Step | Hyp | Ref
| Expression |
1 | | assalmod 21048 |
. . . 4
⊢ (𝐾 ∈ AssAlg → 𝐾 ∈ LMod) |
2 | | assaring 21049 |
. . . 4
⊢ (𝐾 ∈ AssAlg → 𝐾 ∈ Ring) |
3 | 1, 2 | jca 511 |
. . 3
⊢ (𝐾 ∈ AssAlg → (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) |
4 | 3 | a1i 11 |
. 2
⊢ (𝜑 → (𝐾 ∈ AssAlg → (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring))) |
5 | | assalmod 21048 |
. . . 4
⊢ (𝐿 ∈ AssAlg → 𝐿 ∈ LMod) |
6 | | assapropd.1 |
. . . . 5
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
7 | | assapropd.2 |
. . . . 5
⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
8 | | assapropd.3 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
9 | | assapropd.5 |
. . . . 5
⊢ (𝜑 → 𝐹 = (Scalar‘𝐾)) |
10 | | assapropd.6 |
. . . . 5
⊢ (𝜑 → 𝐹 = (Scalar‘𝐿)) |
11 | | assapropd.7 |
. . . . 5
⊢ 𝑃 = (Base‘𝐹) |
12 | | assapropd.8 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠
‘𝐾)𝑦) = (𝑥( ·𝑠
‘𝐿)𝑦)) |
13 | 6, 7, 8, 9, 10, 11, 12 | lmodpropd 20167 |
. . . 4
⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod)) |
14 | 5, 13 | syl5ibr 245 |
. . 3
⊢ (𝜑 → (𝐿 ∈ AssAlg → 𝐾 ∈ LMod)) |
15 | | assaring 21049 |
. . . 4
⊢ (𝐿 ∈ AssAlg → 𝐿 ∈ Ring) |
16 | | assapropd.4 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
17 | 6, 7, 8, 16 | ringpropd 19802 |
. . . 4
⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)) |
18 | 15, 17 | syl5ibr 245 |
. . 3
⊢ (𝜑 → (𝐿 ∈ AssAlg → 𝐾 ∈ Ring)) |
19 | 14, 18 | jcad 512 |
. 2
⊢ (𝜑 → (𝐿 ∈ AssAlg → (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring))) |
20 | 9, 10 | eqtr3d 2781 |
. . . . . . . 8
⊢ (𝜑 → (Scalar‘𝐾) = (Scalar‘𝐿)) |
21 | 20 | eleq1d 2824 |
. . . . . . 7
⊢ (𝜑 → ((Scalar‘𝐾) ∈ CRing ↔
(Scalar‘𝐿) ∈
CRing)) |
22 | 13, 17, 21 | 3anbi123d 1434 |
. . . . . 6
⊢ (𝜑 → ((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring ∧ (Scalar‘𝐾) ∈ CRing) ↔ (𝐿 ∈ LMod ∧ 𝐿 ∈ Ring ∧
(Scalar‘𝐿) ∈
CRing))) |
23 | 22 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → ((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring ∧ (Scalar‘𝐾) ∈ CRing) ↔ (𝐿 ∈ LMod ∧ 𝐿 ∈ Ring ∧
(Scalar‘𝐿) ∈
CRing))) |
24 | | simpll 763 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝜑) |
25 | | simplrl 773 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐾 ∈ LMod) |
26 | | simprl 767 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑟 ∈ 𝑃) |
27 | 9 | fveq2d 6772 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (Base‘𝐹) =
(Base‘(Scalar‘𝐾))) |
28 | 11, 27 | eqtrid 2791 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾))) |
29 | 24, 28 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑃 = (Base‘(Scalar‘𝐾))) |
30 | 26, 29 | eleqtrd 2842 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑟 ∈ (Base‘(Scalar‘𝐾))) |
31 | | simprrl 777 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑧 ∈ 𝐵) |
32 | 24, 6 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐵 = (Base‘𝐾)) |
33 | 31, 32 | eleqtrd 2842 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑧 ∈ (Base‘𝐾)) |
34 | | eqid 2739 |
. . . . . . . . . . . . . . . 16
⊢
(Base‘𝐾) =
(Base‘𝐾) |
35 | | eqid 2739 |
. . . . . . . . . . . . . . . 16
⊢
(Scalar‘𝐾) =
(Scalar‘𝐾) |
36 | | eqid 2739 |
. . . . . . . . . . . . . . . 16
⊢ (
·𝑠 ‘𝐾) = ( ·𝑠
‘𝐾) |
37 | | eqid 2739 |
. . . . . . . . . . . . . . . 16
⊢
(Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐾)) |
38 | 34, 35, 36, 37 | lmodvscl 20121 |
. . . . . . . . . . . . . . 15
⊢ ((𝐾 ∈ LMod ∧ 𝑟 ∈
(Base‘(Scalar‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑟( ·𝑠
‘𝐾)𝑧) ∈ (Base‘𝐾)) |
39 | 25, 30, 33, 38 | syl3anc 1369 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠
‘𝐾)𝑧) ∈ (Base‘𝐾)) |
40 | 39, 32 | eleqtrrd 2843 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠
‘𝐾)𝑧) ∈ 𝐵) |
41 | | simprrr 778 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑤 ∈ 𝐵) |
42 | 16 | oveqrspc2v 7295 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑟( ·𝑠
‘𝐾)𝑧) ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = ((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐿)𝑤)) |
43 | 24, 40, 41, 42 | syl12anc 833 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = ((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐿)𝑤)) |
44 | 12 | oveqrspc2v 7295 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑧 ∈ 𝐵)) → (𝑟( ·𝑠
‘𝐾)𝑧) = (𝑟( ·𝑠
‘𝐿)𝑧)) |
45 | 24, 26, 31, 44 | syl12anc 833 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠
‘𝐾)𝑧) = (𝑟( ·𝑠
‘𝐿)𝑧)) |
46 | 45 | oveq1d 7283 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐿)𝑤) = ((𝑟( ·𝑠
‘𝐿)𝑧)(.r‘𝐿)𝑤)) |
47 | 43, 46 | eqtrd 2779 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = ((𝑟( ·𝑠
‘𝐿)𝑧)(.r‘𝐿)𝑤)) |
48 | | simplrr 774 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐾 ∈ Ring) |
49 | 41, 32 | eleqtrd 2842 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑤 ∈ (Base‘𝐾)) |
50 | | eqid 2739 |
. . . . . . . . . . . . . . . 16
⊢
(.r‘𝐾) = (.r‘𝐾) |
51 | 34, 50 | ringcl 19781 |
. . . . . . . . . . . . . . 15
⊢ ((𝐾 ∈ Ring ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑧(.r‘𝐾)𝑤) ∈ (Base‘𝐾)) |
52 | 48, 33, 49, 51 | syl3anc 1369 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(.r‘𝐾)𝑤) ∈ (Base‘𝐾)) |
53 | 52, 32 | eleqtrrd 2843 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(.r‘𝐾)𝑤) ∈ 𝐵) |
54 | 12 | oveqrspc2v 7295 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ (𝑧(.r‘𝐾)𝑤) ∈ 𝐵)) → (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐾)𝑤))) |
55 | 24, 26, 53, 54 | syl12anc 833 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐾)𝑤))) |
56 | 16 | oveqrspc2v 7295 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(.r‘𝐾)𝑤) = (𝑧(.r‘𝐿)𝑤)) |
57 | 24, 31, 41, 56 | syl12anc 833 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(.r‘𝐾)𝑤) = (𝑧(.r‘𝐿)𝑤)) |
58 | 57 | oveq2d 7284 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤))) |
59 | 55, 58 | eqtrd 2779 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤))) |
60 | 47, 59 | eqeq12d 2755 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) ↔ ((𝑟( ·𝑠
‘𝐿)𝑧)(.r‘𝐿)𝑤) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤)))) |
61 | 34, 35, 36, 37 | lmodvscl 20121 |
. . . . . . . . . . . . . . 15
⊢ ((𝐾 ∈ LMod ∧ 𝑟 ∈
(Base‘(Scalar‘𝐾)) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑟( ·𝑠
‘𝐾)𝑤) ∈ (Base‘𝐾)) |
62 | 25, 30, 49, 61 | syl3anc 1369 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠
‘𝐾)𝑤) ∈ (Base‘𝐾)) |
63 | 62, 32 | eleqtrrd 2843 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠
‘𝐾)𝑤) ∈ 𝐵) |
64 | 16 | oveqrspc2v 7295 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ (𝑟( ·𝑠
‘𝐾)𝑤) ∈ 𝐵)) → (𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐾)𝑤))) |
65 | 24, 31, 63, 64 | syl12anc 833 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐾)𝑤))) |
66 | 12 | oveqrspc2v 7295 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑟( ·𝑠
‘𝐾)𝑤) = (𝑟( ·𝑠
‘𝐿)𝑤)) |
67 | 24, 26, 41, 66 | syl12anc 833 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠
‘𝐾)𝑤) = (𝑟( ·𝑠
‘𝐿)𝑤)) |
68 | 67 | oveq2d 7284 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐿)𝑤))) |
69 | 65, 68 | eqtrd 2779 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐿)𝑤))) |
70 | 69, 59 | eqeq12d 2755 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) ↔ (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐿)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤)))) |
71 | 60, 70 | anbi12d 630 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) ∧ (𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤))) ↔ (((𝑟( ·𝑠
‘𝐿)𝑧)(.r‘𝐿)𝑤) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤)) ∧ (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐿)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤))))) |
72 | 71 | anassrs 467 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) ∧ (𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤))) ↔ (((𝑟( ·𝑠
‘𝐿)𝑧)(.r‘𝐿)𝑤) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤)) ∧ (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐿)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤))))) |
73 | 72 | 2ralbidva 3123 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ 𝑟 ∈ 𝑃) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) ∧ (𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤))) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·𝑠
‘𝐿)𝑧)(.r‘𝐿)𝑤) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤)) ∧ (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐿)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤))))) |
74 | 73 | ralbidva 3121 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (∀𝑟 ∈ 𝑃 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) ∧ (𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤))) ↔ ∀𝑟 ∈ 𝑃 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·𝑠
‘𝐿)𝑧)(.r‘𝐿)𝑤) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤)) ∧ (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐿)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤))))) |
75 | 28 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝑃 = (Base‘(Scalar‘𝐾))) |
76 | 6 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝐵 = (Base‘𝐾)) |
77 | 76 | raleqdv 3346 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (∀𝑤 ∈ 𝐵 (((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) ∧ (𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤))) ↔ ∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) ∧ (𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤))))) |
78 | 76, 77 | raleqbidv 3334 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) ∧ (𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤))) ↔ ∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) ∧ (𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤))))) |
79 | 75, 78 | raleqbidv 3334 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (∀𝑟 ∈ 𝑃 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) ∧ (𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤))) ↔ ∀𝑟 ∈ (Base‘(Scalar‘𝐾))∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) ∧ (𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤))))) |
80 | 10 | fveq2d 6772 |
. . . . . . . . 9
⊢ (𝜑 → (Base‘𝐹) =
(Base‘(Scalar‘𝐿))) |
81 | 11, 80 | eqtrid 2791 |
. . . . . . . 8
⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿))) |
82 | 81 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝑃 = (Base‘(Scalar‘𝐿))) |
83 | 7 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝐵 = (Base‘𝐿)) |
84 | 83 | raleqdv 3346 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (∀𝑤 ∈ 𝐵 (((𝑟( ·𝑠
‘𝐿)𝑧)(.r‘𝐿)𝑤) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤)) ∧ (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐿)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤))) ↔ ∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·𝑠
‘𝐿)𝑧)(.r‘𝐿)𝑤) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤)) ∧ (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐿)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤))))) |
85 | 83, 84 | raleqbidv 3334 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·𝑠
‘𝐿)𝑧)(.r‘𝐿)𝑤) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤)) ∧ (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐿)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤))) ↔ ∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·𝑠
‘𝐿)𝑧)(.r‘𝐿)𝑤) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤)) ∧ (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐿)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤))))) |
86 | 82, 85 | raleqbidv 3334 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (∀𝑟 ∈ 𝑃 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·𝑠
‘𝐿)𝑧)(.r‘𝐿)𝑤) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤)) ∧ (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐿)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤))) ↔ ∀𝑟 ∈ (Base‘(Scalar‘𝐿))∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·𝑠
‘𝐿)𝑧)(.r‘𝐿)𝑤) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤)) ∧ (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐿)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤))))) |
87 | 74, 79, 86 | 3bitr3d 308 |
. . . . 5
⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (∀𝑟 ∈
(Base‘(Scalar‘𝐾))∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) ∧ (𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤))) ↔ ∀𝑟 ∈ (Base‘(Scalar‘𝐿))∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·𝑠
‘𝐿)𝑧)(.r‘𝐿)𝑤) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤)) ∧ (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐿)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤))))) |
88 | 23, 87 | anbi12d 630 |
. . . 4
⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring ∧ (Scalar‘𝐾) ∈ CRing) ∧
∀𝑟 ∈
(Base‘(Scalar‘𝐾))∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) ∧ (𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)))) ↔ ((𝐿 ∈ LMod ∧ 𝐿 ∈ Ring ∧ (Scalar‘𝐿) ∈ CRing) ∧
∀𝑟 ∈
(Base‘(Scalar‘𝐿))∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·𝑠
‘𝐿)𝑧)(.r‘𝐿)𝑤) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤)) ∧ (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐿)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤)))))) |
89 | 34, 35, 37, 36, 50 | isassa 21044 |
. . . 4
⊢ (𝐾 ∈ AssAlg ↔ ((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring ∧
(Scalar‘𝐾) ∈
CRing) ∧ ∀𝑟
∈ (Base‘(Scalar‘𝐾))∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·𝑠
‘𝐾)𝑧)(.r‘𝐾)𝑤) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤)) ∧ (𝑧(.r‘𝐾)(𝑟( ·𝑠
‘𝐾)𝑤)) = (𝑟( ·𝑠
‘𝐾)(𝑧(.r‘𝐾)𝑤))))) |
90 | | eqid 2739 |
. . . . 5
⊢
(Base‘𝐿) =
(Base‘𝐿) |
91 | | eqid 2739 |
. . . . 5
⊢
(Scalar‘𝐿) =
(Scalar‘𝐿) |
92 | | eqid 2739 |
. . . . 5
⊢
(Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿)) |
93 | | eqid 2739 |
. . . . 5
⊢ (
·𝑠 ‘𝐿) = ( ·𝑠
‘𝐿) |
94 | | eqid 2739 |
. . . . 5
⊢
(.r‘𝐿) = (.r‘𝐿) |
95 | 90, 91, 92, 93, 94 | isassa 21044 |
. . . 4
⊢ (𝐿 ∈ AssAlg ↔ ((𝐿 ∈ LMod ∧ 𝐿 ∈ Ring ∧
(Scalar‘𝐿) ∈
CRing) ∧ ∀𝑟
∈ (Base‘(Scalar‘𝐿))∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·𝑠
‘𝐿)𝑧)(.r‘𝐿)𝑤) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤)) ∧ (𝑧(.r‘𝐿)(𝑟( ·𝑠
‘𝐿)𝑤)) = (𝑟( ·𝑠
‘𝐿)(𝑧(.r‘𝐿)𝑤))))) |
96 | 88, 89, 95 | 3bitr4g 313 |
. . 3
⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg)) |
97 | 96 | ex 412 |
. 2
⊢ (𝜑 → ((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring) → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg))) |
98 | 4, 19, 97 | pm5.21ndd 380 |
1
⊢ (𝜑 → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg)) |