, 23, 25, 26, 27, 28, 30 | prdsvscacl 19742 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎( ·𝑠
‘𝑌)𝑏) ∈ (Base‘𝑌)) |
32 | 31 | 3impb 1111 |
. 2
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎( ·𝑠
‘𝑌)𝑏) ∈ (Base‘𝑌)) |
33 | 5 | ffvelrnda 6853 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ LMod) |
34 | 33 | adantlr 713 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ LMod) |
35 | | simplr1 1211 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑆)) |
36 | 29 | fveq2d 6676 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑦))) = (Base‘𝑆)) |
37 | 36 | adantlr 713 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑦))) = (Base‘𝑆)) |
38 | 35, 37 | eleqtrrd 2918 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑦)))) |
39 | 4 | ad2antrr 724 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑆 ∈ Ring) |
40 | 24 | ad2antrr 724 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ V) |
41 | 5 | ffnd 6517 |
. . . . . . . 8
⊢ (𝜑 → 𝑅 Fn 𝐼) |
42 | 41 | ad2antrr 724 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑅 Fn 𝐼) |
43 | | simplr2 1212 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑌)) |
44 | | simpr 487 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼) |
45 | 3, 20, 39, 40, 42, 43, 44 | prdsbasprj 16747 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑏‘𝑦) ∈ (Base‘(𝑅‘𝑦))) |
46 | | simplr3 1213 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑐 ∈ (Base‘𝑌)) |
47 | 3, 20, 39, 40, 42, 46, 44 | prdsbasprj 16747 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦))) |
48 | | eqid 2823 |
. . . . . . 7
⊢
(Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦)) |
49 | | eqid 2823 |
. . . . . . 7
⊢
(+g‘(𝑅‘𝑦)) = (+g‘(𝑅‘𝑦)) |
50 | | eqid 2823 |
. . . . . . 7
⊢
(Scalar‘(𝑅‘𝑦)) = (Scalar‘(𝑅‘𝑦)) |
51 | | eqid 2823 |
. . . . . . 7
⊢ (
·𝑠 ‘(𝑅‘𝑦)) = ( ·𝑠
‘(𝑅‘𝑦)) |
52 | | eqid 2823 |
. . . . . . 7
⊢
(Base‘(Scalar‘(𝑅‘𝑦))) = (Base‘(Scalar‘(𝑅‘𝑦))) |
53 | 48, 49, 50, 51, 52 | lmodvsdi 19659 |
. . . . . 6
⊢ (((𝑅‘𝑦) ∈ LMod ∧ (𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑦))) ∧ (𝑏‘𝑦) ∈ (Base‘(𝑅‘𝑦)) ∧ (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))) → (𝑎( ·𝑠
‘(𝑅‘𝑦))((𝑏‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦))) = ((𝑎( ·𝑠
‘(𝑅‘𝑦))(𝑏‘𝑦))(+g‘(𝑅‘𝑦))(𝑎( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)))) |
54 | 34, 38, 45, 47, 53 | syl13anc 1368 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎( ·𝑠
‘(𝑅‘𝑦))((𝑏‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦))) = ((𝑎( ·𝑠
‘(𝑅‘𝑦))(𝑏‘𝑦))(+g‘(𝑅‘𝑦))(𝑎( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)))) |
55 | | eqid 2823 |
. . . . . . 7
⊢
(+g‘𝑌) = (+g‘𝑌) |
56 | 3, 20, 39, 40, 42, 43, 46, 55, 44 | prdsplusgfval 16749 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑏(+g‘𝑌)𝑐)‘𝑦) = ((𝑏‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦))) |
57 | 56 | oveq2d 7174 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎( ·𝑠
‘(𝑅‘𝑦))((𝑏(+g‘𝑌)𝑐)‘𝑦)) = (𝑎( ·𝑠
‘(𝑅‘𝑦))((𝑏‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦)))) |
58 | 3, 20, 21, 22, 39, 40, 42, 35, 43, 44 | prdsvscafval 16755 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎( ·𝑠
‘𝑌)𝑏)‘𝑦) = (𝑎( ·𝑠
‘(𝑅‘𝑦))(𝑏‘𝑦))) |
59 | 3, 20, 21, 22, 39, 40, 42, 35, 46, 44 | prdsvscafval 16755 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎( ·𝑠
‘𝑌)𝑐)‘𝑦) = (𝑎( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦))) |
60 | 58, 59 | oveq12d 7176 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (((𝑎( ·𝑠
‘𝑌)𝑏)‘𝑦)(+g‘(𝑅‘𝑦))((𝑎( ·𝑠
‘𝑌)𝑐)‘𝑦)) = ((𝑎( ·𝑠
‘(𝑅‘𝑦))(𝑏‘𝑦))(+g‘(𝑅‘𝑦))(𝑎( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)))) |
61 | 54, 57, 60 | 3eqtr4d 2868 |
. . . 4
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎( ·𝑠
‘(𝑅‘𝑦))((𝑏(+g‘𝑌)𝑐)‘𝑦)) = (((𝑎( ·𝑠
‘𝑌)𝑏)‘𝑦)(+g‘(𝑅‘𝑦))((𝑎( ·𝑠
‘𝑌)𝑐)‘𝑦))) |
62 | 61 | mpteq2dva 5163 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑦 ∈ 𝐼 ↦ (𝑎( ·𝑠
‘(𝑅‘𝑦))((𝑏(+g‘𝑌)𝑐)‘𝑦))) = (𝑦 ∈ 𝐼 ↦ (((𝑎( ·𝑠
‘𝑌)𝑏)‘𝑦)(+g‘(𝑅‘𝑦))((𝑎( ·𝑠
‘𝑌)𝑐)‘𝑦)))) |
63 | 4 | adantr 483 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring) |
64 | 24 | adantr 483 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V) |
65 | 41 | adantr 483 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼) |
66 | | simpr1 1190 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆)) |
67 | 19 | adantr 483 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑌 ∈ Grp) |
68 | | simpr2 1191 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌)) |
69 | | simpr3 1192 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌)) |
70 | 20, 55 | grpcl 18113 |
. . . . 5
⊢ ((𝑌 ∈ Grp ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌)) → (𝑏(+g‘𝑌)𝑐) ∈ (Base‘𝑌)) |
71 | 67, 68, 69, 70 | syl3anc 1367 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏(+g‘𝑌)𝑐) ∈ (Base‘𝑌)) |
72 | 3, 20, 21, 22, 63, 64, 65, 66, 71 | prdsvscaval 16754 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠
‘𝑌)(𝑏(+g‘𝑌)𝑐)) = (𝑦 ∈ 𝐼 ↦ (𝑎( ·𝑠
‘(𝑅‘𝑦))((𝑏(+g‘𝑌)𝑐)‘𝑦)))) |
73 | 31 | 3adantr3 1167 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠
‘𝑌)𝑏) ∈ (Base‘𝑌)) |
74 | 4 | adantr 483 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring) |
75 | 24 | adantr 483 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V) |
76 | 5 | adantr 483 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod) |
77 | | simprl 769 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆)) |
78 | | simprr 771 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌)) |
79 | 29 | adantlr 713 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (Scalar‘(𝑅‘𝑦)) = 𝑆) |
80 | 3, 20, 21, 22, 74, 75, 76, 77, 78, 79 | prdsvscacl 19742 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠
‘𝑌)𝑐) ∈ (Base‘𝑌)) |
81 | 80 | 3adantr2 1166 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠
‘𝑌)𝑐) ∈ (Base‘𝑌)) |
82 | 3, 20, 63, 64, 65, 73, 81, 55 | prdsplusgval 16748 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎( ·𝑠
‘𝑌)𝑏)(+g‘𝑌)(𝑎( ·𝑠
‘𝑌)𝑐)) = (𝑦 ∈ 𝐼 ↦ (((𝑎( ·𝑠
‘𝑌)𝑏)‘𝑦)(+g‘(𝑅‘𝑦))((𝑎( ·𝑠
‘𝑌)𝑐)‘𝑦)))) |
83 | 62, 72, 82 | 3eqtr4d 2868 |
. 2
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠
‘𝑌)(𝑏(+g‘𝑌)𝑐)) = ((𝑎( ·𝑠
‘𝑌)𝑏)(+g‘𝑌)(𝑎( ·𝑠
‘𝑌)𝑐))) |
84 | 4 | ad2antrr 724 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑆 ∈ Ring) |
85 | 24 | ad2antrr 724 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ V) |
86 | 41 | ad2antrr 724 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑅 Fn 𝐼) |
87 | | simplr1 1211 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑆)) |
88 | | simplr3 1213 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑐 ∈ (Base‘𝑌)) |
89 | | simpr 487 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼) |
90 | 3, 20, 21, 22, 84, 85, 86, 87, 88, 89 | prdsvscafval 16755 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎( ·𝑠
‘𝑌)𝑐)‘𝑦) = (𝑎( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦))) |
91 | | simplr2 1212 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑆)) |
92 | 3, 20, 21, 22, 84, 85, 86, 91, 88, 89 | prdsvscafval 16755 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑏( ·𝑠
‘𝑌)𝑐)‘𝑦) = (𝑏( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦))) |
93 | 90, 92 | oveq12d 7176 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (((𝑎( ·𝑠
‘𝑌)𝑐)‘𝑦)(+g‘(𝑅‘𝑦))((𝑏( ·𝑠
‘𝑌)𝑐)‘𝑦)) = ((𝑎( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦))(+g‘(𝑅‘𝑦))(𝑏( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)))) |
94 | 33 | adantlr 713 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ LMod) |
95 | 36 | adantlr 713 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑦))) = (Base‘𝑆)) |
96 | 87, 95 | eleqtrrd 2918 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑦)))) |
97 | 91, 95 | eleqtrrd 2918 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑏 ∈ (Base‘(Scalar‘(𝑅‘𝑦)))) |
98 | 3, 20, 84, 85, 86, 88, 89 | prdsbasprj 16747 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦))) |
99 | | eqid 2823 |
. . . . . . 7
⊢
(+g‘(Scalar‘(𝑅‘𝑦))) =
(+g‘(Scalar‘(𝑅‘𝑦))) |
100 | 48, 49, 50, 51, 52, 99 | lmodvsdir 19660 |
. . . . . 6
⊢ (((𝑅‘𝑦) ∈ LMod ∧ (𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑦))) ∧ 𝑏 ∈ (Base‘(Scalar‘(𝑅‘𝑦))) ∧ (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))) → ((𝑎(+g‘(Scalar‘(𝑅‘𝑦)))𝑏)( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)) = ((𝑎( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦))(+g‘(𝑅‘𝑦))(𝑏( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)))) |
101 | 94, 96, 97, 98, 100 | syl13anc 1368 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎(+g‘(Scalar‘(𝑅‘𝑦)))𝑏)( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)) = ((𝑎( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦))(+g‘(𝑅‘𝑦))(𝑏( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)))) |
102 | 29 | adantlr 713 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (Scalar‘(𝑅‘𝑦)) = 𝑆) |
103 | 102 | fveq2d 6676 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) →
(+g‘(Scalar‘(𝑅‘𝑦))) = (+g‘𝑆)) |
104 | 103 | oveqd 7175 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎(+g‘(Scalar‘(𝑅‘𝑦)))𝑏) = (𝑎(+g‘𝑆)𝑏)) |
105 | 104 | oveq1d 7173 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎(+g‘(Scalar‘(𝑅‘𝑦)))𝑏)( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)) = ((𝑎(+g‘𝑆)𝑏)( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦))) |
106 | 93, 101, 105 | 3eqtr2rd 2865 |
. . . 4
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎(+g‘𝑆)𝑏)( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)) = (((𝑎( ·𝑠
‘𝑌)𝑐)‘𝑦)(+g‘(𝑅‘𝑦))((𝑏( ·𝑠
‘𝑌)𝑐)‘𝑦))) |
107 | 106 | mpteq2dva 5163 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑦 ∈ 𝐼 ↦ ((𝑎(+g‘𝑆)𝑏)( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦))) = (𝑦 ∈ 𝐼 ↦ (((𝑎( ·𝑠
‘𝑌)𝑐)‘𝑦)(+g‘(𝑅‘𝑦))((𝑏( ·𝑠
‘𝑌)𝑐)‘𝑦)))) |
108 | 4 | adantr 483 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring) |
109 | 24 | adantr 483 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V) |
110 | 41 | adantr 483 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼) |
111 | | simpr1 1190 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆)) |
112 | | simpr2 1191 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑆)) |
113 | | eqid 2823 |
. . . . . 6
⊢
(+g‘𝑆) = (+g‘𝑆) |
114 | 22, 113 | ringacl 19330 |
. . . . 5
⊢ ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+g‘𝑆)𝑏) ∈ (Base‘𝑆)) |
115 | 108, 111,
112, 114 | syl3anc 1367 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g‘𝑆)𝑏) ∈ (Base‘𝑆)) |
116 | | simpr3 1192 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌)) |
117 | 3, 20, 21, 22, 108, 109, 110, 115, 116 | prdsvscaval 16754 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g‘𝑆)𝑏)( ·𝑠
‘𝑌)𝑐) = (𝑦 ∈ 𝐼 ↦ ((𝑎(+g‘𝑆)𝑏)( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)))) |
118 | 80 | 3adantr2 1166 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠
‘𝑌)𝑐) ∈ (Base‘𝑌)) |
119 | 5 | adantr 483 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod) |
120 | 3, 20, 21, 22, 108, 109, 119, 112, 116, 102 | prdsvscacl 19742 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏( ·𝑠
‘𝑌)𝑐) ∈ (Base‘𝑌)) |
121 | 3, 20, 108, 109, 110, 118, 120, 55 | prdsplusgval 16748 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎( ·𝑠
‘𝑌)𝑐)(+g‘𝑌)(𝑏( ·𝑠
‘𝑌)𝑐)) = (𝑦 ∈ 𝐼 ↦ (((𝑎( ·𝑠
‘𝑌)𝑐)‘𝑦)(+g‘(𝑅‘𝑦))((𝑏( ·𝑠
‘𝑌)𝑐)‘𝑦)))) |
122 | 107, 117,
121 | 3eqtr4d 2868 |
. 2
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g‘𝑆)𝑏)( ·𝑠
‘𝑌)𝑐) = ((𝑎( ·𝑠
‘𝑌)𝑐)(+g‘𝑌)(𝑏( ·𝑠
‘𝑌)𝑐))) |
123 | 92 | oveq2d 7174 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎( ·𝑠
‘(𝑅‘𝑦))((𝑏( ·𝑠
‘𝑌)𝑐)‘𝑦)) = (𝑎( ·𝑠
‘(𝑅‘𝑦))(𝑏( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)))) |
124 | | eqid 2823 |
. . . . . . 7
⊢
(.r‘(Scalar‘(𝑅‘𝑦))) =
(.r‘(Scalar‘(𝑅‘𝑦))) |
125 | 48, 50, 51, 52, 124 | lmodvsass 19661 |
. . . . . 6
⊢ (((𝑅‘𝑦) ∈ LMod ∧ (𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑦))) ∧ 𝑏 ∈ (Base‘(Scalar‘(𝑅‘𝑦))) ∧ (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))) → ((𝑎(.r‘(Scalar‘(𝑅‘𝑦)))𝑏)( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)) = (𝑎( ·𝑠
‘(𝑅‘𝑦))(𝑏( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)))) |
126 | 94, 96, 97, 98, 125 | syl13anc 1368 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎(.r‘(Scalar‘(𝑅‘𝑦)))𝑏)( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)) = (𝑎( ·𝑠
‘(𝑅‘𝑦))(𝑏( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)))) |
127 | 102 | fveq2d 6676 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) →
(.r‘(Scalar‘(𝑅‘𝑦))) = (.r‘𝑆)) |
128 | 127 | oveqd 7175 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎(.r‘(Scalar‘(𝑅‘𝑦)))𝑏) = (𝑎(.r‘𝑆)𝑏)) |
129 | 128 | oveq1d 7173 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎(.r‘(Scalar‘(𝑅‘𝑦)))𝑏)( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)) = ((𝑎(.r‘𝑆)𝑏)( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦))) |
130 | 123, 126,
129 | 3eqtr2rd 2865 |
. . . 4
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎(.r‘𝑆)𝑏)( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)) = (𝑎( ·𝑠
‘(𝑅‘𝑦))((𝑏( ·𝑠
‘𝑌)𝑐)‘𝑦))) |
131 | 130 | mpteq2dva 5163 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑦 ∈ 𝐼 ↦ ((𝑎(.r‘𝑆)𝑏)( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦))) = (𝑦 ∈ 𝐼 ↦ (𝑎( ·𝑠
‘(𝑅‘𝑦))((𝑏( ·𝑠
‘𝑌)𝑐)‘𝑦)))) |
132 | | eqid 2823 |
. . . . . 6
⊢
(.r‘𝑆) = (.r‘𝑆) |
133 | 22, 132 | ringcl 19313 |
. . . . 5
⊢ ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(.r‘𝑆)𝑏) ∈ (Base‘𝑆)) |
134 | 108, 111,
112, 133 | syl3anc 1367 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(.r‘𝑆)𝑏) ∈ (Base‘𝑆)) |
135 | 3, 20, 21, 22, 108, 109, 110, 134, 116 | prdsvscaval 16754 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(.r‘𝑆)𝑏)( ·𝑠
‘𝑌)𝑐) = (𝑦 ∈ 𝐼 ↦ ((𝑎(.r‘𝑆)𝑏)( ·𝑠
‘(𝑅‘𝑦))(𝑐‘𝑦)))) |
136 | 3, 20, 21, 22, 108, 109, 110, 111, 120 | prdsvscaval 16754 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠
‘𝑌)(𝑏(
·𝑠 ‘𝑌)𝑐)) = (𝑦 ∈ 𝐼 ↦ (𝑎( ·𝑠
‘(𝑅‘𝑦))((𝑏( ·𝑠
‘𝑌)𝑐)‘𝑦)))) |
137 | 131, 135,
136 | 3eqtr4d 2868 |
. 2
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(.r‘𝑆)𝑏)( ·𝑠
‘𝑌)𝑐) = (𝑎( ·𝑠
‘𝑌)(𝑏(
·𝑠 ‘𝑌)𝑐))) |
138 | 29 | fveq2d 6676 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) →
(1r‘(Scalar‘(𝑅‘𝑦))) = (1r‘𝑆)) |
139 | 138 | adantlr 713 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) →
(1r‘(Scalar‘(𝑅‘𝑦))) = (1r‘𝑆)) |
140 | 139 | oveq1d 7173 |
. . . . 5
⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) →
((1r‘(Scalar‘(𝑅‘𝑦)))( ·𝑠
‘(𝑅‘𝑦))(𝑎‘𝑦)) = ((1r‘𝑆)( ·𝑠
‘(𝑅‘𝑦))(𝑎‘𝑦))) |
141 | 33 | adantlr 713 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ LMod) |
142 | 4 | ad2antrr 724 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝑆 ∈ Ring) |
143 | 24 | ad2antrr 724 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ V) |
144 | 41 | ad2antrr 724 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝑅 Fn 𝐼) |
145 | | simplr 767 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑌)) |
146 | | simpr 487 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼) |
147 | 3, 20, 142, 143, 144, 145, 146 | prdsbasprj 16747 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → (𝑎‘𝑦) ∈ (Base‘(𝑅‘𝑦))) |
148 | | eqid 2823 |
. . . . . . 7
⊢
(1r‘(Scalar‘(𝑅‘𝑦))) =
(1r‘(Scalar‘(𝑅‘𝑦))) |
149 | 48, 50, 51, 148 | lmodvs1 19664 |
. . . . . 6
⊢ (((𝑅‘𝑦) ∈ LMod ∧ (𝑎‘𝑦) ∈ (Base‘(𝑅‘𝑦))) →
((1r‘(Scalar‘(𝑅‘𝑦)))( ·𝑠
‘(𝑅‘𝑦))(𝑎‘𝑦)) = (𝑎‘𝑦)) |
150 | 141, 147,
149 | syl2anc 586 |
. . . . 5
⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) →
((1r‘(Scalar‘(𝑅‘𝑦)))( ·𝑠
‘(𝑅‘𝑦))(𝑎‘𝑦)) = (𝑎‘𝑦)) |
151 | 140, 150 | eqtr3d 2860 |
. . . 4
⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → ((1r‘𝑆)(
·𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦)) = (𝑎‘𝑦)) |
152 | 151 | mpteq2dva 5163 |
. . 3
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (𝑦 ∈ 𝐼 ↦ ((1r‘𝑆)(
·𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦))) = (𝑦 ∈ 𝐼 ↦ (𝑎‘𝑦))) |
153 | 4 | adantr 483 |
. . . 4
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑆 ∈ Ring) |
154 | 24 | adantr 483 |
. . . 4
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝐼 ∈ V) |
155 | 41 | adantr 483 |
. . . 4
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑅 Fn 𝐼) |
156 | | eqid 2823 |
. . . . . . 7
⊢
(1r‘𝑆) = (1r‘𝑆) |
157 | 22, 156 | ringidcl 19320 |
. . . . . 6
⊢ (𝑆 ∈ Ring →
(1r‘𝑆)
∈ (Base‘𝑆)) |
158 | 4, 157 | syl 17 |
. . . . 5
⊢ (𝜑 → (1r‘𝑆) ∈ (Base‘𝑆)) |
159 | 158 | adantr 483 |
. . . 4
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (1r‘𝑆) ∈ (Base‘𝑆)) |
160 | | simpr 487 |
. . . 4
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 ∈ (Base‘𝑌)) |
161 | 3, 20, 21, 22, 153, 154, 155, 159, 160 | prdsvscaval 16754 |
. . 3
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ((1r‘𝑆)(
·𝑠 ‘𝑌)𝑎) = (𝑦 ∈ 𝐼 ↦ ((1r‘𝑆)(
·𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦)))) |
162 | 3, 20, 153, 154, 155, 160 | prdsbasfn 16746 |
. . . 4
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 Fn 𝐼) |
163 | | dffn5 6726 |
. . . 4
⊢ (𝑎 Fn 𝐼 ↔ 𝑎 = (𝑦 ∈ 𝐼 ↦ (𝑎‘𝑦))) |
164 | 162, 163 | sylib 220 |
. . 3
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 = (𝑦 ∈ 𝐼 ↦ (𝑎‘𝑦))) |
165 | 152, 161,
164 | 3eqtr4d 2868 |
. 2
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ((1r‘𝑆)(
·𝑠 ‘𝑌)𝑎) = 𝑎) |
166 | 1, 2, 9, 10, 11, 12, 13, 14, 4, 19, 32, 83, 122, 137, 165 | islmodd 19642 |
1
⊢ (𝜑 → 𝑌 ∈ LMod) |