Step | Hyp | Ref
| Expression |
1 | | eqid 2728 |
. . . 4
⊢
(mulGrp‘𝑌) =
(mulGrp‘𝑌) |
2 | | eqid 2728 |
. . . 4
⊢
(Base‘𝑌) =
(Base‘𝑌) |
3 | 1, 2 | mgpbas 20079 |
. . 3
⊢
(Base‘𝑌) =
(Base‘(mulGrp‘𝑌)) |
4 | | eqid 2728 |
. . . 4
⊢
(1r‘𝑌) = (1r‘𝑌) |
5 | 1, 4 | ringidval 20122 |
. . 3
⊢
(1r‘𝑌) = (0g‘(mulGrp‘𝑌)) |
6 | | eqid 2728 |
. . . 4
⊢
(.r‘𝑌) = (.r‘𝑌) |
7 | 1, 6 | mgpplusg 20077 |
. . 3
⊢
(.r‘𝑌) = (+g‘(mulGrp‘𝑌)) |
8 | | xpsringd.s |
. . . . . 6
⊢ (𝜑 → 𝑆 ∈ Ring) |
9 | | eqid 2728 |
. . . . . . 7
⊢
(Base‘𝑆) =
(Base‘𝑆) |
10 | | eqid 2728 |
. . . . . . 7
⊢
(1r‘𝑆) = (1r‘𝑆) |
11 | 9, 10 | ringidcl 20201 |
. . . . . 6
⊢ (𝑆 ∈ Ring →
(1r‘𝑆)
∈ (Base‘𝑆)) |
12 | 8, 11 | syl 17 |
. . . . 5
⊢ (𝜑 → (1r‘𝑆) ∈ (Base‘𝑆)) |
13 | | xpsringd.r |
. . . . . 6
⊢ (𝜑 → 𝑅 ∈ Ring) |
14 | | eqid 2728 |
. . . . . . 7
⊢
(Base‘𝑅) =
(Base‘𝑅) |
15 | | eqid 2728 |
. . . . . . 7
⊢
(1r‘𝑅) = (1r‘𝑅) |
16 | 14, 15 | ringidcl 20201 |
. . . . . 6
⊢ (𝑅 ∈ Ring →
(1r‘𝑅)
∈ (Base‘𝑅)) |
17 | 13, 16 | syl 17 |
. . . . 5
⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅)) |
18 | 12, 17 | opelxpd 5717 |
. . . 4
⊢ (𝜑 →
〈(1r‘𝑆), (1r‘𝑅)〉 ∈ ((Base‘𝑆) × (Base‘𝑅))) |
19 | | xpsringd.y |
. . . . 5
⊢ 𝑌 = (𝑆 ×s 𝑅) |
20 | 19, 9, 14, 8, 13 | xpsbas 17553 |
. . . 4
⊢ (𝜑 → ((Base‘𝑆) × (Base‘𝑅)) = (Base‘𝑌)) |
21 | 18, 20 | eleqtrd 2831 |
. . 3
⊢ (𝜑 →
〈(1r‘𝑆), (1r‘𝑅)〉 ∈ (Base‘𝑌)) |
22 | 20 | eleq2d 2815 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) ↔ 𝑥 ∈ (Base‘𝑌))) |
23 | | elxp2 5702 |
. . . . . 6
⊢ (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) ↔ ∃𝑎 ∈ (Base‘𝑆)∃𝑏 ∈ (Base‘𝑅)𝑥 = 〈𝑎, 𝑏〉) |
24 | 8 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑆 ∈ Ring) |
25 | 13 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring) |
26 | 12 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (1r‘𝑆) ∈ (Base‘𝑆)) |
27 | 17 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (1r‘𝑅) ∈ (Base‘𝑅)) |
28 | | simprl 770 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘𝑆)) |
29 | | simprr 772 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘𝑅)) |
30 | | eqid 2728 |
. . . . . . . . . . 11
⊢
(.r‘𝑆) = (.r‘𝑆) |
31 | 9, 30, 24, 26, 28 | ringcld 20198 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r‘𝑆)(.r‘𝑆)𝑎) ∈ (Base‘𝑆)) |
32 | | eqid 2728 |
. . . . . . . . . . 11
⊢
(.r‘𝑅) = (.r‘𝑅) |
33 | 14, 32, 25, 27, 29 | ringcld 20198 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r‘𝑅)(.r‘𝑅)𝑏) ∈ (Base‘𝑅)) |
34 | 19, 9, 14, 24, 25, 26, 27, 28, 29, 31, 33, 30, 32, 6 | xpsmul 17556 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (〈(1r‘𝑆), (1r‘𝑅)〉(.r‘𝑌)〈𝑎, 𝑏〉) = 〈((1r‘𝑆)(.r‘𝑆)𝑎), ((1r‘𝑅)(.r‘𝑅)𝑏)〉) |
35 | | simpl 482 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑎 ∈ (Base‘𝑆)) |
36 | 9, 30, 10 | ringlidm 20204 |
. . . . . . . . . . 11
⊢ ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆)) →
((1r‘𝑆)(.r‘𝑆)𝑎) = 𝑎) |
37 | 8, 35, 36 | syl2an 595 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r‘𝑆)(.r‘𝑆)𝑎) = 𝑎) |
38 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑏 ∈ (Base‘𝑅)) |
39 | 14, 32, 15 | ringlidm 20204 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ 𝑏 ∈ (Base‘𝑅)) →
((1r‘𝑅)(.r‘𝑅)𝑏) = 𝑏) |
40 | 13, 38, 39 | syl2an 595 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r‘𝑅)(.r‘𝑅)𝑏) = 𝑏) |
41 | 37, 40 | opeq12d 4882 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 〈((1r‘𝑆)(.r‘𝑆)𝑎), ((1r‘𝑅)(.r‘𝑅)𝑏)〉 = 〈𝑎, 𝑏〉) |
42 | 34, 41 | eqtrd 2768 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (〈(1r‘𝑆), (1r‘𝑅)〉(.r‘𝑌)〈𝑎, 𝑏〉) = 〈𝑎, 𝑏〉) |
43 | | oveq2 7428 |
. . . . . . . . 9
⊢ (𝑥 = 〈𝑎, 𝑏〉 →
(〈(1r‘𝑆), (1r‘𝑅)〉(.r‘𝑌)𝑥) = (〈(1r‘𝑆), (1r‘𝑅)〉(.r‘𝑌)〈𝑎, 𝑏〉)) |
44 | | id 22 |
. . . . . . . . 9
⊢ (𝑥 = 〈𝑎, 𝑏〉 → 𝑥 = 〈𝑎, 𝑏〉) |
45 | 43, 44 | eqeq12d 2744 |
. . . . . . . 8
⊢ (𝑥 = 〈𝑎, 𝑏〉 →
((〈(1r‘𝑆), (1r‘𝑅)〉(.r‘𝑌)𝑥) = 𝑥 ↔ (〈(1r‘𝑆), (1r‘𝑅)〉(.r‘𝑌)〈𝑎, 𝑏〉) = 〈𝑎, 𝑏〉)) |
46 | 42, 45 | syl5ibrcom 246 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑥 = 〈𝑎, 𝑏〉 →
(〈(1r‘𝑆), (1r‘𝑅)〉(.r‘𝑌)𝑥) = 𝑥)) |
47 | 46 | rexlimdvva 3208 |
. . . . . 6
⊢ (𝜑 → (∃𝑎 ∈ (Base‘𝑆)∃𝑏 ∈ (Base‘𝑅)𝑥 = 〈𝑎, 𝑏〉 →
(〈(1r‘𝑆), (1r‘𝑅)〉(.r‘𝑌)𝑥) = 𝑥)) |
48 | 23, 47 | biimtrid 241 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) → (〈(1r‘𝑆), (1r‘𝑅)〉(.r‘𝑌)𝑥) = 𝑥)) |
49 | 22, 48 | sylbird 260 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) → (〈(1r‘𝑆), (1r‘𝑅)〉(.r‘𝑌)𝑥) = 𝑥)) |
50 | 49 | imp 406 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → (〈(1r‘𝑆), (1r‘𝑅)〉(.r‘𝑌)𝑥) = 𝑥) |
51 | 9, 30, 24, 28, 26 | ringcld 20198 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(.r‘𝑆)(1r‘𝑆)) ∈ (Base‘𝑆)) |
52 | 14, 32, 25, 29, 27 | ringcld 20198 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑏(.r‘𝑅)(1r‘𝑅)) ∈ (Base‘𝑅)) |
53 | 19, 9, 14, 24, 25, 28, 29, 26, 27, 51, 52, 30, 32, 6 | xpsmul 17556 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (〈𝑎, 𝑏〉(.r‘𝑌)〈(1r‘𝑆), (1r‘𝑅)〉) = 〈(𝑎(.r‘𝑆)(1r‘𝑆)), (𝑏(.r‘𝑅)(1r‘𝑅))〉) |
54 | 9, 30, 10 | ringridm 20205 |
. . . . . . . . . . 11
⊢ ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑎(.r‘𝑆)(1r‘𝑆)) = 𝑎) |
55 | 8, 35, 54 | syl2an 595 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(.r‘𝑆)(1r‘𝑆)) = 𝑎) |
56 | 14, 32, 15 | ringridm 20205 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑏(.r‘𝑅)(1r‘𝑅)) = 𝑏) |
57 | 13, 38, 56 | syl2an 595 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑏(.r‘𝑅)(1r‘𝑅)) = 𝑏) |
58 | 55, 57 | opeq12d 4882 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 〈(𝑎(.r‘𝑆)(1r‘𝑆)), (𝑏(.r‘𝑅)(1r‘𝑅))〉 = 〈𝑎, 𝑏〉) |
59 | 53, 58 | eqtrd 2768 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (〈𝑎, 𝑏〉(.r‘𝑌)〈(1r‘𝑆), (1r‘𝑅)〉) = 〈𝑎, 𝑏〉) |
60 | | oveq1 7427 |
. . . . . . . . 9
⊢ (𝑥 = 〈𝑎, 𝑏〉 → (𝑥(.r‘𝑌)〈(1r‘𝑆), (1r‘𝑅)〉) = (〈𝑎, 𝑏〉(.r‘𝑌)〈(1r‘𝑆), (1r‘𝑅)〉)) |
61 | 60, 44 | eqeq12d 2744 |
. . . . . . . 8
⊢ (𝑥 = 〈𝑎, 𝑏〉 → ((𝑥(.r‘𝑌)〈(1r‘𝑆), (1r‘𝑅)〉) = 𝑥 ↔ (〈𝑎, 𝑏〉(.r‘𝑌)〈(1r‘𝑆), (1r‘𝑅)〉) = 〈𝑎, 𝑏〉)) |
62 | 59, 61 | syl5ibrcom 246 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑥 = 〈𝑎, 𝑏〉 → (𝑥(.r‘𝑌)〈(1r‘𝑆), (1r‘𝑅)〉) = 𝑥)) |
63 | 62 | rexlimdvva 3208 |
. . . . . 6
⊢ (𝜑 → (∃𝑎 ∈ (Base‘𝑆)∃𝑏 ∈ (Base‘𝑅)𝑥 = 〈𝑎, 𝑏〉 → (𝑥(.r‘𝑌)〈(1r‘𝑆), (1r‘𝑅)〉) = 𝑥)) |
64 | 23, 63 | biimtrid 241 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) → (𝑥(.r‘𝑌)〈(1r‘𝑆), (1r‘𝑅)〉) = 𝑥)) |
65 | 22, 64 | sylbird 260 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) → (𝑥(.r‘𝑌)〈(1r‘𝑆), (1r‘𝑅)〉) = 𝑥)) |
66 | 65 | imp 406 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → (𝑥(.r‘𝑌)〈(1r‘𝑆), (1r‘𝑅)〉) = 𝑥) |
67 | 3, 5, 7, 21, 50, 66 | ismgmid2 18627 |
. 2
⊢ (𝜑 →
〈(1r‘𝑆), (1r‘𝑅)〉 = (1r‘𝑌)) |
68 | 67 | eqcomd 2734 |
1
⊢ (𝜑 → (1r‘𝑌) =
〈(1r‘𝑆), (1r‘𝑅)〉) |